Introduction: Derivative instrument
A security or contract whose value is dependent on or derived from the value of some underlying asset. The
main classes of derivative instruments are: forwards, futures, options (and their securitized equivalents,
warrants) and swaps. There are derivative contracts on currencies, commodities, equities, interest rates,
credit or default events, indices and baskets in all these asset classes and combinations of all of them.
Derivatives can be exchange-traded or traded over-the-counter (OTC). The latter are contracts between
counterparties and are telephone and screen traded by banks outside the regulated exchanges.
The definitions in this chapter cover the general terms used in the derivatives markets to describe the
instruments, positions, markets, contracts and risks found in them. Some are used in the cash markets, some
are used only in the derivatives markets. Some have different meanings in derivatives markets than in the
underlying cash markets.
This chapter does not include the majority of option-specific terminology. The basic terms in the options
market are covered in chapter two, and the more complex concepts from mathematics and statistics that are
integral to the modelling of option prices are covered, along with many of the best known pricing models,
in chapter three.
Accreting:
The notional principal amount of accreting instruments increases over their life according to a preset schedule or pre-defined index. Accreting instruments are useful for hedging liabilities expected
to grow predictably, for example to hedge the step-by-step drawdown of a syndicated loan
agreement. Accretion has been applied to caps, collars, floors and swaptions. See amortizing.
Accrual:
The accumulation of interest or other payoffs between payment or reset dates.
Aggregation:
The netting of positive and negative values of swaps affected by early termination allowed by
some swap master agreements.
Amortizing:
The notional principal amount of amortizing instruments decreases over their life according to a
pre-set schedule or pre-defined index. Instruments that have been structured in this way include
caps, collars, floors, swaps and swaptions. See accreting.
Arbitrage:
Instruments that have identical characteristics and so are perfect substitutes should trade at the
same price. If they do not, a risk-free profit can be generated by simultaneously selling the higherpriced asset and buying the lower-priced asset. True arbitrage is the identification and exploitation
of such price anomalies. More generally the term arbitrage is used to mean profiting from
differences in price between similar securities or from trades which are undertaken when prices
have moved from some historical or theoretical path or relationship in the expectation of a move
back to the statistical norm. This is also known as statistical arbitrage.
Assignment:
Notice to an option writer that an option has been exercised. In the swap market, assignment is the
transfer of a swap obligation to another counterparty.
Back contract:
The longest maturity futures contract currently trading.
Backwardation: A commodity market is in backwardation when commodity futures prices are
lower than spot prices to produce a negatively sloped forward curve. See contango.
Barstrier:
Many derivatives, combinations of derivatives and structured notes have a payout determined or
altered by the underlying trading at or through a particular level. The barrier, or trigger, is the price
or rate at which these instruments are activated (knocked-in) or deactivated (knocked-out), or
more generally, change their character in some pre-determined way. See chapter 11.
Basis:
(i) In futures markets, the price of the futures contract minus the spot price. That is, the difference
between the forward price/yield and spot price/yield of futures (and also options). Basis is divided
into carry basis and value or excess basis. Carry basis is the theoretical price of the future, minus
the spot price of the underlying asset, and is equal to the net cost of carry. Value/excess basis is
the difference between the theoretical price of the future and its market price. (ii) More generally,
the relationship between prices/yields in related markets. (iii) The basis upon which interest rates
are calculated for bond and money market instruments.
Basis risk:
The risk that prices in the underlying cash market are not exactly correlated with prices in the
futures market. Consequently basis risk is used more generally for the risk that hedges composed
of offsetting positions in the cash and derivatives markets become unbalanced.
Basis trading: Trading the spread between the futures (or more generally derivatives) markets and
the underlying cash market. See cash-and-carry arbitrage.
Basket:
A selection of stocks, indices, commodities, currencies or interest rates which can either be traded
as a unit themselves or which can be used as the underlying for a derivative product.
Beta:
A measure of the sensitivity of an asset’s return to the market. The returns on a security with a
beta of one will move in line with the market; if beta is greater than one the security will
exaggerate market returns; if it is less than one it will under-reflect market moves; and if beta is
negative, security and market returns move in opposite directions. Beta is a measure of the
systematic risk of a security relative to the market. Betas are additive, hence the beta of a portfolio
is the weighted average of all the individual betas in a portfolio. The capital asset pricing model
states that unique or unsystematic risk can be diversified away so that only systematic risk
commands a risk premium. See Capital Asset Pricing Model.
Binary:
Any derivative or derivative-linked instrument whose fixed payout is either made (‘on’) or not
made (‘off’) depending on the level of the underlying. Also known as digital. See chapters 12 and
13.
Bootstrapping:
A general term for techniques used to decompose the prices of instrument in the market to the
prices of instruments that are simpler, more fundamental or analytically more tractable.
It is most commonly used to describe the process of mapping a yield curve defined through a
series of market instruments into a series of zero-coupon bonds but also applies, for instance, to
the mapping of a term volatility surface indexed by strike and time to maturity to a local volatility
surface indexed by spot and elapsed time. See implied forward rate, spot rate.
Breakeven:
The price or time at which a derivative strategy has no gain or loss relative to another strategy,
usually a cash position or a ”do-nothing strategy“. For example, the breakeven price of purchasing
a call option is the strike price plus the option premium paid. The breakeven time of purchasing a
knockout option is the time at which, if the option knocks out, the strategy of buying a knockout
option and then a vanilla option for the remaining time is the same price as having bought a
vanilla option to the maturity date in the first place.
Callable:
Terminable early. Callable bonds can be redeemed at their pre-set dates and prices by the issuer.
Callable swaps allow the fixed-rate payer to terminate the swap. Where the fixed-rate receiver has
the right to terminate, the swap is known as puttable.
Capital Asset Pricing Model (CAPM):
A model describing the relationship between expected risk and expected return for financial assets.
At its simplest, it takes the form of a linear relationship: Rj = rf + ßj (Rm – rf) where Rj is the
expected return of a security ßj is the beta of the security Rm is the expected return of “the
market”, e.g. the stock market rf is the return on riskless assets
Capped:
The payout of options, warrants or swaps is capped if it is limited to a maximum specified amount.
The opposite of floored.
Carry:
The benefit or cost of maintaining a position in the cash market due to interest rate differentials.
For example, in a positive yield curve environment carry is positive if one receives the long
(swap) rate and pays the short rate (Libor). In the FX markets, carry is positive if the interest rate
of the borrowing (short) currency is less than the interest rate of the lending (long) currency.
Cash-and-carry arbitrage: A basis trade involving a long cash position exactly offset by a short
futures position. The holder of the position believes that the futures contract is expensive. He
shorts the future, borrows at money market rates to finance a long position in the underlying and
either delivers the asset into the futures contract or waits for a narrowing of the basis and closes
out the positions, in which case he effectively collects the yield on a synthetic money market
instrument. Also called buying the basis. This arbitrage and its opposite, reverse cash-and-carry,
ensure that cash and derivatives markets do not diverge too far. The currency market version is
called covered interest arbitrage – the arbitrage that keeps the interest rate differential between two
currencies equal to the difference in the spot and forward exchange rates.
Cash settlement:
The closing of a derivatives position by marking it to market and settling outstanding obligations
in cash instead of by physical delivery of the underlying asset. Most financial derivatives and
almost all over-the-counter derivatives are settled in this way, except for OTC foreign-exchange
options, which tend to be physically settled.
Cheapest to deliver:
In some futures contracts the seller has a choice of which of a variety of underlying securities to
deliver. The security which it is most advantageous for the seller to deliver is called the cheapest
to deliver.
<STRONG
Constant maturity swap (CMS):
For each maturity for which it is available the CMS rate is an index consisting of swap rates
adjusted to that constant maturity.
Constant maturity treasury (CMT):
CMT rates are indices consisting of the semi-annual yield of outstanding US Treasuries adjusted
to a constant x-year maturity. So the 10-year CMT would be adjusted to a constant 10-year
maturity. CMT rates are calculated daily by the Federal Reserve and published weekly.
Contango: A commodity market is in contango when futures prices are above spot prices. The
opposite of backwardation.
Contingency:
Dependence of strike price, payout or existence of a derivative product upon one or more
uncertain variable. See chapter 13, also contingent swap.
Convertibility risk:
The risk that a given currency cannot be freely exchanged/delivered for another freely
exchangeable ‘hard’ currency. See sovereign risk guarantee.
Convexity:
In a fixed-income instrument convexity is a measure of the way duration changes as interest rates
change. An instrument is said to have positive convexity if its value increases by more than
duration predicts when interest rates drop and decreases by less than duration predicts when
interest rates rise. An instrument for which the opposite is true is said to have negative convexity.
For options convexity is a measure of the way delta (or, more generally, any of the first-order
derivatives or greeks) changes as the underlying changes. In this context (that is, with respect to
delta) it is more commonly known as gamma.
Country risk:
The political or legal risks incurred by a transaction in a particular country, particularly an
emerging market.
Cover: A long or {shot} position in an instrument that offsets partially or wholly a short {long}
position in another. So a covered call is the sale of call options while long the underlying
instrument. Also known as a buy-write. The covered call creates a synthetic short put position (see
put-call parity) and the writer of the option gives up any upside potential beyond the strike of the
calls in exchange for the premium income. If he believes that the price of the underlying will
exceed the strike, then this is a form of forward sale. The covered put is the sale of put options
while short the underlying. Also known as targeted put selling because the writer is effectively
targeting a price at which he will buy the underlying while increasing its yield by taking in option
premium. A covered warrant is covered either by other warrants or by holdings of the underlying
which entitles the holder to buy existing securities in a company at a preset price for a given
period. Originally a feature of the Japanese cum-warrant bond market where warrants were
stripped from bonds and then re-packaged, covered warrants have become popular in Germany
and Switzerland (where they are sometimes known as stillhalter warrants.)
Credit event:
An occurrence that leaves an obligor unable to fulfil its financial obligations. Particularly in credit
derivatives transactions these events are carefully specified and the International Swap Dealers’
Association has its own definitions for events that may constitute a credit event. These are: failure
to pay, insolvency, cross-default, restructuring, repudiation, merger and downgrade.
Credit risk:
The risk that the obligor or counterparty in a financial transaction defaults on their obligations
under the terms of the transaction. The holder of a bond is exposed to the risk that the issuer
undergoes a credit event and defaults on a coupon or principal payment. The counterparty to a
swap is exposed to the risk that the other counterparty does not make payments due under the
swap agreement.
Currency protected:
Used for instruments which give the buyer exposure to a foreign index or asset without the
exposure to the foreign currency that would normally follow. Derivatives that incorporate this
feature are also variously described as currency translated, quantized/quantoed or differential. See
currency protected option, currency protected FRA, currency protected swap, differential interest
rate fix.
Curve lock:
Any instrument or combination of instruments that locks in the spread between two different
points on a yield curve.
Deleveraged:
Used for derivatives or notes with embedded derivatives whose payoff is linked to a fraction of
some index or variable, just as leveraged is used for instruments whose payoff is linked to a
multiple of an index, spread or variable.
Digital:
The same as binary. See chapters 12 and 13.
Duration:
Modified duration is the percentage change in the price of a fixed-income instrument per basis
point change in yield. For a 1% change in yield, an instrument with a modified duration of 1.5 will
change 1.5% in price in the opposite direction. Macaulay duration is the present value weightedaverage term to maturity of a fixed-income instrument expressed in years. It is calculated as the
average life of the present values of all future cashflows of an instrument with the time delay until
receipt of each cashflow weighted by the contribution of that cashflow to the total present value of
the instrument. Both are measures of price sensitivity to interest rate changes. The longer its
duration, the more sensitive an instrument is to interest rates. Instruments whose price rises as
rates rise are said to exhibit negative duration.
Duration leverage:
Concentrating the risk of a longer-dated instrument (e.g. 10 years) in a shorter-dated one (e.g.
three years). The principal amount of the three-year notes loses or gains value according to the
mark-to-market value of the longer-dated one. See duration enhanced notes.
Duration matched hedge:
A risk offsetting position constructed from a long position in one instrument, such as a
government bond, and a short position in another instrument, such as an interest rate swap, which
may have a different maturity, coupon, yield to maturity and equivalent life to the first but which
has an equal and opposite duration.
Embedded option:
An option implicit in another instrument. The commonest are: call options embedded in bonds,
which allow the issuer to redeem the bond early; the options implicit in bonds with sinking funds;
the embedded put provisions in some bonds, that allow investors to put the bond back to the issuer
at a predetermined price; the caps in capped FRNs; the equity call options in convertibles and
exchangeables; the mortgagee’s prepayment options in mortgage-backed bonds; the options
represented by attached debt or equity warrants; the currency options in dual-currency bonds and
the debt or interest rate options in pay-in-kind bonds. Embedded currency, commodity, equity and
interest rate options have become commonplace in both the private and public debt markets.
Exchange-traded contract:
A futures or option contract traded on an organized exchange by exchange members. Exchangetraded contracts tend to be short-term, standardized and limited in complexity though innovation is
changing this.
Extendible:
Used of instruments whose life can be extended beyond an original term at the option of one or
both of the counterparties. See extendible swap, extendible floater.
Factor sensitivity:
The impact on a portfolio of assets of movements in the underlying risk parameter of an individual
asset.
Fair:
The fair price is usually either the theoretical price an instrument should fetch or the no-arbitrage
price.The fair price of a future or forward contract is the price at which arbitrage between the
derivative and the underlying asset just breaks even. The fair value of an option is what should be
the price of that option in an efficient market with reference to theoretical option models.
Floored:
The payout of options, warrants or swaps is floored if it is guaranteed to be at least a minimum
specified amount. The opposite of capped.
Forward-forward [interest] rate:
An interest rate that will apply to a loan or deposit beginning on a future date and maturing on a
second future date. For example, a 6s/12s forward forward is an interest rate agreement fixing the
rate payable on a loan starting in six months time and maturing six months later. Two forwardforward rates are used to calculate an FX forward-forward swap.
Forward outright rate:
The actual forward exchange rate used in a forward contract. The forward foreign exchange rate
for two currencies assuming simple interest rates can be expressed by: Ft = St * [1 + (rd x T - t) x
1/Dd] / [1 + (rf x T - t) x 1/Df] where Ft = the forward rate St = the spot rate (direct quotation) rd=
the domestic interest rate rf = the foreign interest rate Dd= the daycount in the domestic currency
Df= the daycount in the foreign currency T = the maturity of the forward contract t = the current
time such that T - t = the life of the forward. (Note: rd and rf are spot i.e. zero coupon interest
rates. For indirect quotations the foreign currency interest rate would be the numerator and the
domestic currency rate would be the denominator.) The formula is the result of interest rate parity,
the theory that states that investors will transfer funds from low-interest currencies to high-interest
currencies until the advantage of doing so the interest rate differential is offset by the cost of
covering the exchange risk in the forward market - that cost being the forward exchange margin.
That is, the ratio of the forward rate to the spot is a reflection of the interest rates in the two
countries.
Forward points:
The number added to or subtracted from the spot exchange or interest rate to calculate a forward
price, adjusted for point size convention. For example 100 forward points in USD/DEM is equal to
DEM0.01 per USD. See swap rate.
Futures contract:
An agreement to buy or sell a given quantity of a particular asset at a specified future date at a preagreed price. Like forwards, futures differ from options in that they represent an obligation to buy
or sell the underlying. Unlike forwards, they have standard delivery dates, trading units and terms
and conditions. They are available on a wide range of financial and commodity assets, generally
expire quarterly and can be cash or physically settled. Most importantly, they are traded on
exchanges which act as counterparties to all transactions and which run margining systems.
Margin is the collateral futures traders must set aside against their positions. First, an initial
margin must be deposited with the clearing house on entering a trade. Thereafter futures positions
are marked-to-market daily and a variation margin is paid/received to maintain the required level
of collateralization. The role of the exchange and the margin system significantly limit credit risk.
Hedge:
To offset the potential risks and returns of one position by taking out an opposing position to
create an outcome of greater certainty.
Hybrid:
An instrument whose returns depend on a combination of risk types or which has been constructed
from several different instruments to produce returns which mimic those of another instrument. A
common hybrid is the combination of interest rate or currency swaps and barrier or digital options
on other asset classes. See commodity-linked interest rate swap, hybrid barrier option.
Implied forward rate:
A forward interest rate that can be implied from the par or zero coupon yield curves. Not only do
the expectations embedded in the yield curve indicate what the yields on varying maturities of
fixed-income instrument should be, they contain all the information needed to calculate, say, the
one-year rate in one year’s time from the two-year rate and the one-year rate.
So, if six-month Libor is 5.00% (180 days) and three-month Libor is 4.00% (90 days) the implied
rate for three-month Libor in three months’ time must be 6.01%, since this rate satisfies the
condition that an investor {borrower} would be indifferent between receiving {paying} 4.00% for
three months and reinvesting {rolling over} at 6.01% for a further three months, and receiving
{paying} 5.00% for six months. The curve plotted by these rates, known as the implied forward
curve, is steeper than the yield curve. That is, when the yield curve is positive, implied forward
rates are higher than spot rates and in a negatively sloped curve implied forward rates are lower
than spot rates. Mathematically:
(1 + r0,1 t0,1/D) (1 + r1,2 t1,2/D) = (1 + r0,2 t0,2/D) where r1,2 is the implied forward rate. The
implied forward curve is central to an understanding of derivative products as they are priced off
it, rather than the spot rate even if they are struck with reference to the spot. Therefore by
definition, when using derivatives to profit from a market view, potential users must first compare
their view with the implied forward. If they are the same, there is no opportunity to profit from
that view. A number of derivative instruments have been devised that modify the payout from
their vanilla versions by allowing users to take advantage of a view that differs from the implied
forward, for example the Libor in arrears swap. Implied repo rate: The short-term financing rate
that will make a cash-and-carry arbitrage involving the cheapest-to-deliver bond break even. It is
equal to the return earned by buying the cheapest-to-deliver bond for a bond futures contract and
selling it forward via a short position in the futures contract.
Leverage:
The ability to control or gain exposure to a large nominal amount of an underlying asset with a
relatively small amount of capital. Futures and options are leveraged because with relatively small
down payments (of margin or premium) the buyer gains exposure to large amounts of the
underlying.
Leveraged:
When used for derivatives or structured notes ‘leveraged’ indicates that the instruments payoff
formula includes a greater than one multiple of some underlying index or asset price. In the case of
leveraged notes this is generally achieved using embedded swaps or options whose notional
principal is greater than the nominal principal of the bond. See leveraged swap, leveraged diff
floater, leveraged floater, total return index notes.
The New York Federal Reserve has defined leveraged derivatives transactions (LDTs) much more
broadly as a derivative transaction (i) in which a market move of two standard deviations in the
first month would lead to a reduction in value to the counterparty of the lower of 15% of the
notional amount or $10 million and (ii) for notes or transactions with a final exchange of principal,
where counterparty principal (rather than coupon) is at risk at maturity, and (iii) for coupon swaps,
where the coupon can drop to zero (or below) or exceed twice the market rate for that market and
maturity, and (iv) for spread trades that include an explicit leverage factor, where a spread is
defined as the difference in the yield between two asset classes. This definition bears no strict
relevance to the general concept of leverage but means the reclassification of many previously
plain vanilla transactions as LDTs.
Mark-to-market:
The process of determining the present market value of a security or derivative position. See
market contingent credit derivative, mark-to-market swap, mark-to-market cap, swap guarantee.
Margin:
See futures contract.
Market risk:
The risk of mark-to-market losses associated with portfolios of financial instruments.
Monetization:
In derivatives markets monetization refers to the realization of the value of the options embedded
in puttable and callable bonds. This can be done using forward swaps or by selling call options on
government bonds. Most commonly though swaptions are used. The issuer of a callable bond has
in effect bought a receiver swaption with a notional principal equal to the bond principal, an
exercise date equal to the call date of the bond and with the underlying swap maturity equal to the
maturity date of the bond. Selling this swaption monetizes the value of the call feature.
Naked:
The opposite of covered. A long or short derivatives position initiated without any corresponding
position existing in the underlying. So, naked positions would include being long puts without an
underlying position to hedge or being long a swap with no underlying liability or a smaller
liability portfolio than the notional principal of the swap.
Net present value (NPV):
The difference between the present values of two different cashflows. For example, because there
is no upfront premium payable on a standard interest rate swap, the present value at initiation of
the future fixed- and floating-rate payment streams due under the swap must be equal hence the
NPV is zero.
Notional principal [amount]:
The nominal value used to calculate the cashflows on swaps and other cash-settled derivatives. In
an interest rate swap, for example, each period‘s interest rates are multiplied by the notional
principal amount and the daycount to determine the actual amount each counterparty must pay. In
interest rate swaps the notional amounts are not exchanged, so any credit risk is limited to the net
amount payable plus a potential future exposure factor. Descriptions of the size of the derivatives
market almost always refer to notional principal amounts when in fact the amount of money at risk
is a tiny fraction of that.
Novation:
The replacement of one or more derivative contracts with new ones, often also with one of the
counterparties replaced by a new one. One common use of novation is in the creation of chains of
swaps which, having been cancelled and reassigned, can be used to provide loans in circumstances
where straightforward lending would be expensive or not permitted.
Off-market:
Below or above the market rate.
Operational risk:
Any risk that is not market risk or credit risk related. This includes the risk of loss from events
related to technology and infrastructure failure, from business interruptions, from staff related
problems and from external events such as regulatory changes.
Over-the-counter:
The market for securities or derivatives created outside organized exchanges by dealers trading
directly with one another or their counterparties by telephone, screen, telex or other computerdriven means.
Par swap yield curve:
The term structure of swap rates, that is, a yield curve that plots swap rates against maturity and
that is derived from the zero-coupon yield curve.
Par yield curve: The curve formed by the yields to maturity associated with bonds currently selling
at par. The par curve is important as the yields on bonds selling at par are likely to be more
representative of the underlying term discounting rates implicit in the market. Bonds selling at a
substantial discount or premium to par are often subject to special forces which distort their prices.
For example, a high-coupon bond may be considered an especially desirable investment in an
environment where interest rates have bottomed out. Gaps in the curve caused by a lack of
available bonds are filled by interpolating from existing bonds the coupons necessary for bonds at
those maturities to be priced at par.
Parallel shift:
A parallel shift in the yield curve, assumed by many hedging strategies, is a movement of each
point on the yield curve by the same amount at the same time.
Participating derivative:
Derivatives that allow the holder to buy participation in the upside of the instrument either by
giving up protection on the downside or by limiting the upside of products that in their vanilla
form have unlimited profit potential. See participating forward, participating swap.
Power derivative:
Applied to any structure that incorporates leverage that is unusual either on account of its
magnitude or non-linearity. The most common form is an option whose payout is a power (usually
the square) of the intrinsic value at maturity. See power libor swap, power option.
Quanto:
A currency-protected derivative product that is, one denominated in a currency other than that of
the underlying to which exposure is sought. The name refers to the variable notional principal of
these products which reflects the fact that the face amount of currency coverage they contain rises
or falls to cover changes in the foreign currency value of the underlying. An example is a
EUR/USD option with a payout in Japanese Yen embedded in a Japanese yen deposit. See
currency protected.
Quantize:
To give a derivative quanto features. See currency protected.
Ratchet:
Used for a variety of derivative structures in which key variables such as strike price are
resettable, usually in an assymetrical manner. In some cases this leads to investors’ gains being
locked in regardless of future movements in the underlying asset. See ratchet floater and chapter
10.
Second generation structured assets:
Bonds and notes incorporating design complexity in addition to simple embedded options. These
include notes containing index maturity to reset frequency mismatch (such as a CMT FRN with
coupons linked to 10-year Treasury rates but that are reset and paid on a quarterly basis), notes
that pay a coupon based on the differential or sum of a number of indices, notes that include
embedded exotic options, notes incorporating quantization and notes containing very highly
leveraged formulae.
Short-rate volatilities by mean reversion:
short rates tend to be pulled back towards a long-term average. The volatilities of each spot rate
are modelled to produce a term structure of volatility – that is volatility plotted against term to
maturity. This is an important input into term structure pricing models. The term structure of
volatility is also sometimes a reference to the differing implied volatilities of options with
different maturities. The implied volatilities of short-dated options change faster than those of
longer-dated options. Volatility itself also exhibits mean reversion.
Spot rate:
In currency markets, today’s market exchange rate for a transaction now with standard immediate
delivery (usually two business days after trade date). In interest rate markets, the spot rate is the
rate at which a single future payment is discounted back to the present. That is, where observable,
the n-year spot rate is the yield to maturity of a zero coupon bond with a maturity of n-years. For
maturities at which zero coupon bonds are not available, the spot rates can be bootstrapped from
coupon paying bonds at those maturities since the price of these bonds is the present value of all
their cash flows with each cash flow discounted at the appropriate spot rate. See bootstrap.
Spot yield curve:
The curve that plots spot rates against term to maturity.
Spread:
The difference between the yields or volatility on two financial assets (aside from the bid-offer).
For example, in the oil markets, the crack spread is the spread between the price of crude oil and
the refined (‘cracked’) distillates such as gasoil and naphtha. In fixed income markets the credit
spread is the difference in yields between fixed-income instruments of different credit qualities.In
the options markets an option spread, sometimes just called a spread, is the purchase of one or
more options and sale of others on the same underlying.
Spread trade:
In derivatives either a trade designed to profit from movements in the spread between two or more
underlying indices or an options trade involving the simultaneous purchase and sale of different
options on the same underlying.
Step-up {-down}:
A general concept that can be applied to any of the terms of a contract which determine the size of
payments but not their timing, step-up or step-down features involve a schedule of values indexed
in time which determine how payments will vary. So a bond with a step-up or step-down coupon
will have a schedule of coupons which increase or decrease over time. Similarly a step-up or down
barrier option is a barrier option whose barrier increases or decreases over time.
Stepped:
When used for bonds denotes a bond with a fixed first coupon which then reverts to a
predetermined floating rate formula. Differs from a step-up coupon in that only the first coupon
acts as a step. Usually this first coupon is extremely attractive in comparison with vanilla FRN
rates and is an encouragement to the investor to buy highly structured assets that take strong
directional views. For example, leveraged inverse floaters and leveraged floaters often have a
fixed first coupon.
Structured note:
Bonds/notes whose performance is linked to that of a conventional security and an embedded
derivative. Also known as embeddos, derivative-linked securities. See chapter 17.
Swap rate:
(i) The yield to maturity of the swap. That is, the price of the swap which, when used both as a
fixed-rate payment and an internal rate of return, will equate the present value of the two payment
streams. On a vanilla interest rate swap, the bid swap rate is the fixed rate a marketmaker will pay
to receive Libor and the offer is the fixed rate a counterparty must pay to receive Libor. Swap rates
are mathematically equal to the weighted average of all relevant FRAs and so are determined by
the term structure of interest rates, credit and transaction costs. (ii) In currency markets, the swap
rate is the forward points on a currency rate that is, the adjustment to the spot exchange rate that
has to be made to compensate for interest rate parity differences between currencies. When it
refers to the difference between spot and forward foreign exchange rates the swap rate is also
known as the forward exchange margin.
Swap spread:
The difference (positive) between swap rates and the relevant government bond market. The
spread reflects the credit differential between the swap and government markets but in practice is
also heavily influenced by supply and demand factors in the swap market. A glut of fixed payers
will widen the spread. A glut of swapped new issuance will reduce it. The spread in any individual
transaction will also be affected by the relative credit qualities of the counterparties to the
transaction: a triple-A bank marketmaker will quote a wider swap spread to a single-B corporate
than to a double-A supranational entity.
Synthetic:
In financial contexts used for any instrument constructed from others so that its cashflows and
sometimes risk-reward characteristics replicate those of another asset or liability. Such instruments
are created either because certain users cannot buy the components separately or because an
arbitrage opportunity allows the synthetic to be purchased {sold} more >cheaply {expensively}
than the straightforward product. Almost any position or instrument can be constructed in this
way. For example: a synthetic call option can be constructed by the simultaneous purchase of a put
option and the underlying; a synthetic put from a long call and short position in the underlying;
and a synthetic forward from a long European-style call and short European put with the same
expiration and strike price. See conversion [arbitrage], put-call parity, static replication, synthetic
forward, delta hedging.
Systemic risk:
The risk that derivatives permit the transmission of risk across previously unrelated markets, thus
making it more likely that a large shock in one will be transmitted (with negative consequences) to
others. The term is also used for the risk supposedly inherent in the concentration of derivatives
business at a small number of large financial institutions. If – so the argument runs – one of these
were to fail, the whole financial system would be threatened.
Term structure:
The interrelationship of underlying assets of different maturities. The term structure of interest
rates is the interrelationship of interest rates of different maturities. It relates spot rates to the term
to maturity in the form of the spot or zero coupon yield curve. Modelling the relationships
between spot rates at different points in the curve is crucial to the pricing of interest rate
derivatives since even a short-term instrument will span several spot rates and so its price will
depend on how they interact with each other and the rest of the term structure. The dynamic nature
of the term structure has led to the development of multi-factor pricing models where the factors
represent changes in the level, slope and curvature of the term structure.
The term structure of volatility is the volatility of the prices or rates of the underlying at different
maturities. For example, studies of the term structure of interest rates show that spot rates at
different maturities have different volatilities. A basic observation is that long rates are less
volatile than short rates and that long-rate volatilities are linked to current
Termination:
Cancellation of a risk management agreement or derivative transaction upon an agreed event and
on previously agreed terms and conditions.
Total return:
All the cashflows and capital gains or losses associated with an investment.
Transaction risk:
The currency risk incurred by an entity with certain or near-certain cashflows in currencies other
than the domestic accounting currency of the entity.
Trigger:
See barrier, capped call [option], trigger forward, chapter 11.
Value at risk (VAR):
A measure of the maximum potential change in the value of a portfolio of financial instruments
with a given probability over a specified time period.
Yield curve:
A plot of interest rates versus time.
Zero-coupon yield curve:
The spot rate curve of the observed or interpolated yields to maturity of default-free zero coupon
bonds plotted against maturity. From this a forward rate curve or forward term structure can be
implied to give the markets current expectation of future spot rates.
Basic options terms
Key concepts
The options market, like other financial markets, has developed its own jargon. Some of this is necessary to
express the basic operation of options contracts (such as ‘premium’ or ‘exercise price’) and some of it is the
result of the mathematical complexity of option pricing. In particular, it is impossible for any potential user
of options to avoid contact with the ‘Greeks’ a set of Greek letters used to denote variables used in option
valuation.
However, although the underlying mathematics used in today’s option pricing models can be complicated
and well beyond the grasp of non-mathematicians, it is not necessary to understand advanced calculus nor
even the key variables such as delta and gamma respectively the first and second derivatives of the option
premium with respect to the price of the underlying. For an end-user who needs to hedge an underlying
cash position, or an investor who wishes to take a directional view on a market, the concepts that the
‘Greeks’ represent and their impact on the price of any particular option are intuitive and easy to grasp.
This chapter explains all the terms an end-user of options (as opposed to a professional trader) is likely to
encounter in putting together an options trade. For easy reference, ‘Greeks’ are listed below with a brief
explanation. For more complete details, read the individual entries overleaf.
Delta (³): The change in option value for a given change in the value of the underlying.
Gamma (G): The change in the delta of an option for a one-unit change in the price of the underlying.
Rho (R): The change in option value for a one percentage point change in interest or discount rates.
Sigma (•): The standard deviation or volatility of the instrument underlying an option.
Theta (Q): The change in option value over (usually) one day keeping strike, volatility and discount rate the
same.
Vega: (V) The change in option value for a small movement in volatility.
Lambda (L): The change in option value for a small change in the dividend rate (equity options) or foreign
interest rate (foreign exchange options)
American-style:
An American-style option can be exercised at any point during its life. In cases where early
exercise is beneficial (for example, deep in-the-money calls {puts} on underlying stocks with
large {small} dividends), American-style options are more expensive than European-style options.
However for options on non-dividend-paying stocks the American-style call option is the same
price as the European-style. See Bermudan-style, European-style, option.
Assignment:
Notice to an option writer that an option has been exercised. In the swap market, assignment zis
the transfer of a swap obligation to another counterparty.
Asymmetric payoff:
The skewed profit pattern associated with options that gives profit sharing on the upside
(appreciation of the underlying for a call, depreciation for a put) while limiting liability on the
downside. Contrast with the symmetrical payoff associated with forwards and futures.
At-the-money:
An option is at-the-money forward if its strike price option is equal to the current implied forward
price of the underlying. A useful rule of thumb for the approximate price of an at-the-money
forward option is Price = 0.4 * volatility * time * discount factor. For example, a three-month
EUR Call/USD Put with a strike of 1.0370 and with a forward rate at 1.0370 and volatility of 10%
would cost approximately 0.4*0.1*sqrt(0.25)*0.992 = 1.90%. The Black-Scholes price is 1.92%.
Options are often struck at-the-money forward but can also be struck at-the-money spot. This is
the point at which the strike is equal to the prevailing spot price of the underlying. An interest rate
cap struck at the current Libor level is at-the-money spot; one struck at the current swap rate for
the period of the cap (or the FRA rate for a caplet) is at-the-money forward.
An option is in-the-money if it has positive intrinsic value because the market price of the
underlying is above {below} the strike price of a call {put}. The reference rate to determine
whether an option is in-the-money can be either the spot (in which case the option is said to be inthe-money spot) or the forward (in which case the option is said to be in-the-money forward). If an
option is not in-the-money and is not at-the-money then it is said to be out-of-the-money.
Bermudan-style:
An option that can be exercised on a number of predetermined occasions. So, for example, a
bermudan receiver swaption would allow the buyer to enter into an interest rate swap as fixed-rate
receiver on a number of pre-determined occasions as a hedge for a step-up fixed-rate callable bond
in which the bond coupon stepped up annually and the bond was cancellable at each annual
coupon payment. Also known as limited-exercise or quasi-American.
Buy-Write:
A covered call position created by simultaneously buying the underlying asset and selling a call
option on it. This synthetically creates a short put position – see put-call parity.
Call option:
An option that grants the holder the right but not the obligation to buy the underlying at a
predetermined price at or by a predetermined time. The buyer of a call is expressing a bullish view
of the underlying and also implicitly, since he is long an option, believes either that volatility will
rise or at least that it will not fall. See chapter 8.
Delta (³): Delta is defined in three, interrelated ways: (i) The rate of change of the value of an
option for a given change in the value of the underlying asset. An option with a delta of 0.5 (50%)
is expected to change in value 50 cents for every $1 move in the underlying.
(ii) Delta can also be interpreted as a rough measure of the probability of a vanilla option expiring
in-the-money: an at-the-money-forward option has a delta of 0.5, since there is an equal
probability that the underlying will end up above or below this level. The option therefore has a
50% chance of expiring in-the-money and a 50% chance of expiring out-of-the-money.
(iii> Delta also measures the ’hedge ratio‘ – that is the amount of the underlying asset that needs
to be bought or sold to immunize the option to small changes in the price of the underlying. So if
if a call option on a particular stock had a delta of 0.5, then 0.5 shares are required to immunize
that one call.
For European-style options delta increases in a non-linear fashion from zero to one as an option
moves from far out-of-the-money to deep in-the-money. This is because a deeply in-the-money
option has a high probability of expiring that way and so will act as a proxy for the underlying,
rising and falling in a 1:1 ratio with it. A deeply out-of-the-money option will have little
probability of being exercised, so a small change in the price of the underlying will do little to
close the gap between asset and strike price. In addition, the closer an option is to the money, the
faster delta changes. So for our 0.5 delta call, as the stock price rises the probability that the option
will expire in the money rises, so the delta rises, so the more stock has to be bought to immunize
the position. This helps explain why a high delta means greater sensitivity of the option price to
the price of the underlying: the higher the delta, the greater the replicating portfolio’s stake in the
underlying. It also shows how simple option replication requires purchasing the underlying from a
rising market and selling it into a falling market.
For interest rate options delta can be calculated with respect to the underlying bond price, with
respect to each underlying forward interest rate (as sometimes with cap deltas), or with respect to a
small parallel shift in the zero coupon yield curve so that delta is the change in the option price for
a small change in all zero-coupon rates. See delta hedging, dynamic hedging, static replication,
replication.
Delta hedging:
Delta is the neutral hedge ratio derived from the Black-Scholes model the ratio of underlying asset
to options necessary to create the risk-free portfolio that is at the heart of the Black-Scholes option
pricing formula. So the delta of a stock option indicates the number of shares needed to hedge a
position in an option on that stock for example a portfolio long 100 stock call options with delta of
0.3 is delta hedged by a short position of 30 shares and the delta of an interest rate option indicates
the notional amount of interest rate swap required to hedge it against small movements in interest
rates. Delta hedging is the application of this concept to the hedging of options portfolios. A true
delta hedge is the combination of underlying asset and money market instrument that creates the
riskless hedge Black-Scholes says will exactly replicate the pay-off of the option to be hedged. See
delta, dynamic hedging, static replication, replication.
Delta neutral:
An option portfolio delta-hedged such that it has no exposure to small moves in the price of the
underlying. In practice, since delta is altered by all but the very smallest changes in the price of the
underlying, by the volatility of that price, by the maturity of the option, by how close-to-themoney the option is and by interest rates, the ratio of options to underlying must be constantly rebalanced to maintain delta neutrality.
Delta positive:
Call options are said to be delta positive because their value increases by the value of delta for a
one unit rise in the price of the underlying. Put options are said to be delta negative because their
value decreases in value by delta for every one unit rise in the price of the underlying. This
relationship can be upset in barrier options. An in-the-money knock-out call {put} will behave
normally until, at a point near to the knock-out, any further increase {decrease}) in the underlying
will cause the value of the option to drop because the probability of its being knocked-out is more
significant than the fact that it is moving further into the money. At this point puts become delta
positive and calls become delta negative.
Dynamic hedging:
Replication of the payoff of a portfolio long the underlying and long a put by continuous delta
hedging. It started as a theory of Hayne Leland and Mark Rubenstein on the back of the BlackScholes model. It was used to provide put protection for equity portfolios at a time when portfolio
puts were not available. The theory assumed that an option position could be replicated by
continuously adjusting the fraction of funds invested in the underlying equities with the remainder
invested in a risk-free asset. An initial hedge of treasury bills was created, its size depending on
the level of protection required. If the portfolio value fell, stocks had to be sold and the hedge
position increased; the opposite had to be done if its value rose. The theory worked as long as
volatility was predictable and low and while markets did not gap dramatically. Since it relied on a
large amount of trading in the underlying, it also required liquid markets and low bid/offer
spreads. The price discontinuity experienced in the 1987 crash caused such strategies to lose
money and credibility. Also known as portfolio insurance. See delta hedging, static replication,
replication.
Elasticity:
Properly a measure of the percentage change in the option premium for a 1% change in the asset
price. Sometimes loosely used as a synonym for delta (delta strictly measures the absolute change
in the option premium for a one unit change in the underlying). Because elasticity is usually
significantly positive (a 1% change in the asset price can give rise to more than a 1% change in the
option price) it is also sometimes used as a synonym for gearing. This is most common in the
warrant market, where it is calculated as delta times the price of the underlying divided by the
option price.
European-style option:
An option which can only be exercised on expiration.
Exercise:
Of an option, to put into effect the right to buy or sell at the strike price.
Expected value (EV):
The pay-off of an event multiplied by the probability of its occurring summed over all possible
events. For example, the probability of rolling a six on one die is 1¼6 or 16.67%. The EV of a
game in which one is paid $100 for rolling a six and nothing for any other roll is (1¼6 x $100) +
(5¼6 x $0) = $16.67.
EV is a key concept in option pricing, since the calculation of option value relies heavily on
probability theory. The present value of the EV of an option will be the same as its premium if it is
fairly priced. The EV of an option is a function of the size of two things: the relevant distribution
of probabilities for the underlying asset price (itself determined by time to expiry and volatility)
and by the location of the distribution versus the strike price (determined by the relationship
between the strike and the current implied forward rate). The former establishes the range of
possible outcomes, the latter defines the pay-off value of each outcome. See premium.
Forward intrinsic value:
The intrinsic value of an option plus the basis of the forward underlying it. In an efficient market a
European option does not typically trade at less than its forward intrinsic value. An exception is a
deeply in-the-money put option where the inability to exercise early and earn interest on the
proceeds means that the option’s value is the intrinsic value times the discount factor. This is by
definition less than the intrinsic value. See intrinsic value, parity.
Greeks:
The Greek letters used to represent key concepts in pricing derivatives. See delta, gamma, theta,
rho, sigma, vega.
Gamma (G):
Mathematically the second derivative of the option premium with respect to the price of the
underlying. Gamma measures the change in the delta of an option for a one-unit change in the
price of the underlying. If an option has a delta of 0.49 and a gamma of 0.04, the delta would be
expected to rise to around 0.53 if the underlying moved one unit in price. (This relationship is
made more complex because gamma itself changes with movements in the underlying). Gamma is
important to anyone hedging a portfolio of options because it is an indicator of the frequency with
which a delta-neutral portfolio should be re-balanced. Gamma is highest for close-to-the-money
forward options and decreases the further away from the money the option is. Gamma also
increases as volatility decreases for an option which is at-the-money forward. See convexity.
Historical volatility:
The volatility in the underlying’s price, rate or return over a specified period in the past, usually
measured as the standard deviation of the natural log of the underlying price relatives. It is used to
check whether the implied volatility of an option is cheap or expensive by historical standards. N
Ö1/N x S (xi - m) x Ann i=1 Where: xi = log of business daily returns N = total number of
business daily returns m = mean of business daily returns Ann = annualization factor
Implied volatility:
The value for volatility embedded in the market price of an option that will equate that market
price to the fair or model price of the option. Since option pricing models normally require an
input for volatility to derive an option’s price, they can use the market price of the option to derive
the level of volatility implied in it. In theory, since the price of options should depend significantly
on future views of volatility, the implied volatility should contain some indication of the market’s
views of this. In practice option prices are driven by supply and demand factors, themselves
heavily dependent on directional views. In general, the higher the implied volatility, the higher the
price of the option. Many option prices (particularly foreign exchange options) are quoted in
volatility terms, as opposed to ‘live’ price terms.
In-the-money:
See at-the-money.
Intrinsic value:
The amount by which an option is in-the-money and so the cashflow that the holder would realize
if he exercised it. It can only be zero or positive, reflecting the asymmetric payout profile of
options. [The discounted value of the difference between the strike of an option and the forward.
Options struck at-the-money-forward by definition have zero intrinsic value.
Lambda (L):
The change in option value for a small change in the dividend rate (equity options) or foreign
interest rate (foreign exchange options).
Limit/extremum dependent option:
An option where the payoff is a function of the maximum or minimum achieved by an asset
during a reference period. One example is a lookback option.
Option: A contract giving the holder the right but not the obligation to buy {call} or sell {put} a
specified underlying asset at a pre-agreed price at either a fixed point in the future (Europeanstyle) or at a number of specified times in the future (Bermudan-style) or at a time chosen by the
holder up to maturity (American-style). Options are available in exchange-traded and over-thecounter form and can also be packaged as securities either separately (where they are known as
warrants) or embedded in bonds.
Out-of-the-money:
See at-the-money.
Parity: Used in several different senses in the warrant and option markets. Of options generally, parity is
the condition in which an option’s value in the market is the same as its intrinsic value. In the warrant
market though parity can be positive (the warrant has intrinsic value) or negative (it has no intrinsic value).
In the convertible bond market, parity is the market value of the shares of common stock into which the
convertible can be converted. It is calculated by multiplying the stock price by the conversion ratio. For inthe-money knock-out and digital options, parity is the intrinsic value at the barrier level.
Path-dependent option:
An option whose payoff is a function of the path the underlying rate or price has taken over the
life of the option. This contrasts with straightforward European options whose payoff is usually a
function of the price of the underlying at only one point: expiry. Path dependent options are
typically not priced off analytical solutions and to arrive at a price for the discounted expected
value of their terminal payoff over all possible paths, computationally intensive numerical
methods are needed. Many non-vanilla options are path-dependent including: average rate options,
average price options, average strike options, lookback options, cumulative options, ratchet
options, ladder options, digital options, barrier options, shout options and periodic reset options.
Premium:
In derivatives terminology, the amount paid by an option buyer for the option. An option’s
premium, technically, equals the probability-weighted sum of all its possible payoffs at expiry,
discounted to the present. Option pricing models use formulae to calculate this premium or
expected value. Vanilla options are paid for upfront. Many exotic options are paid for in
instalments or have premiums whose payment or the timing of whose payment is contingent upon
some event. In the UK warrant market, warrant premium is the negative intrinsic value of a
warrant if exercised immediately. And in the convertible bond market the conversion premium of
a convertible is the difference between the market value of the convertible and its parity value.
This is usually expressed as a percentage of parity. See expected value.
Put option:
The right but not the obligation to sell a pre-agreed amount of a specified underlying at a predetermined price or rate at or by a predetermined time. The buyer of a put is expressing a bearish
view of the underlying and also implicitly, since he is long an option, believes either that volatility
will rise or at least that it will not fall. See chapter 8.
Put-call parity:
The proposition that the value of a put option is equal to the value of a call option with the same
strike price and time to expiration plus a riskless investment of the discounted value of the
exercise price and a short position in the underlying. That is, the value of a long call option and
short put option both struck at-the-money forward is zero. For European options, an arbitrage
opportunity will exist if this condition is not fulfilled since a put purchased alongside a long
forward position will synthesize a call and a call purchased alongside a short forward will
synthesize a put. Arbitrage prevents the synthetic version of a contract from costing more or less
than the original.
Replication:
To duplicate the pay-out of an option by buying or selling the underlying or futures in proportion
to its delta. To replicate a call option, the hedger must buy an increasing amount of the underlying
if its price is rising and sell increasing amounts if the price is falling because calls are delta
positive. The opposite is true of put replication. Volatility and substantial price gapping makes
replication difficult in practice. This kind of dynamic hedging is central to the theory of portfolio
insurance. See delta, dynamic hedging and static replication.
Rho (R):
The change in option premium for a one percentage point change in interest rates or the discount
rates applied to an asset (rho is the Greek letter ‘r’ and ‘r’ is usually the symbol used to represent
interest rates). So an option with a Rho of 0.30 USD/% will rise by USD0.30 if interest rates rose
instantaneously by 1%.
In general, the higher interest rates are in the denominating currency of the underlying asset, the
higher will be the value of the call option and the lower will be the value of a put option. This is
because the higher interest rates are the higher the forward price and the lower the present value of
the exercise price of an option and so the higher the value of a call and the lower that of a put. In
buying a call instead of the asset, the buyer releases capital to be invested in a risk-free asset.
Sigma (•):
The annualized standard deviation or volatility of the instrument underlying an option.
Spread:
See chapter one.
Static replication:
In some cases, a complex option can be hedged through portfolios of more standard options
designed to replicate, either approximately or exactly, the payoff characteristics of the complex
option. In contrast to dynamic hedging the idea of this approach is that the hedge would not have
to be changed over the life of the option. For this reason the technique is known as static
replication. See delta hedging, dynamic hedging, replication, synthetic.
Strike price/rate/level:
The pre-determined level at which an option can be exercised. For example, the owner of a
European-style three-month USD call/JPY put with a strike at 140 has the right to buy dollars and
sell yen at an exchange rate of 140 yen to the dollar in three-months time.
Theta (Q):
The sensitivity of option price to the passage of time. The longer the maturity of an option, the
greater the value in having the right to exercise or not and so the more valuable the option. The
amount of the option‘s value that is derived from this phenomenon is its time value and the rate at
which this decreases as the option’s life shortens is called theta or time decay. An option with a
one-day theta of 0.075 will lose 0.075 of its value as the number of days to expiry decreases by
one. Theta is greatest for at-the-money options close to expiry. Theta is closely related to gamma.
Time value:
In options terminology, that part of the premium that is not intrinsic value – that is, the part of the
value of an option made up primarily of its time to expiry, strike level and volatility. Time value
represents the value of the right to choose to exercise an option against the obligation. The time
value of an option decreases at a faster rate the closer it is to expiry.
Vega:
The change in an option’s price for a small movement in volatility. It is expressed either as the
absolute change in the value or price of an option for a percentage point change in the standard
deviation of the underlying or in points per percentage change in volatility. At-the-money forward
options are most sensitive to changes in volatility (their vega is highest) while deep in-the-money
and deep out-of-the-money options are relatively insensitive. Options are also more sensitive to
volatility the longer their time to maturity.
Vega is important in hedging options positions because implied volatility and therefore the
expected hedging costs and value of the options can change, reflecting a change in views about
future volatility, without any change in the theoretical price of the underlying. This means that the
option premium may change, and so a hedge position may change in value, even if the position is
delta and gamma hedged. The total exposure to volatility of a position is measured by the
weighted average of vega. A positive vega position is used if a rise in volatilities is predicted and a
negative vega if a fall is foreseen.
Volatility:
The measure of how quickly a price varies over time. Annualized volatility is the commonest
measure and is usually calculated as the annualized variance or standard deviation of the
underlying price, rate or return. Volatility is at the core of all option pricing models because the
more volatile the price, rate or return on an asset is, the more likely it is to exceed the option strike
price and so the more valuable the option.
Option pricing models differ in their approach to volatility which affects the prices they generate.
Black-Scholes and other early single-factor models assume constant volatility. Newer models
remedy this error by assuming volatility to be stochastic. This helps explain the volatility smile
effect as it increases the value of out-the-money forward options relative to the at-the-money
forward options. This is because models that incorporate this assumption allow a greater
probability to large movements in the underlying than simpler models. However, as stochastic
volatility is a non-traded source of risk, using it as an input into pricing models loses their
completeness – that is the ability to hedge options with the underlying asset. Other models assume
that the continuously compounded returns of the asset are normally distributed with a variance that
is proportional to the time over which the price change takes place. This implies that volatility will
increase indefinitely with time. In fact, financial assets exhibit tend to exhibit mean reversion – at
a given price extreme it is more likely for the price to move back towards the mean than it is for it
to move to a new and more extreme price.
Volatility skew:
The asymmetrical distribution of implied volatility in many markets. Out-of-the-money puts can
have higher implied volatilities than calls and vice versa, a fact explained in market terms by
supply and demand. When traders talk of trading the skew, they are generally talking about trying
to predict the slope of the implied volatility curve plotted against strikes or deltas and choosing an
option position that profits if their view is correct. A negatively sloped implied volatility curve
implies a negatively skewed probability distribution for the level of the underlying. The skew
implied by the Black-Scholes model is zero. In extreme cases the smile can create a two humped
or bi-modal probability distribution, unlike the one-humped probability distribution predicted by
Black-Scholes. Skew is generally largest in pegged or managed exchange rates where the
probability of a large move in one direction is virtually zero, whereas in the other direction it is
non-zero or possibly quite large. See risk reversal.
Warrant:
A securitized, generally medium- to long-term, option – often listed on a stock exchange.
Write:
To sell an option.
Advanced option terms & pricing models
Introduction
Most end-users of options want answers to a handful of key questions about an option position: how much
is the premium? Is the structure the best way to hedge a position or to take a view on a market? What are
the tax, legal and regulatory implications of the trade? However, anyone trading options, anyone who has to
mark option positions to market, anyone looking to close out an option position before maturity, anyone for
whom the ongoing efficiency of a hedge is important, anyone who has bought a security with embedded
optionality in fact just about any user of options is exposed to the complexities of option valuation.
Understanding how and why the value of an option can change as the underlying changes is critical if
mistakes and disappointments are to be avoided.
These complexities have led to a proliferation of different pricing models. Some of these have been
designed for a particular type of option foreign exchange options, or American-style options for example.
Some are responses to the known weaknesses of other models. And many were created to incorporate more
sophisticated assumptions about the behaviour of the price of the underlying assets and the volatility of that
price in an attempt to reflect more accurately observed market movements.
Explaining these models is beyond the scope of this book. The descriptions are included to enable options
users to compare the different assumptions underlying the models and to highlight the fact that the price of
an option depends not just on the conditions they observe in the markets but on the ways in which option
dealers choose to price them.
Because so much of option pricing theory is concerned with understanding the behaviour of the underlying
variables, much of the terminology employed comes straight from statistics. This chapter therefore also
explains the key concepts and defines the technical terms most commonly used by derivatives experts terms
that may not be familiar to those outside risk management.
Analytic model:
An option pricing model which, like the Black-Scholes model and its later variants, finds an
explicit solution to the problem of pricing a particular option or options using mathematical
functions. Black-Scholes and others, for example, specify and solve a stochastic differential
equation. While these models are simple, they cannot handle the early exercise feature of
American-style options. This is because the decision to exercise before expiration depends on the
behaviour of the price of the underlying security throughout the life of the option and cannot be
reduced to a single parameter. They are also increasingly inaccurate as the term of the option
lengthens because they cannot easily take into account variations in short-term interest rates or the
time-dependence of volatility. The analytical solutions on which these models are based are also
known as closed-form solutions and so the models are known as closed-form [option pricing]
models.
Analytic approximation models:
One of the three main classes of option pricing model (along with analytic and numerical models).
Analytic approximation models may be used when traditional analytic approaches do not bear
fruit. They involve a combination of theoretical analysis and simplifications judiciously chosen to
make the solution tractable. One example is provided by the Barone-Adesi-Whaley model for
pricing American options. Here the premium for early exercise is estimated and then added to the
price of a European option which may be obtained analytically.
Antithetic variables:
This is a technique used in Monte Carlo valuation for reducing the variance of the estimate of
derivative security. In its simplest form it involves creating two paths from the same set of random
numbers. These paths would be mirror images of each other. The value of the derivative is
calculated for each of these paths. The average of these two values has lower variance than the
individual values themselves and is hence a better estimator of the value of the derivative.
Arch:
Acronym for autoregressive conditional heteroscedasticity, an econometric technique developed
by Robert Engle in 1982 to model economic variables. It is an estimation procedure developed on
the basis of a model of economic variables that allows the covariance matrix of these variables to
change with time. It assumes that variance is stochastic and is a function of the variance of the
previous time period and the absolute level of the underlying variable. Specifically, the conditional
variance of a time series is allowed to depend on lagged squared residuals in an autoregressive
manner. This means that during periods in which there are large unexpected shocks to the variable,
its estimated variance will increase, and during periods of relative stability, its estimated variance
will decrease.
Arch has found much favour in the options world as the basis for models which do not assume that
volatility is constant. Most of the older option pricing models do despite the evidence to the
contrary.
Instead, Arch-based models assume that volatility follows clear patterns; that today’s depends on
yesterday’s and so historical volatility contains information that can be used to estimate future
volatility; and in particular that volatility should regress back to its long-term average. Several
other variations exists, including Garch, AGarch, EGarch and QGarch. See Garch.
Arbitrage-free model:
Option pricing models that do not allow arbitrage of the underlying variable. Most commonly
applied to models developed by Cox-Ingersoll-Ross, Ho-Lee, Heath-Jarrow-Morton and HullWhite. These were originally developed to price interest rate options and incorporate constraints
on the movement of interest rates designed to avoid arbitrage possibilities caused by yield curve
movements. They differ essentially only in their assumptions about spot rate movements.
Autocorrelation:
The correlation between changes in a single variable over non-overlapping time periods. If a price
or rate were negatively autocorrelated a move down in one period would suggest a move up the
next (and vice versa). If it were positively autocorrelated then a move down would suggest a
following move down (and vice versa).
Backward Induction:
This is a mathematical technique fundamental to the valuation of derivatives by tree or finite
difference methods. It assumes that when making a decision an agent will maximize value based
on the future expectation of returns accruing from each alternative. As an illustration of the basic
principle consider the holder of an American option. At each point in time the holder will exercise
the option if the value of the payoff he receives on exercise is greater than the discounted expected
values of future payoffs if he does not exercise. (That is, the value of the payoff on exercise is
greater than the value of the American-style option).
Barone-Adesi-Whaley:
An analytic approximation option pricing model devised in 1987 by Giovanni Barone-Adesi and
Robert Whaley which incorporates a quadratic approximation approach in a very accurate model
for haluing American-style calls and puts on assets which pay continuous dividends.
Binomial distribution:
The most important discrete probability distribution in options pricing. (Discrete probability
distributions are those in which the underlying variable can only have certain discrete values. Most
option pricing models assume continuous probability distributions such as lognormal and normal
distributions.) To satisfy a binomial distribution a discrete random variable must satisfy four
conditions: only two possible values can be taken on by the variable in a given time period (known
as a binomial trial); for each of a succession of trials the probability of each of the two outcomes
must be the same; each trial is identical; each trial is independent.
Binomial option pricing model:
An option pricing model which uses binomial trees to model the price of the underlying. This is
the most common type of numerical model. The key to the binomial or binomial lattice-based
model is the binomial trial process.
This divides the time until option maturity into discrete intervals or steps and presumes that during
each of these intervals the key parameter typically the price or yield of a security follows a
binomial process moving from its initial value S, either up to value Su with probability p or down
to value Sd with probability 1 minus p. Representations of the resulting distribution resemble trees
or lattices and so the series of values generated by the binomial trial process is known as a
binomial tree. (More complex versions exists: a trinomial tree would allow three possible
movements, and a multinomial model more than that).
The binomial process is usually specified as being path-independent that is, a move up followed
by a move down results in the same price as a move down followed by a move up so that the
branches recombine. Trees that do not incorporate this feature are said to be non-recombining,
bushy or exploding. They are much more computationally demanding. By working backward
through the lattice from expiration, at which time the value of the option is known, options can be
evaluated by backward induction to discount the terminal payoff through the tree: the value of the
option is that which avoids an arbitrage profit. The advantage of binomial models is that they can
deal with a range of different assets, options or market conditions. So, a lattice-based model gives
rise to an algorithm rather than a closed formula for determining the option value. Such models are
particularly useful for valuing American-style options. The best-known is the Cox-RossRubinstein model. See backward induction, Cox-Ross-Rubinstein.
Black’s Model:
This, like Garman-Kohlhagen’s model for foreign exchange, is a derivative of the original BlackScholes model. Originally this was a model for the pricing of options on futures but it has been
extended in scope to include any situation in which it is necessary to value a European option on a
variable which can be assumed to be lognormally distributed about a forward price and where
interest rates are non-stochastic. In particular it is frequently used to price interest rate caps and
swaptions.
Black-Derman-Toy:
A single-factor (in this case short-term interest rates) term structure option pricing model proposed
by Fisher Black, Emanuel Derman and William Toy in 1990 which expanded on the
Ho-Lee model by specifying a time-varying structure for volatility and incorporating it into a
binomial tree of possible forward short rates.
Black-Karasinski:
A single-factor model of the term-structure where the logarithm of the short-term interest rate is
assumed to be a ‘mean-reverting’ process with time-varying coefficients.
Black-Scholes:
Developed by Fischer Black and Myron Scholes in 1973, this is the classic modern option pricing
model and the first general equilibrium solution for the valuation of options. The model provides a
no-arbitrage value for European-style call options on shares as a function of the forward price, the
exercise price of the option, the risk-free interest rate and the variance of the stock price which is
assumed to follow a lognormal distribution. It does this by recognising that stocks and calls on
them can be combined to construct a risk-free portfolio and that options on equities can therefore
be valued using a dynamic hedging argument. That is, the option writer can exactly offset his
exposure to the underlying stock by continuously buying or selling it. The model shows that, by
combining the underlying stock and a money market instrument, a riskless hedge (the delta hedge)
can always be formed that exactly replicates the payoff of the option to be hedged. This means that
a portfolio formed by the combination of the option and its riskless hedge must appreciate at the
risk-free interest rate. This riskless hedge method circumvents the difficulties of specifying
investors’ risk preference and allows the risk-free interest rate to be used in the valuation process
rather than some other discount rate that reflects the appropriate risk level. For any time period,
the value of such a portfolio can be computed as its value at the end of the period discounted back
one period at the risk-free rate. Because the price of an option is a deterministic function of the
price of the underlying asset at that time, given that the distribution of asset prices is known for
each time period (and in this model it is assumed to be lognormal), then the initial value of the
option can be deduced by working backwards in time. For a European call option, the BlackScholes pricing formula is:
C = SN(d1) - Ee-rT (N(d2) where d1 = ln (s/e) + (r + 0.5s2)T sT d2 = d1 - sT, N(d1) and N(d2) are
the cumulative probability for a unit normal variable z That is it is the probability N(d1) = ¥òd1
f(z)dz where f(z) is distributed normally with mean zero and standard deviation of one. 1n is the
natural logarithm e is the exponential T is time to maturity s2 is the instantaneous variance of the
stock price which is the measure of volatility of the stock.
The equation states that the value of the call is equal to the stock price, S, minus the discounted
value of the exercise price, Ee-rT, each weighted by a probability. The stock price is weighted by
N(d1) which is also the hedge ratio. For each call written, the riskless hedge portfolio contains
N(d1) shares of stock. On the other hand, the discounted value of the exercise price is weighted by
N(d2) which is the probability that the option will finish in the money. The value of a European
put, P, can be derived in a similar way as: P = Ee-rT N(-d2) - SN(-d1). The model’s great
achievement is completeness: it provides a method for hedging options with the underlying asset,
which allows for arbitrage pricing and hedging. Its drawbacks are that it assumes no dividends, no
taxes or transaction costs, constant short-term interest rates, no penalties for short sales, that
volatility and interest rates are constant, that the market operates continuously and that stock price
distribution is lognormal. The generalizations of Black-Scholes address these problems, while
extensions to it apply it in a modified form to options on futures (Black’s Model), options on
currencies (Garman and Kohlhagen’s Model) and to exotic options.
The basic model has problems pricing short-dated options because volatility is not timehomogeneous and long-dated options because it fails to take into account mean reversion. It
systematically undervalues near-maturity options, deeply out-of-the-money options, options on
low volatility stocks. It systematically overvalues long-dated options, deeply in-the-money options
and options on high volatility stocks. All these problems are due to the model’s assumption of the
uniformity of variance across time. Other types of models address these problems.
Brownian bridge:
A Brownian Bridge defined between two points in time t0 and t1 is an arithmetic Brownian
motion conditioned to take specific values x0 and x1 at those points. The distributional properties
of the Brownian Bridge may sometimes be used in option pricing when time is approximated as
being discrete (e.g. in binomial trees or Monte Carlo techniques) to smooth out the effect of the
discretization.
Brownian motion:
Archetypal random motion. Variants of this are used as the assumed path of securities prices in
many financial models.
CEV- model:
The CEV, or constant elasticity of variance, model was original proposed by Cox and Ross in
1976 to try to account for some empirical observations about equity price volatility. The
specification of volatility is given as sS–a so for a=0 we have the classical model of stock prices
used within Black-Scholes. With values of a greater than 0 the equity prices have high volatility
when the equity price is low and low volatility when the equity price is high.
Confidence interval:
A range of values in which, with some specified probability, the value taken by a variable will lie.
Continuous variable: A variable such as time that can be subdivided into an infinite number of sub-units
for measurement. The unit of measurement can therefore be increased or decreased infinitesimally. See
discrete variable.
Continuous probability distribution: A distribution where the variable can take any value within a
specified range such as the gain or loss from a position in a financial asset over a specified interval of time.
Continuous probability distributions do not consider the probability of the variable taking on a specific
value. See discrete probability distribution.
Control variates:
This is a technique available to all numerical valuation techniques. It consists of using the same
technique to value not just the option whose value is required but also an option, the control,
whose characteristics are close to the original option but whose value can be calculated in some
more accurate manner (preferably analytically). The difference between the control’s values
calculated analytically and numerically is an estimate of the error inherent in the numerical
approach and is added on to the numerically calculated value of the original option. Choosing an
appropriate control can result in a greatly reduced variance of the estimate.
Convexity adjustment:
The most common use of this term is to describe the adjustment that has to be made to the interest
rate implied by the price of a Euro-deposit future to obtain the corresponding forward interest rate.
It has come to mean any adjustment to a price obtained under simplifying assumptions to account
for real-world non-linearities, for example adjusting the price of a barrier option to take account of
non-uniform volatility over different deltas.
Correlation:
A measure of the degree to which changes in two variables are related. The standard measure of
correlation is the correlation coefficient, a number between minus one and plus one that indicates
the strength and direction of a linear relationship between two variables. A correlation coefficient
of minus one indicates that they are perfectly negatively correlated, zero that they are not
correlated at all and one that they are perfectly correlated.
Correlation risk is the risk that two variables or instruments are unfavourably correlated.
Identifying and quantifying correlation risk has become a key element in pricing and hedging
certain derivative instruments. In some options such as spread options and cross-currency caps, the
correlation between the underlying assets is a first-order effect as it directly affects the option
price. In quanto products, such as differential swaps, there is a second order or indirect effect, in
that case between interest rates and exchange rates. See second-order effect.
Covariance:
A measure of how two random variables behave in relation to each other. Matrices of covariances
are used in several different financial models, the most famous of which is Sharpe’s Capital Asset
Pricing Model. It differs from correlation in that it incorporates measurements of the magnitude of
the variations as opposed to the correlation coefficient which is dimensionless. The correlation
coefficient between two random variables is equal to the covariance between them divided by the
product of their standard deviations.
Cox-Ingersoll-Ross:
A generalization of the Black-Scholes option pricing model incorporating the work of John Cox,
Stephen Ross and Jonathan Ingersoll. The model represents one of the two approaches followed
by term structure option pricing models. It models the expected returns from movements in the
term structure in order to price them. The second approach, followed by Ho-Lee, Heath-JarrowMorton, Black-Derman-Toy, and Hull-White utilizes the volatilities of the various sectors of the
term structure to derive a probability distribution for an arbitrage-free binomial, trinomial or
multinomial lattice of the term struc?ure.
These models all have one thing in common:
they allow for the whole-term structure to be stochastic instead of the price of a single underlying
instrument or a single interest rate. The whole-term structure is represented at each node of the
lattice. This methodology allows both long-term and short-term interest rate instruments to be
priced with an internal consistency not possible if different models are used to price different
instruments.
Cox-Ross-Rubinstein model:
The classical binomial option pricing approach first proposed by Cox, Ross and Rubinstein in
1979. It requires that u (the up-jump multiplier) is 1/d (the down-jump multiplier).
Crank-Nicholson technique:
Technique for the numerical solution of partial differential equations, which is particularly useful
for diffusion equations. Since the equation satisfied by option prices is a diffusion equation it is
frequently used in pricing by finite differences.
Discrete probability distribution:
A distribution of the probabilities that a variable takes certain discrete values.
Discrete variable:
A variable that can only take certain discrete values – such as whole numbers.
Differential equation:
Equation where the values of variables are implicitly related to each other through their derivatives
(or rates of change). The analysis performed by Black and Scholes resulted in a differential
equation where the derivatives of the value of an option with respect to time and the spot price of
the underlying are related together.
Diffusion process:
A continuous-time model of the behaviour of a random variable that uses geometric Brownian
motion as its basic assumption. In the Black-Scholes model, the price of the underlying follows a
pure diffusion process – that is, it is assumed to move continuously from one point to another. The
consequence of this assumption is that the terminal distribution of share prices is lognormal. Other
models, particularly discrete-time models, use modifications of the process.
Equilibrium model:
An equilibrium model of the yield curve makes assumptions about economic variables and
economic behaviour and uses the requirement of equilibrium of the economic system to deduce
the process followed by the yield curve. From this process the processes followed by discount
bonds and options can be deduced. A fundamental obstacle to using this approach for pricing
interest rate derivatives is that there is no guarantee that the initial term structure will be matched.
Fauré sequence:
A particular type of quasi-random number sequence. See Halton sequence.
Finite difference methodology:
An option pricing approach based on finding a numerical solution to the differential equation that
the option valuation must satisfy. It does this by converting the differential equation into a series
of difference equations which are then solved iteratively.
Garch:
Acronym for generalized autoregressive conditional heteroscedasticity. A variation of the pure
Arch that generalizes the univariate Arch models into allowing the whole covariance matrix to
change with time instead of just the variance. Several other variations exist. See Arch.
Garman-Kohlhagen:
The classic and commonly used extension of the Black-Scholes option pricing model to pricing
currency options. Mark Garman and Steven Kohlhagen showed that much the same arguments
apply to pricing currency options as apply to pricing stock options with adaptations to allow for
the two interest rates and the fact that a currency can trade at a premium or discount forward
depending on the interest rate differential. (The dividend yield is replaced by the foreign interest
rate).
Gaussian distribution:
See normal distribution.
Geometric Brownian motion:
Describes the movements in a variable or asset price when the proportional
change in its value in a short period of time is normally distributed. The
proportional changes in two non-overlapping periods of time are uncorrelated,
hence the alternative name for the process random walk. The term geometric
refers to the fact that it is the proportional change in the asset price (not the
absolute level) that is normally distributed. This means that the future value of a
variable following geometric Brownian motion has a lognormal probability
distribution and is always positive, unlike a variable following a Wiener process,
whose value can become negative. This makes it mathematically useful and
consequently it is the most common assumption for the movement of stock prices,
stock indices, currencies and futures contracts. It is the assumption made for stock
prices in the original Black-Scholes options pricing model.
Geske-Johnsonng
The Roll-Geske-Whaley model values call options on dividend-paying assets but
is not applicable to American-style puts on such assets. Indeed there is no
analytical solution. The Geske-Johnson model, an extension of the Roll-GeskeWhaley model, notes that there is a positive probability of early exercise of in-themoney puts which means that an American-style option can be viewed as an
infinite sequence of options to exercise a European-style option. However, when
the put is on an asset that pays dividends, the valuation procedure is simplified
because it will not be optimal to exercise prematurely the option at any time near
to but prior to an ex-dividend date. Because of its complexity, it uses trivariate
normal density functions. Many market practitioners use binomial models instead.
Halton Sequence:
A particular type of quasi-random number sequence. See Fauré sequence.
<STRONG
Heath-Jarrow-Morton:
A multi-factor term structure option pricing model that uses all the information in
the term structure and can handle multiple causes of term structure movement.
This means that the returns on zero-coupon bonds of differing maturities are not
assumed to be perfectly correlated (as is assumed, for example, by the Ho-Lee
model). The most common form is a two-factor version, where the two factors are
an underlying (in this case the entire term structure which is an input into the
model in the same way that the current stock price is an input into Black-Scholes)
and volatility – that is, a description of how the term structure fluctuates over
time. This means that the model does not have to assume that all bond prices (in
fact the model uses stochastic forward rates not zero coupon yields) are perfectly
correlated. Instead, it assumes a random term structure of interest rates and is
designed to be automatically consistent with both the observed term structure and
the volatility functions input by the user. As a result of using a multi-factor model
of the term structure, the model employs a multinomial instead of binomial model
of term structure movement. The key difference between it and the spot rate
models of Black-Derman-Toy, Vasicek, Hull-White and Cox-Ingersoll-Ross is
that these models treat the spot interest rate as the underlying variable. Besides the
current spot rate, these models include various parameters used to describe the
possible future paths of the spot rate. Since the current term structure is not a
direct input, these models try to fit the term structure by searching for parameter
values which cause calculated zero coupon bond prices to match the market.
Heteroscedastic:
In simple linear regression, an error term compensates for the fact that in
modelling the relationship between two variables, one of which is assumed to be
the major factor in the movements of the other, movements in one will in fact be
imperfectly described by movements in the other because of factors not captured
by the regression model. This error term is normally distributed with a mean of
zero so that its effects cancel each other out. If the variance of the error terms is
constant, the regression is said to be homoscedastic. If it is not, it is said to be
heteroscedastic.
Ho-Lee:
The first whole-term structure option pricing model, proposed by Thomas Ho and
Sang-Bin Lee in 1986. Using a discrete-time binomial approach this single-factor
model incorporates the whole term structure rather than just changes in a long or
short interest rate. Given the term structure as known today, in the next time
period the whole term structure can move up or down. However, the model makes
a number of assumptions not borne out by empirical observations: it assumes that
the returns of zero coupon bonds of different maturities (which it uses to represent
the term structure at each node on the binomial lattice) move in a perfectly
correlated manner and it requires that all interest rates both spot and forward have
the same volatility. It also allows for negative interest rates as it does not
incorporate mean reversion. Unlike Black-Scholes-type models, Ho-Lee
establishes no explicit link between hedging and pricing.
Hull-White:
A single factor model developed using a trinomial lattice. It is a yield-curve based
model in the same mould as the Ho-Lee, Vasicek, Heath-Jarrow-Morton and
Black-Derman-Toy models. A key feature of Hull-White is that it treats mean
reversion as time-dependent.
Implied correlation:
The correlation coefficient implied by the price of a two-factor option on the
assumption that volatilities for the two assets involved are available. This concept
is most relevant in the foreign exchange markets where there are liquid markets
in, for instance, dollar-mark, dollar-yen and mark-yen options. The option prices
quoted on these options give implied volatilities, the exchange rates and implied
correlations between them.
Ito Process:
A generalized Wiener process where the drift and dispersion parameters are a
function of state and time.
Ito’s Lemma:
An equation giving the process followed by a function of an Ito process. This is
also an Ito process whose coefficients are given in terms of the derivatives of the
original process with respect to state and time. This lemma (proposition) is
fundamental to Black-Scholes since an option price is a function of an underlying
spot price whose behaviour is conventionally assumed to be an Ito process.
Knowledge of the relationship between the dispersion parameters of the two
processes given by the lemma enables a risk-neutral portfolio to constructed and
hence a risk-neutral price for the option to be derived as a solution to a partial
differential equation whose form is also determined by the lemma.
Jump diffusion process:
The process proposed by Robert Merton whereby the price of the underlying
neither simply jumps nor follows a pure diffusion process but moves by a
combination of a jump followed by continuous diffusion. Option pricing models
have been extended to incorporate these kinds of jump price dynamics with
directional bias but there are still theoretical problems associated with jump
diffusion models. For example, the underlying asset in a foreign exchange option
is an exchange rate which can be denominated in either of two currencies.
However, jump diffusion models do not give the same prices when compared in a
common currency.
Jump process:
A stochastic process for movements in the price of the underlying proposed by
John Cox and Stephen Ross. In it the price of the underlying does not follow the
pure diffusion process assumed by the Black-Scholes model but rather jumps
from one point to another in steps larger than traditional random processes would
generate. This idea was expanded in the Cox-Ross-Rubinstein binomial model.
Kurtosis:
A measure of the extent to which probability is concentrated more around the
mean and in the tails rather than in the mid-range relative to a normal distribution.
A normal distribution has a kurtosis coefficient of three. Kurtosis of less than that
indicates a distribution with a fat midrange on either side of the mean and a low
peak – called platykurtic . A kurtosis coefficient greater than that indicates a high
peak, thin midrange and fat tails – called leptokurtic. Empirically the latter is the
phenomenon most frequently observed in financial prices and markets. It results
in implied volatilities of vanilla options which are higher for strikes above and
below the forward than for options struck at the forward - the well-known
volatility smile. See volatility smile.
Least squares regression:
One of a number of types of regression analysis that measure the relationship
between variables.
Linear regression:
Linear regression is a regression where the dependent variable is modelled as a
linear combination of the dependent variables plus the error term. The simplest
case is simple linear regression where there is only one dependent variable. Here
the relationship is: y = a + bx + u, where a is a constant; b is the regression
coefficient and u is the error or disturbance term.
Lognormal distribution:
A variable has a lognormal distribution if the natural logarithm of the variable is
normally distributed. A consequence of the classical assumption about the process
followed by assets, that it is geometric Brownian motion with drift, is that the
asset price at a particular point in time in the future has a lognormal distribution.
Thus when valuing an option using the risk-neutral expectation approach, to
obtain for instance the classical Black-Scholes option pricing formula, we would
take the expectation with respect to that lognormal distribution.
Low-Discrepancy sequences:
See quasi-random numbers.
Markov process:
A class of stochastic processes or models which define a finite set of states. The
essential property of the Markov process is that the future behaviour of the
process (the progression of the set of states from one state to the next) is
independent of past behaviour and determinable solely from the current state.
Most option pricing models assume that movements in the price of the underlying
or, in the case of interest rate options, the zero-coupon curve, are determined by a
Markov process.
Mean:
The sum of observation values divided by the number of observations. A
statistical measure of central tendency.
Mean reversion:
The statistical tendency of variables, most relevantly stock prices, interest rates,
and volatility, to trend away from extremely high or low values and to gravitate
towards a long-term average level. When the value of a mean-reverting variable
reaches a very high level, it is more likely to go down than to go up. When it
reaches a very low level, it is more likely to go up than to go down. Mean
reversion is important in option pricing because it contradicts an assumption of
many early models that the variance of the price of the underlying asset of an
option is directly proportional to the option’s term to expiration. This assumption
implies that the statistical dispersion of asset prices will widen indefinitely further
and further into the future. In interest-rate option pricing models it means that
interest rates can become negative. (Interest rate models are further constrained in
absolute terms: in a normal economy 100% rates are extremely unlikely.)
The practical consequence for pricing is that the longer-dated an option, the more
seriously it will be mispriced by models that ignore mean reversion. To account
properly for mean reversion and hence estimate the volatility of an economic
variable that demonstrates it, a more complicated underlying model than
geometric Brownian motion is needed. Models such as Vasicek and CoxIngersoll-Roll incorporate mean reversion to account for the term structure of
volatility. The Hull-White model goes further by proposing that mean reversion is
time-dependent.
Monte-Carlo simulation:
A generic technique involving the generation of random numbers to solve
deterministic problems. It is often used by numerical option pricing models as an
alternative to the binomial process as a simulation of the underlying asset price.
Using computers, a Monte-Carlo simulation attempts to simulate the process that
generates future movements in the price of the underlying. Each simulation results
in a terminal asset value and several thousand computer simulations give a
distribution of terminal asset values from which the expected asset value at option
expiration can be extracted. This method is used to value complex options,
particularly path-dependent options for which there is no analytical solution.
Multi-factor model:
An option pricing model in which there are two or more sources of randomness
contributing to the option price. The reasons for using multi-factor models can be
split into two: either the option payout itself is a function of two variables (e.g. a
spread option on A or B, or an option on A that knocks out on B), or the processes
of the variables appearing in the option payout are defined in terms of many
factors (e.g. stochastic volatility models). See single-factor model.
Non-uniformity:
In option pricing used to refer to the fact that volatility is expected to be higher on
certain days than on others.
Normal distribution:
The most widely occurring frequency distribution. The normal (or Gaussian)
distribution is distinguished by its symmetrical bell shape and has the statistically
desirable characteristics of being completely described by the mean and standard
deviation of the distribution. The mean indicates the position of the centre of the
bell, the standard deviation how spread out it is. If a variable is normally
distributed, 68.27% of its values will fall within plus or minus one standard
deviation of the mean; 95.45% will fall within plus or minus two standard
deviations and 99.73% will fall within plus or minus three standard deviations
from the mean.
Numerical model:
An option pricing model which avoids the requirement to solve a stochastic
differential equation by specifying a particular process for the underlying asset
price and then using an iterative approach to solve the value of the option.
Numerical models can be divided into three main classes: the binomial models,
the finite difference models, and Monte Carlo simulations.
Poisson Process:
A process useful for describing events which happen discretely but randomly in
time, e.g. crashes, central bank rate hikes. It is frequently used as a component of
jump diffusion processes to describe the occurrence of the discrete jumps.
Pseudo-random numbers:
Monte Carlo techniques for valuing derivatives require a supply of random
numbers which are uniformly distributed and independent. Implementation of
these techniques is performed on computers which are deterministic machines and
therefore not intrinsically good at creating randomness on demand. There are
deterministic algorithms which generate sequences of numbers which appear to
have the appropriate properties of randomness. However, because they are not
truly random they are termed pseudo-random. See quasi-random numbers.
Probability density [function]:
A function associated with any distribution which specifies mathematically the
way in which probability is distributed over different possible values for that
distribution. So, the density function of a normal distribution is the well-known
bell-shape which implies that there is a high probability of being close to the
mean and low probability of being a long way from it.
Quasi-random numbers:
These sequences, like pseudo-random numbers, may be used as the random
number generating process in Monte Carlo models. Successive elements of a
sequence of pseudo-random numbers are designed to be independent whereas
successive elements of a sequence of quasi-random numbers are not. In fact they
are designed to have a certain structure which allows them to cover the
probability space in as uniform a manner as possible and hence to improve the
convergence of the estimation. See Fauré sequence, Halton sequence, Sobol
sequence.
Rendleman and Bartter:
This is a single-factor model of the yield curve where the short rate is defined to
follow Geometric Brownian Motion with drift. It is not in common use since it
does not incorporate any of the mean reversion that is empirically observed.
Risk-neutral pricing principle:
Developed by John Cox and Stephen Ross, the theory that stock options may be
valued as if the underlying stock’s mean rate of growth is equal to the riskless
rate. In particular, the value of a European option is the discounted present value
of the payoff under the risk-adjusted probability distribution for the stock price at
expiry.
Roll-Geske-Whaley:
An extension to the Black-Scholes model incorporating the independent work of
Richard Roll (1977), Robert Geske (1979) and Robert Whaley (1981) and
providing a solution for the pricing of American-style call options on assets
paying dividends. Behind the model is the observation that an American-option
can be viewed as a portfolio of three options: a European option on the
underlying; a European option to exercise the first option which will not be
exercised until the instant before the ex-dividend date; and a compound option
written on the first option (to incorporate the cost incurred by exercising the first
two options of forfeiting the remaining life of the first option). The model can
also be used to value calls on stock indices and American puts on stocks that do
not pay dividends but cannot be used to value American puts on assets that pay
dividends.
Second order effect:
Error term which occurs due to non-linear effects in a model which has been
approximated by a simpler linear model. Option gamma is a second order effect.
See correlation risk.
Single-factor model:
An option pricing model that incorporates only one uncertain parameter, the
future price of the underlying. Such models make fixed assumptions about other
variables such as the term structure of interest rates and volatility. Multi-factor
models which can accept more than one parameter are better able to model
interest rates and volatility and are necessary to price options on a number of
underlying assets (such as spread assets) correctly. See multi-factor model.
Skew:
In statistics skew is the asymmetry of a distribution around its mean. Positive
skew is an asymmetric tail extending toward positive values (right-hand side).
Negative skew is an asymmetry toward negative values (left-hand side). In
options skew is commonly used to refer to the volatility skew. See volatility skew.
Sobol sequences:
A particular type of quasi-random number sequence.
Stochastic volatility model:
A multi-factor model where the volatility of the process followed by the
underlying is itself a stochastic, usually mean-reverting, process.
Tail:
The end (left or right hand section) of a probability distribution. Also used by
futures traders either for the change in the number of futures contracts needed to
hedge a position because of variation margin flows or for the number of excess
futures contracts in a basis trade. Also used in the bond or note markets of a
security with only a short time to maturity.
Tri-nominal tree:
An extension of the binomial method of option pricing in which the variable
being modelled (the price of the underlying) is allowed three possible outcomes
instead of just two: move up, move down or stay the same. This provides greater
flexibility and is useful in pricing more complex products.
Variance:
The statistical measure of how widely a variable is dispersed around the mean.
Standard deviation is the square root of the variance.
Volatility smile:
Refers to the influence of the out-of-moneyness of an option of a given maturity
on its quoted implied volatility. Generally the implied volatility of out-of-themoney options (that is options with low deltas) is greater than that of at-themoney options. If the implied volatilities are plotted (Y-axis) versus the strike (Xaxis), a curved line resembling a smile is obtained. This is due to option sellers
needing a premium for selling low delta (disaster insurance) options.
This phenomenon is not consistent with the basic Black-Scholes model which
implies that asset volatility is constant. If true, the implied volatility from
European options of all strikes and maturities would be identical. In fact, implied
Black-Scholes’ volatilities depend on the maturity and strike of the European
option in question. That is, the market may believe that extreme upward and
downward movements are more likely than allowed by the Black-Scholes model.
In this case it is said that the implied market distribution is more leptokurtotic
than that implied by Black-Scholes. This can be seen when the implied volatility
smile is convex in the strike price. See kurtosis.
Volatility trading:
Taking options positions that will profit not from moves in the price of the
underlying but from changes in implied volatility. Traders can take views on
absolute levels of implied volatility by buying and selling combinations of options
– classically delta-hedged straddles and strangles. They can also trade future
actual or realized volatility versus present implied volatility, profiting if future
actual volatility is more or less than the implied volatility of the position when the
trade is put on. So if they believe that the volatility implied by an option is too
low, then the option is cheap and they will buy it. Buyers {sellers} of realized
volatility against implied volatility profit when the underlying is more {less}
volatile than the implied volatility predicted. These trades are non-directional, that
is they are hedged against absolute price moves in the underlying. Traders can
also trade between the different implied volatilities of options at different
maturities or with different strikes (smile or skew trading).
Whole-term structure pricing model:
An interest rate option pricing model that takes into account the relationships
between spot rates at different points in the curve. By using the information
contained in the current term structure of interest rates and also the volatilities of
each of the spot rates as inputs into binomial, trinomial or multinomial trees
which value the underlying debt instrument at each node, the models provide the
basis for a valuation of an option on that instrument. The Ho-Lee and HeathJarrow-Morton models are of this type. Such models are designed to enable the
exposure on all interest-rate derivative products to be aggregated. For example,
the volatility exposure created by a long position in swaptions should be able to
be offset by a short position in caps so that only the net volatility is hedged.
Wiener process:
The description of movements in a variable when the change in its value in a short
period of time is normally distributed and the changes in two non-overlapping
periods of time are uncorrelated. Also known as arithmetic Brownian motion.
Adjustable strike options
Adjustable strike option
Key concepts
Adjustable strike options are options whose strike is reset either automatically or by the holder, depending
on the path/level of the underlying. Depending on the terms of the reset mechanism they are also known as
moving/floating strike options, indexed-strike options, periodic (reset) options, ratchet/ladder options and
step-up/step-down options.
They are often combinations of vanilla and digital or barrier options and in the two chapters covering those
instruments a number of products are explained which, because they actually consist of a number of options
packaged together, appear to have similar resettable strikes. In this section we also include a small number
of options whose unusual exercise conditions make them similar to adjustable strike options, namely fixedstrike lookbacks, lookforwards and shout options. All these options share one common characteristic: they
enable the holder to create strike price conditions that more exactly suit their views especially their views
on the dynamic path of spot not just its final resting place than conventional options.
Definitions
Deferred strike price option
Also known as a forward start option, this is an option that allows the holder to set the strike price
at a predetermined time or during a predetermined period after its trade date. The strike price is
usually expressed as a fixed ratio to or percentage of spot. The option’s premium is usually set on
the trade date. These options allow the holder to lock in current levels of volatility in the
expectation that volatilities will rise or fall without setting the delta of the option until the strike is
set. These are more commonly embedded in structured assets than used as naked options.
Example
An investor might want a three-year bond whose annual coupon captured the appreciation of a
currency, say sterling against the dollar, in each year. This would be constructed from a strip of
two-one year forward start USD put/GBP call options plus a one year vanilla option. The first
could be struck at-the money spot with the two forward starting options setting at 100% of spot on
the first business day of the year and an expiry on the last business day of the year.
Hi-lo option
An option which pays out the difference between the high and the low price or rate reached by the
underlying over the term of the option. Constructed from a combination of a lookback call and
lookback put, the buyer is taking a view that the volatility of the underlying will be greater than
the implied volatility of the component options. Because the expected payout is high, the premium
is high, and the option buyer is taking a large, long position in gamma and vega.
Indexed-strike option
Also known as a periodic reset option, this is an option whose initial strike price moves up or
down according to a preset schedule or depending on the path of a reference asset or index. The
size, timing and direction of the reset mechanism can depend on almost any contingency required
by the buyer/seller. It may rely simply on pre-set trigger points being hit by the underlying at any
time during the life of the option; it may require that the underlying move a certain amount
relative to the last fix within a given sub-period of the option’s life (sometimes called momentum
or gap options); or it can be automatic with the option’s strike price resetting at a pre-agreed
spread above or below the reference index or at a series of pre-agreed absolute levels for each
successive period without the underlying having had to hit any predefined level (sometimes
known as a moving strike option).
Many of these products are combinations of vanilla and exotic options. The holder of a
momentum cap is long a conventional cap and short a series of digital caps. The benefits to him
versus a vanilla cap will depend on the value of the sold options. In a positive yield curve
environment, the steeper the curve the higher the chance that the trigger Libor rise will be
breached and the higher the probability that the strike will be raised and the higher their value.
Also known as step-up/-down options.
Example
Momentum options and gap options illustrate the subtleties available with resettable strike options.
They enable the holder to hedge against or benefit from dramatic movements in the price of the
underlying. An option on Libor struck at 7.50% would pay out if Libor rises by more than, say, 75
bp in the next three months. It therefore has two triggers, the gap trigger (75bp) and the speed
trigger (one month). Regardless of whether Libor did rise by 75bp in the first three-month period,
the strike would then be reset to current Libor at the beginning of the next payment period. This
structure is usually altered so that the strike price ratchets up by a predetermined amount. So a
borrower with a three-year US dollar loan based on three-month Libor could buy a three-year
momentum cap with a 7.50% strike and a 75bp trigger amount. If In any three-month period threemonth Libor rises by more than 75bp, then the cap strike is reset 25bp higher with a maximum cap
rate of 8.50%.
Ladder option
An option whose strike resets automatically when the underlying hits predetermined levels
(‘rungs’). When the strike is reset the intrinsic value of the option is automatically locked in
regardless of whether the underlying subsequently moves disadvantageously. Ladder options are
strips of capped/exploding options with the cap level of one option set equal to the strike level of
the next and each cap level a rung in the ladder. Every time a rung/cap level is reached that option
is exercised for its intrinsic value locking in that gain and a new option is triggered with a strike
equal to the previous cap level and a new cap level higher (call)/lower (put) than the previous
one.) As these options are sometimes known as cliquet options (because cliqueter is French for to
knock and the automatic exercise became known as the cliquet clause) cliquet option can also be
used as a name for ladder option. These options are more expensive than vanilla options,
particularly if the put asset has a lower interest rate than the call asset.
Example
A ladder call on the EUR/USD rate with a strike of 1.0500 might have a rung every 1 cent up to a
maximum of 1.0800 and have a payout of the greater of (i) the closing spot less the original strike
and (ii) the highest rung reached less the strike. The more frequent the rungs, the more expensive
it is. Other ladder options have only a minimum settlement level. Once the underlying has risen
by, say, 10%, that gain is locked in regardless of the future path of the underlying price. If it
subsequently rises above 10%, the investor still participates, but he also has a floor at 110. In
exchange for this downside protection the maximum return is generally capped.
Lookback option
An option that allows the buyer, at maturity, to choose the most advantageous exercise conditions
that have occurred over the life of the option, or in the case of a partial lookback option, during a
pre-set sub-period (usually between one and three months) of the life of the option after which it
becomes a standard European- or American-style option. A lookback period limited to the first
part of the option’s life will help improve the timing of any market entry; one limited to the last
part of the option’s life will help with market exit timing.
Lookbacks come in two varieties. The lookback strike option/floating-strike lookback, instead of
having a specified strike price, allows the buyer at expiration to look back over the life of the
option and set as the strike the most favourable price that has occurred during that time. A
lookback call {put} allows the buyer to choose the lowest {highest} price that has occurred over
the life of the option. These strikes are then compared with the spot price at expiration to
determine the option’s payoff. The lookback spot option/fixed-strike lookback has a strike set at
the outset but then at maturity allows the buyer to look back over the life of the option and choose
the most favourable exercise point to maximize profit between strike and exercise. Lookbacks,
like conventional options, are most profitable to the buyer (net of premium) if the realized
volatility of the underlying price is higher than the implied volatility. If a buyer believes that there
will be a sharp move in price but is not sure when and for how long the price will move, lookbacks
are attractive. Because they allow the buyer to choose the best exercise conditions with perfect
hindsight, lookbacks command much higher premiums than conventional options. Also known as
hindsight options and lookforward options.
A fixed-strike lookback struck at-the-money spot is sometimes called a lookforward option. This
gives the buyer the difference between the asset price at the beginning of the period covered by the
option and its high (call) or low (put) over that period.
Ratchet option
A type of indexed strike option whose strike price resets favourably if the underlying moves outof-the-money relative to the initial strike and hits certain trigger or ratchet levels but which does
not reset in the other direction if the underlying subsequently moves in the other direction. A
ratchet call option is a call struck at the ratchet option’s strike price, plus a series of bought knockin put options each struck at a ratchet level and a series of sold knock-out puts with strikes
staggered one rung behind the purchased options. Confusingly, like ladder options, ratchet options
are also sometimes known as cliquet options because vilbréquin à cliquet is French for ratchet
brace.
Roll-up option
An option whose strike price is favourably reset at the same time as the option itself is converted
into knock-out option if the price of the underlying asset trades through a predetermined trigger
point, usually struck at a point where the underlying has moved significantly against the original
option. So, a roll-up put with an original at-the-money strike of 80 might be converted into an outof-the-money knock-out (up-and-out) put with a strike of 100 and a knock-out level of 110 if the
underlying traded to 100. The holder has a new, more favourable put strike, but if the underlying
continues to rise (i.e. in his favour as long as his put is hedging an existing position) then the put is
knocked-out (at a point where he does not need protection).
The roll-up put outperforms the standard put if the roll-up trigger is reached but the trigger is not.
If the roll up trigger is not reached, then the roll-up put and vanilla put behave the same. Only if
the roll-up trigger and the trigger are reached does the roll-up underperform the vanilla instrument.
The trigger price for the up-and-out put is set in advance and is above the roll-up strike.
Shout option
Confusingly there are two completely different types of option that are called shout options. (i) A
path dependent option that combines the features of lookback, ratchet and ladder options. A shout
option allows the purchaser to lock in a minimum payout (the intrinsic value of the option at the
time of the ‘shout’) while retaining the right to benefit from further upside. So-called because
when the option holder thinks the market has reached a high (call) or low (put), he ‘shouts’ and
locks in that level as the minimum and, with a one-shout option, still holds a European option with
the original strike price for the remaining life of the option. If the market finishes higher (call)
lower (put) than the shout level, the holder benefits further i.e. the payout of a shout option is the
greater of the intrinsic value locked in by the shout and the intrinsic value at maturity.
This type of shout option is similar to a ladder option in which profits are locked in when the
underlying rises/falls sufficiently to hit a pre-determined ‘rung’ level, but in the shout option the
rungs are not set in advance. This makes the shout option more expensive than the ladder option,
the more so when multiple shouts are allowed (multiple shout options are very expensive). As
with a ladder option, the more shouts that are allowed, the more like a lookback the shout option
becomes. The ability to lock in gains before expiry makes the shout more expensive than a
standard European option, and the fact that even after a shout, the option holder effectively has
another option struck at the shout level, makes it more expensive than an American-style option.
Example
A corporate treasurer might be bullish on EUR/USD rates but also expects the cross to be very
volatile. He is worried that using a vanilla option will mean that he misses out on temporary highs.
A shout call solves the problem. If the EUR/USD rate rises above the strike price, but ends up
below the shout level (wherever the treasurer eventually chooses that to be), the treasurer receives
a profit of the shout level less the strike level. If the exchange rate closes above the shout level, the
investor will receive that additional profit as well. The payout is therefore the maximum of (shout
strike) and (close strike).
(ii) The second type of shout option is a call or put option that gives the buyer or seller the right
once and only once during a pre-specified period to ‘shout’ the option and reset the strike to the
then prevailing spot rate (or some percentage thereof)). These shout options are therefore, in a
sense, halfway between a vanilla and a lookback option. They are more expensive to buy and
generate less premium when sold.
Surge options
An option whose strike price is reset on a daily basis to a fixed spread above or below a moving
average. This hedges against the risk of rapid price changes rather than absolute price trends over
longer periods. Commonest in the commodity markets, a put surge option on the price of crude oil
could work like this whenever the spot oil price falls below the 45-day moving average less two
cents, the option is in-the-money. The settlement amount is determined by the difference between
the spot price and the strike price multiplied by the number of barrels to be priced each day. A call
would move into the money if the spot price moved above the moving average plus a fixed spread.
Asian options
Key concepts
Asian or average rate options are options whose payout or strike price is based on an average of the price of
the underlying over the life of the option.
The averaging process can begin at any point during the option period (for example a one-year option
whose payout depends on the average underlying price in the final month). The sampling process frequency
and interval of underlying price observations can also be tailored. The number and timing of price (strike)
observations is determined in advance and may start at the beginning or near the end of the life of the
option. Observations may also be weighted in favour of prices (strikes) observed on specific dates.
Unlike a straight American- or European-style option, an average option can be settled more than once over
its life. So for example, the holder of a one-year average option can choose to settle the option monthly
versus the average price of the underlying the previous month. Average options are nearly always cheaper
than conventional options because the averaging process smoothes out the underlying price movements
thereby reducing volatility and hence the premium of the option.
Definitions
average price/rate option (ARO)
Unlike a conventional option, which is settled by comparing the strike with the spot rate at
expiration, an average rate option’s payout is the difference, if positive, between the
predetermined, fixed strike price and the average of spot rates observed over the option’s life. This
hedges against the average prevailing spot over the life of the trade. It also removes the reliance of
the option’s expiration value to the underlying cash price on a particular day. Typically the
volatility of an average rate option is about 58% of the volatility of a conventional option and so is
cheaper.
AROs are cash-settled, not deliverable, so when hedging an underlying exposure, cash flows need
to be converted in the underlying market on the relevant fixing dates. This ensures that the hedge
instrument effectively offsets the aggregate FX rate of the cashflow conversions. There are three
main varieties:
i.
ii.
iii.
Arithmetic Asian options are the most common. The arithmetic average is the sum of the
price observations divided by the number of observations. These options cannot be priced
using a closed-form model because the sum of lognormal components has no explicit
representation the arithmetic average is not lognormally distributed even if its underlying
is.
Geometric Asian options’ payout is based on the geometric average price of a series of
observed underlying spot rates. The geometric average is the nth root of the product of n
quantities. These options can be priced using a closed-form option-pricing model because
the product of lognormal prices is itself lognormal. They are rarer than arithmetic Asian
options.
Weighted Asian options are also available in which the weighting of each periodic price
or rate used in the averaging process varies according to a predetermined schedule. These
options are useful if the timing and magnitude of cash flows is known but the price or
rate is unknown. A simple weighting scheme is normally used in which the weights add
up to one.
Example
A hedger short EUR/long USD wishes to hedge on average at 1.0500 buys an ARO EUR
call/USD put struck at 1.5000 and a fixing frequency of weekly every Friday for six
months. With the forward at 1.0445 and 10% volatility the premium would be 1.12%
EUR versus the 2.52% EUR of a vanilla European-style option. If the average were
above 1.0500 on expiry, the underlying would be hedged at an effective rate of 1.0616
(strike + premium). If the average were below 1.0500, then the underlying benefits below
an average rate of 1.0385 (strike - premium). If spot trades above the strike early on in
the life of the option and then trades back down, the payoff from this ARO will exceed a
vanilla option. However, if the spot is greater at expiry than its average until expiry, then
the payoff of the ARO will be less than a vanilla. In general, the expected payoff of the
ARO is lower, and this results in the lower premium.
Average strike option (ASO)
A moving/floating strike option whose payoff is determined by comparing the underlying price at
expiration with a strike computed as the average of spot rates over the option’s life. The option is
exercised against the spot rate prevailing at expiry and can be cash or physically settled. An ASO
limits exposure and benefits to large movements of spot. It is equivalent to a strip of daily options
struck at the average spot during each day and where the maximum loss on all these transactions is
equal to the premium of the ASO. So an average strike call has a payoff equal to the difference
between the asset price at expiry and its average over the option’s life if this difference is positive
or zero otherwise.
Barrier or trigger options
Key concepts
Barrier or trigger options are conventional options except that they are cancelled or activated or, more
generally, changed in a pre-determined way when the underlying trades at predetermined barrier/trigger
levels. So a knock-in option pays nothing at expiry unless at some point in its life the underlying reaches a
pre-set barrier and brings the option to life as a standard call or put. A knock-out option is a conventional
option until the price of the underlying reaches a pre-set barrier price, in which case it is extinguished and
ceases to exist. Barrier options have a strike price and a barrier price and the barrier can be above or below
the strike price. In all variations the barrier can be made to be active for either part or all of the option’s
life.
Because of the importance of the barrier event in determining the value of the option, users must ascertain
at the outset of the transaction the definition of a barrier event. For example, is it to be based on quoted
rates or transactions; how is the issue of crosses to be dealt with in illiquid currency pairings; when can
barrier events occur (outside normal trading, only on hourly fixes, at the end of day fix, only on certain
dates and so on. Where barrier events can only occur at certain times (and under certain circumstances)
there is said to be barrier discontinuity. This makes the options more difficult to price and value).
The concept can be applied to every type of option and some option combinations- caps, floors, collars,
digitals, swaptions and in any asset class. The two basic classes of barrier options are the standard (or outof-the-money) barrier options and the reverse (or in-the-money) barrier options.
These options should not be confused with capped {floored} calls {puts} which are also sometimes known
as trigger options. They are described in the chapter on ‘Vanilla Options.
Definitions
Balloon option
An option whose notional principal increases if a preset trigger level is breached. For example, an
equity investor might believe that the FTSE-100 will rise from 4900 to 5000 and then, if it
breaches this resistance level, rise strongly again. He could buy a 4900 call with a trigger of 5000
and a multiple of two, meaning that if the index stays below 5000 the option behaves like a vanilla
call but if it rises above 5000 then the option’s notional principal doubles. The balloon option’s
premium is more expensive on the original notional principal than a vanilla option because it is a
combination of two options a vanilla call struck at 4900 and a knock-in call struck at 4900 with a
knock in at 5000. However, if the trigger is reached, the premium on the ballooned premium is
cheaper. The greater the ballooning the higher the premium; the further the trigger level is relative
to the strike, the cheaper the premium.
Double barrier option
A general term for any barrier option incorporating two knock-out or knock-in levels, one either
side of spot. These are commonest in the FX markets where users may have strong views on both
a support and a resistance level. The illustrated of a knock-out rebate option in this chapter is an
example of a double barrier option.
in-the-money/Reverse barrier option
A barrier option whose barrier is in-the-money relative to the strike. So, the barrier level would be
above the strike for a call (up-and-in/out calls) and below it for a put (down-and-in/out puts).
These are priced and behave very differently from standard barrier options since they have
intrinsic value when they are knocked-in or out, making knock-ins relatively more expensive and
knock-outs relatively cheaper for a given proximity to the strike. So, unlike standard options,
which become more valuable as volatility increases, in-the-money knock-outs become cheaper.
That is, they have negative vega: the probability of knock-out increases with increasing volatility,
reducing the chance that the option will pay out and making them cheaper. In-the-money barrier
options can be used both to hedge an underlying position (as in example 1 below) and to take
outright speculative positions (as in example 2).
Example 1
In-the-money barrier option could be used by a USD-based exporter wishing to reduce the cost of
protection against Euro weakness. The company could purchase a EUR put/USD call with an inthe-money knock-out struck at 1.0200 with the knock-out set at 1.0000. This option could be up to
half as cheap again as the out-of-the-money knock-out, but carries the risk that if EUR/USD does
trade down to the knock-out level, then the corporation has not only lost its hedge, but also has to
re-hedge at much worse levels than with the standard knock-out. The hedger has to have a stronger
view on exchange rate movements than with either vanilla options or standard knock-outs.
Example 2
An investor believes that over the next six months the dollar will strengthen against the yen by
around 10% and by at the very most 12%. If dollar yen is trading at 125 he could buy a reverse
knock-out USD call/JPY put struck at the money forward at, say, 121.50, with a trigger at 140.50,
0.5 yen above his predicted high of 140 and another knock-out at 119. Then if the dollar does
strengthen, but trades either below 119 or above 140.5 over the life of the option, the call will
disappear. If the dollar strengthens, but never reaches 140.50 or 119 over the life of the option, the
call will behave like an ordinary call and the investor will exercise the call and make the same
profit as the ordinary call. If the dollar does not close above the call strike, the option will expire
worthless like an ordinary option.
Knock-in cylinder/collar
Barrier options are often used to modify simpler option spreads. The knock-in collar is an
alternative to selling an out-of-the-money call {put} to finance the purchase of an out-of-themoney put/(call) to create a standard collar. Instead the holder sells a knock-in call {put} instead
of a vanilla option. This allows the holder of the position to participate fully in the upside
{downside} of any moves in the underlying, until the trigger level is breached.
Example
A USD-based exporter has EUR receivables due in six months. The current spot EUR/USD rate is
1.0310 and the six-month forward points are 130 so the six-month forward outright is 1.0440. The
company wants protection against Euro weakening and favours options over selling Euros forward
because of the high potential opportunity cost of locking in an outright forward rate. However it
does not wish to incur any upfront premium at all or does not wish to be long volatility. In this
case they could execute a zero premium standard risk-reversal/collar, selling a call at 1.0550 to
finance the purchase of put protection at 1.0340. This risk-reversal can be as narrow or as wide as
the corporation wishes. Alternatively, instead of selling a vanilla EUR call the hedger could sell a
knock-in call struck at 1.0500 with a 1.0900 trigger level. Although the strikes of the call and put
would be less advantageous than a regular collar, the corporation would have full protection and
only give up its upside if the knock-in were triggered.
Knock-out cylinder/collar
Similar in concept to the knock-in collar, this is the substitution of the short call of a simple zero
cost collar with a knock-out call. In the example above, the corporation would buy a 1.0240 EUR
put and sell a 1.0500 EUR call with a knock-out trigger at 1.0000. If the knock-out were triggered,
then upside is no longer limited.
Knock-out trigger option
A capped call {floored put} (see chapter 7) which incorporate an out-of-the-money knock-out
level. If the underlying trades through this, the whole option is cancelled.
One factor barrier option
A barrier option whose barrier event and payout are based on the same underlying. Also known as
inside barrier options. In a two-factor/outside barrier option the barrier event and the payout are
based on two different underlying assets. So, the payout might be a function of a foreign exchange
rate but the barrier event may be the breach of a level in the price of a commodity. See chapter 15.
Out-of-the-money/Standard barrier option
A barrier option whose barrier is set out-of-the-money relative to the strike. So the barrier level
would be below the strike at the start of the option contract for a call (down-and-in/out calls) and
above it for a put (up-and-in/out puts). These options cost less than standard options because the
price of a vanilla option takes the entire probability distribution of possible prices for the
underlying into account, while the knock-out feature removes many of those possible values. The
exact premium reduction is determined by how likely the barrier event is to occur. The more likely
the option is to be knocked out or the less likely it is to be knocked-in, the greater the premium
reduction, and vice versa. The likelihood of the barrier event depends on how near the
spot/forward level the extinguishing trigger level is, on the maturity of the option and on volatility.
Example
A USD-based exporter finds a vanilla EUR put/USD call too expensive. A knock-out EUR
put/USD call struck at 1.0340 with a barrier level at 1.0600 costs less (and so has a lower breakeven), gives the a guaranteed minimum of 1.0340 USD for every EUR of receivables and only
disappears when the underlying is moving in the hedger’s favour, giving it the flexibility to
examine other hedging strategies. The main risk is that if the option is knocked-out, the
corporation is exposed to any subsequent dollar weakness and so may have to put on a second
forward or option-based hedge that may cost more than the original vanilla put option.
Rebate option
An option that pays a fixed amount if it would otherwise have expired worthless due to some
barrier event. So a knock-in rebate option pays a pre-set fixed amount if it has never been
knocked-in (even if the option then expires out-of-the-money) and a knock-out rebate option pays
the option holder a pre-set fixed amount if it is knocked out. The commonest structure is the
knock-out rebate option a call or put with an in-the-money knock-out level and rebate almost
always set equal to the initial premium paid. This is usually just called a rebate option. A second,
out-of-the-money knock-out level can be incorporated which, if traded, cancels the entire option
(see second diagram). This structure is sometimes, confusingly, called a knock-out rebate option.
Sloping/moving/jumping barrier option
A barrier option whose knock-in/knock-out level changes during the life of the option, for
example to match moving technical levels in the underlying. This change may be either linear (i.e.
sloping) or move in discrete steps (i.e. jumps). For example, our USD-based exporter could set the
knock-out level for the first three months at 1.0500 and then 1.0600 for the last three months.
Step-up {down} barrier option
A barrier option whose barrier increases {decreases} over time.
Switchback option The simultaneous purchase of both a capped call {floored put} and a knock-in put {or
call}. The trigger levels of the knock-in barrier options typically equal the cap/floor strike prices. If the
underlying hits the trigger levels, the capped option is automatically locked-in and the knock-ins activated.
The holder would typically set the strikes at a point he believed to be around a peak {trough} in the
underlying. The position benefits from that level being reached and then switches back from call to put (or
vice versa) as the underlying itself switches back, retreating {rising} from its peak {trough}.
Complex swaps
Key concepts
Although the standard swap structures meet the needs of many hedgers and investors, there are occasions
on which the circumstances of the parties require more highly-structured swap products. Many of these
swaps take the standard interest rate swap structure and, by incorporating foreign exchange rate linkages,
complex indexing of the payment streams and notional principal amounts, options and other contingencies,
allow swap users to define their views on particular markets or the relationship between markets more
precisely than with vanilla swaps and more economically than using a basket of standard derivatives.
Indeed in some cases, the payoffs cannot be constructed with standard derivatives. In the same way that
structured assets are created by combining vanilla fixed-income instruments with derivatives, so many of
these swaps are constructed by taking standard swap structures and combining them with the vanilla and
exotic options defined elsewhere in this book.
Definitions
accrual swap
An interest rate swap under which a counterparty pays a vanilla floating reference rate, usually
three- or six-month Libor, and receives Libor plus a significant spread. Interest payments to this
counterparty will only accrue on days (or another pre-set period) when rates stay within a certain
range dictated by pre-set upper and lower boundaries. A more aggressive variant, the binary
[coupon] accrual swap (also known as a one-touch swap), is also available in which any breach of
the range boundaries cancels all further potential for accrual. The simple accrual swap is
constructed from a conventional interest rate swap plus a short digital strangle (short a digital put
at the lower boundary of the range, short a call at the upper limit) with a maturity equal to the
Libor fixings (usually daily, weekly or monthly). Versions are available where the upper and
lower ranges step-up or down over the life of the swap and where the holder can reset or cancel
the range at specified times.
Callable swap
An interest rate swap in which either the fixed-rate payer or the fixed-rate receiver has the right to
terminate the swap at one or more predetermined points during its life. These points are either
defined in terms of time or in terms of points on the swap curve. Most usually, a callable swap is
one in which the fixed-rate payer has the right to terminate the swap, that is has bought the call. A
callable swap is the combination of an interest rate swap and a receiver swaption. Also known as a
cancellable swap, collapsible swap, retractable swap. A swap in which the fixed-rate receiver has
the right to terminate, that is has bought a put, is known as a puttable swap. It is the combination
of a standard swap and a payer swaption and allows the holder usually an investor to benefit from
a rise in interest rates.
Example
A treasurer paying 10% fixed and receiving Libor flat under a five-year swap might like to cancel
the swap if rates decline. A cancellable swap gives him to option to stop paying fixed (and so
effectively to start paying floating) and he pays for this option by paying a fixed rate on the
cancellable swap that is higher than prevailing vanilla swap rates. The counterparty with the right
to terminate has effectively bought a swaption from the other counterparty which protects them
against adverse moves in interest rates. In this case the treasurer has bought a receiver swaption
(say two year) that gives him the right to receive a fixed rate of 10% against paying Libor flat for a
period of three years. In two year’s time if the prevailing three-year swap rate is below 10%, the
treasurer can exercise the option to enter into a three-year swap. This new swap effectively cancels
the existing swap since the cash flows of one are offset by the other. This allows the treasurer to
lock in a lower swap rate at a future point in time.
The structure can be used by investors too. An investor is paying Libor and receiving 7% fixed in
a five-year swap. He purchases a payer swaption giving him the right to enter a three-year swap in
which he pays 7% fixed against receiving Libor at the end of two years. If in two years’ time
three-year swap rates rise above 7% he can exercise the swaption and enter into a three-year swap
paying 7% fixed and receiving Libor. This swap offsets the existing swap that has three years to
run and the investor can now enter into a new three-year swap under which he receives the
prevailing three-year swap rate against paying Libor.
Clean index principal swap
A path-dependent version of the index principal swap. In the standard IPS the notional principal
can accrete or amortize, and once the process of accretion or amortization has started it either
continues at the level set by the initial barrier or is accelerated as rates move to the next barrier. In
the clean IPS the notional principal is reset according to the Libor rate prevailing at the beginning
of each calculation period. It is clean in the sense that for each calculation period the swap
notional is totally independent of previous settings. This means that the swap’s notional amount is
far more directly linked to the direction of Libor than is the case for a generic index principal
swap.
Example Clean index principal swaps can be used by hedgers thus: A corporate decides to pay
fixed and receive six-month Libor and the amortization factors are set such that, if Libor is below
5.0%, the notional principal on the swap is zero. This means that if Libor is below 5.0% at the
beginning of a calculation period, then for that period the hedger simply pays Libor the swap is
deactivated. The higher Libor rises, the more of the hedger’s outstanding liability is swapped into
fixed until, at a predetermined point, the full liability is capped at the fixed rate payable on the
swap. The product allows clients to fix without being affected by the cost of carry associated with
a steep yield curve. In exchange for this, before the swap is fully activated the corporate pays a
blended rate made up of Libor on the unswapped portion of the liability plus an above-market
fixed rate on the remainder.
Collared swap
An interest rate swap combined with an interest rate collar on the floating leg. Also known as a
floor/ceiling swap.
Commodity-linked interest rate swap
A hybrid swap in which an interest rate index such as Libor is exchanged for a commodity-price
linked fixed rate. A user of aluminium might wish to link the price of his major cost, aluminium,
to the price of his debt. He could elect to receive Libor and pay an aluminium-linked rate such that
as the price of aluminium rises, the fixed rate he pays declines. It is also possible to swap a
commodity price itself for Libor see commodity swap.
Contingent swap
A swap activated by a specified event and usually paid for with a premium. Swaptions can be
viewed as contingent swaps.
Currency protected swap
A quantized interest swap. That is, an interest rate basis swap in which the buyer pays an interest
rate in one currency, usually his domestic Libor, and receives a second currency’s Libor plus or
minus a spread with all payment streams denominated in the same usually the buyer’s domestic
currency. They can be used to reduce funding costs: floating-rate liabilities in high interest-rate
currencies with inverted yield curves can be swapped into low current reference Libors plus a
spread where the yield curve is steep without the risk of currency exchange on coupons or
principal amounts.
The benefit arises from the implied forward curve in each currency. High forward Libors implied
in the low interest reference yield curve combined with lower forward Libors implicit in the high
interest domestic curve result in a current interest saving. So the longer the maturity of the quanto
swap the more attractive the upfront benefit in interest savings. This should be balanced against
the increased uncertainty of actual future Libor settings. Liability hedgers who use the structure
take the view that the yield curves overstate the future path of one or both of the Libors over the
life of the swap.
Example
A borrower of dollars wants to benefit from the fact that floating money market rates in yen are
currently well below equivalent floating money market rates in dollars. He enters into a currency
protected swap under which he pays yen Libor denominated in dollars plus a margin and receives
dollar Libor. As long as the interest rate differential between yen and dollars exceeds the margin
he is paying, then he is benefiting from low yen rates.
It was the boom in this product that spurred the search for correlation risk because of its
importance in pricing these swaps. A swap writer paying yen Libor in dollars and receiving dollar
Libor in dollars funds the yen Libor payout through the swap market. He is therefore hedging US$
denominated yen interest rate risk using yen denominated instruments. So, even if interest rates
remain the same, he is exposed to the risk that the dollar will strengthen, leaving him too few yen
to pay his dollar liability. Although the prevailing exchange rate will determine the initial size
(‘quantum’) of the hedge, ongoing changes in exchange rates will vary the size of the hedge
required. Hedging this risk means taking a view on the correlation covariance between interest
rates (yen Libor) and yen/US$ exchange rates. That is, to what extent will a rise in yen interest
rates, and so the amount of money the swap writer must pay out to the buyer, be offset by a
strengthening of the yen against the dollar?
Because the view taken is that the yield curves overstate the future path of one or both Libors they
can also be used by investors to take yield curve views when they believe that the convergence or
divergence of FRA curves is too acute without taking FX risk. So:
Example
An investor wants to implement his views on the spreads between two markets without taking
direct foreign exchange exposure. He enters a quanto CMS swap receiving US dollar 10-year
CMS rates minus a spread and pays yen 10-year CMS in return, both payment streams
denominated in dollars. He can take a view on more than one spread. For example, a quanto swap
could be purchased that paid two-year US dollar CMS rate in exchange for 50% of the two-year
sterling CMS and 50% of the Deutschmark, paid in dollars.
One variant is the limited risk differential swap. This is a differential swap combined with a crosscurrency cap {floor}. The combination allows the user to benefit from the interest rate differentials
between two currencies while capping {flooring} the maximum loss incurred if the differentials
move adversely. The basic instrument it is known by a large number of names including
differential swap, guaranteed exchange rate swap, cross-indexed basis (CRIB) swap, cross-rate
swap, index differential swap, interest rate index swap, Libor rate differential swap, quanto swap.
Crack spread swap
A commodity swap that enables refiners to lock in a margin by paying the floating price of the
refined product or products, calculated as an average over a pre-set period, and receiving the
floating price of its chosen crude oil feedstock plus a fixed margin the crack spread. By locking in
this margin, refiners can hedge against a narrowing in the differential between crude oil prices and
the prices of the refined products it produces. However, in so doing they give up the right to profit
from any widening of the spread. Also known as the refinery margin swap.
Curve/roll lock swap
A curve lock is any instrument which locks in the spread between two different points on a yield
or price curve. A curve lock swap is a swap that locks in this spread for one of the swap
counterparties. They are used either as outright speculation on future curve movements or to
benefit from a favourable curve shape when the absolute level of the underlying market makes
entering into a swap outright unappealing. Also known as a trigger swap because the counterparty
wishing to lock in the spread can trigger it at any time during a predetermined trigger or lock-in
period.
Example
Instead of entering a contango swap, an oil producer unwilling to fix the price of his production at
current low swap rates can enter a curve lock swap. The swap rate is set at a differential to a
nearby futures contract before the expiry of that futures contract. If his belief that the contango
will diminish proves correct and spot prices rise, the futures price will rise and he will be able to
trigger the swap at a significantly higher level than was available in the swap market originally.
The differential provides a cushion if spot prices fall.
For example, say the price of the December 1997 future is US$14.90 and the price for a calendar
1998 Brent crude swap is US$15.40. An oil producer wants to hedge some of its 1998 production
by receiving fixed in an oil swap but believes the price is too low. However, it likes the level of
contango and thinks it will diminish. A trigger swap would lock in the differential (in this case
US$0.50) between the futures price and the swap price while delaying the moment at which the
swap rate is finally fixed. The producer therefore enters into a calendar 1998 trigger swap the price
of which is set at a differential of US$0.50/bbl to the price of the December 1997 IPE Brent future.
At some time between the deal date and the expiry of the futures contract the producer must lock
in the absolute level of the futures contract resulting in an all-in swap price of US$0.50 more than
this chosen level. If the price of the December 1997 future on October 12 is US$15.80 and the
producer pulls the trigger, the resulting swap price will be US$16.30. If the trigger is not pulled
then the price immediately before expiry of the futures contract is used as the basis for the swap
price. Alternately the basis is rolled forward to a subsequent futures contract with the differential
adjusted accordingly. This structure gives the producer an opportunity to lock in the shape of the
curve when it is favourable and fix the price of the swap when the market is at a more satisfactory
level.
Double-up swap
A fixed-for-floating (usually commodity) swap in which the fixed-rate payments are set lower
than the market rate. In exchange, the fixed-rate payer grants the floating-rate payer a put option to
double the notional amount of the swap if the spot price of the underlying falls below a pre-set
strike price, usually the same as the discounted swap rate. The difference between the off-market
and market rates represents the premium for this embedded option. If the strike is hit, then not
only is the fixed-rate payer paying a higher price for the underlying than the current spot rate, he is
paying it on twice as much as the original notional principal of the swap. If a commodity
user/producer uses double-up swaps to hedge more than half their real requirements/production
and the option is exercised, he ends up over-hedged. That is effectively speculating, since he has
fixed prices on more of the commodity.
Down-and-out floored swap
The combination of receiving floating under an interest rate swap and the sale of a down-and-in
knock-in floor with the trigger set well below the fixed rate on the swap and with a strike at the
swap rate.
Example
A down-and-out floored swap might fix a floating-rate borrower’s cost of funds at 5.90% if rates
rise above 5.90% while allowing him to benefit from rate falls down to 4%. If rates do hit 4%
though, the down-and-in floor is exercised against him at 5.90%. So if Libor is above 5.90% or
below 4% the borrower is fixed at 5.90%.
Drop-lock swap
A deferred-start interest rate swap in which the fixed-rate payment is reset to a lower {higher} preagreed level if, between the time of the agreement and the commencement of the swap, the
floating reference rate drops below {rises} above a predetermined level.
Dual coupon swap
A fixed-for-floating interest rate swap in which one counterparty has the right to alter the currency
in which payments are made contingent upon a predetermined move in exchange rates usually if
rates move against the swap’s base currency. The structure combines an interest rate swap
subsidized by the sale of a strip of currency options whose holders can choose the currency of the
coupon payments. Also known as a currency indexed/linked swap.
Example A borrower wishes to swap a floating rate borrowing into fixed. To reduce the fixed rate
swap payments he agrees to pay coupons in either his base currency or another at the option of the
swap counterparty depending on whether the currencies’ exchange rate is above or below a pre-set
index level (the strike of the options). If the options are in-the-money, the holder will exercise
them against the borrower forcing him to deliver the coupons at an off-market (expensive)
exchange rate. If they are out-of-the-money then the coupon is delivered in the borrower’s base
currency and he benefits from the lower swap rate.
Dual currency swap
A currency swap in which the holder receives one currency on initial exchange, pays coupons in
another and makes the final exchange of principal in a third currency. This type of swap is used by
borrowers willing to lower their nominal interest cost by taking currency risk. A dollar borrower
will achieve lower borrowing costs by agreeing to pay the principal at maturity in, say, Euros at
the prevailing spot rate when Euro swap rates are below dollar swap rates.
Example A borrower who normally achieves sterling Libor less 0.25% wants to reduce his
funding costs. He issues a US$100 million bond and enters a dual currency swap. In the initial
exchange of notional amounts he pays the swap counterparty the US$100 million in exchange for
the equivalent amount in sterling. He receives fixed-rate dollars (to service the initial dollar bond
borrowing) and pays sterling Libor less a spread. On maturity he receives the US$100 million but
pays the swap counterparty the equivalent amount in, say, Euros.
Extendible swap
A swap in which one counterparty has the right to extend a swap beyond its original term. It is the
combination of a vanilla swap with a swaption (payer or receiver) whose expiry date coincides
with the maturity date of the existing interest rate swap. Most commonly it is the fixed-rate payer
who has the option. However, in the commodity markets, it is often the floating-rate payer.
Swaptions can also be used to create reversible swaps. These are swap that allows the user to
switch from being a floating rate payer in a swap to becoming a fixed rate payer. It is the
combination of an interest rate swap with a receiver swaption with a notional principal twice that
of the underlying swap. Half the swaption if exercised allows the holder to cancel an existing swap
(in the same way as swaptions are used in callable and puttable swap) and the other half results in
a new swaps position the same size as the old one with opposite interest obligations. See callable
swap.
Example
In an extendible swap an oil consumer who wishes to fix the price of his oil purchases can enter
into a fixed-for-floating commodity swap under which he pays a fixed rate that is lower than the
going swap rate and that is approximated to his budgeted rate and in exchange grants the floatingrate payer the option to, say, double the life of the swap if the price falls below a certain point. If it
does, the consumer is paying his budgeted rate and the option writer is benefiting from paying out
a lower floating price than he is receiving fixed.
Flex[ible] swap
An interest rate swap in which the buyer receives a floating rate and pays the higher of a fixed rate
lower than the current swap rate or Libor minus a pre-set spread, at the option of the swap
counterparty.
Example
A corporation has US$100 million of floating dollar debt paying Libor plus 0.5% with three-year’s
remaining life. Three-year swap rates are 7.59%. The company expects US rates to fall gradually
over the three years to 7.00%. The company enters into a flexible swap under which it receives US
dollar Libor and pays, at the counterparty’s option, the higher of 7.05% fixed or dollar Libor less
54bp. The company benefits most if dollar Libor stays above 7.05% (below which it would have
been better to stay floating) and below 8.13% (above which it would have been better to fix at the
7.59% swap rate available at the outset. Even then the flexible swap gives the borrower a margin
under the straight floating rate.) The borrower has effectively sold a floor at the implicit fixed rate
under the flexible swap to the bank. The premium received is incorporated into the margin under
the floating rate index and adjustment to the flexible fixed rate to a level below the prevailing
swap rate for the maturity.
Incremental fixed {floating} swap
A swap in which the fixed {floating) rate is only payable on a certain percentage of the notional of
the swap the rest staying floating {fixed}. In an incremental fixed {floating} swap the fixed
{floating} portion of the swap increases with Libor according to a pre-set ratchet table. So in an
incremental fixed swap, the fixed-rate payer pays the IFS swap rate on a resettable notional
principal amount. Because this rate is not always paid on the full notional amount, it is much
higher than vanilla swap rates. The IFS therefore appeals to floating-rate borrowers who believe
that rates will stay considerably below the level at which the fixed rate is payable on a large
proportion of the notional principal (which would push the blended rate well above swap rates.
The IFS therefore performs a similar function to an interest rate cap, in that it fixes a maximum
cost of funds, but instead of paying an upfront premium, users pay for the insurance against
catastrophic rate rises in the form of a higher swap rate. In most cases the attraction of the blended
rate at the outset of the swap is that it offers hedgers a way to protect themselves cheaply against
rate rises when absolute rates are low but the implied forward curve is steep.
The index floating swap achieves much the same in the opposite way. Instead of starting with an
off-market high IFS fixed rate and a low percentage of the notional on which it is payable, the
swap is initiated with an off-market low IFS fixed rate payable on a high percentage (often 100%)
of the notional, usually protected for a defined start period. After the protection period, the fixedrate payer pays the IFS rate on a certain portion of the notional and Libor plus or minus a spread
on the remainder. If Libor rises, the percentage of notional on which the floating rate is paid
increases and the percentage on which the low fixed-rate is payable decreases. Also known as an
index fixed {floating} swap, blended interest rate swap, self-regulating swap. Not to be confused
with index principal swaps which also result in changes in effective rates paid depending on rate
movements but whose notional principal actually changes.
Example In a five-year incremental fixed swap with a notional principal of US$100 million, the
fixed portion of the swap could be determined as follows.
If Libor is above:
but below or equal to:
then the fixed portion is:
8.50%
N/A
100%
8.00%
8.50%
80%
7.50%
8.00%
60%
7.00%
7.50%
40%
6.50%
7.00%
20%
0%
6.50%
0%
The IFS swap rate is 11.16% when the normal five-year swap rate is 9.50%. The payer then pays
X* 11.16% +(1-X)*Libor and receives six-month Libor, where X = the percentage of the notional
principal fixed in the swap. If the Libor setting on a reset date were 7.76%, the payer would pay
60% fixed at 11.16% and 40% floating at 7.66%, resulting in a rate of 10.07%. When Libor is
above 8.50%, the IFS becomes a normal fixed rate swap at 11.16%.
The price of the IFS results from its complex structure. It is constructed from a cap on 20% of the
notional principal at each strike (or 10% or 5% depending on the number of reset bands). The
swap provider then writes digital options at these strikes for 20% of the notional principal, so that
the premiums of the options cover the cost of the cap. So, if Libor is 7.20% the swap provider
would have capped 20% at 6.50% and 20% at 7.00% and the swap would be 60% in floating. The
losses on each digital in each Libor range are then calculated and the present value of these losses
is spread over the life of the swap to calculate the IFS rate. By incorporating different
combinations and types of digital options, these swaps can be tailored to meet a wide variety of
different needs. The swap can even be indexed to a different asset class – linking interest
payments with commodity prices, for example.
Index amortizing (rate) swaps (IAS)
The commonest types of index principal swap, IASs have a notional principal that can only
amortize and which amortizes more quickly as rates fall. (A swap which amortizes more quickly
as rates rise is called a reverse index amortizing swap.)
Typically in an IAS the fixed-rate is higher than would be payable on a swap with a fixed notional
principal because the notional principal amount will amortize as interest rates fall (or as
prepayment rates rise). The fixed-rate receiver obtains this high coupon because he has sold the
fixed-rate payer a series of put options on an interest rate index which effectively gives the fixedrate payer an option to shorten the swap’s life if rates move against him. The notional principal is
generally fixed for an initial two-year period of the swap, known as the lock-out period during
which time the buyer is protected against amortization. After that period, the notional of the swap
will decrease as a function of the level of the index chosen.
Sample terms might state that if Libor stays between 5.0% and 5.5% the swap amortizes by 75%.
If it rises to between 5.5% and 6.0% the swap amortizes by 50%. Between 6.0% and 6.5% it
amortizes by 25%. And above 6.5% the swap notional remains at 100%. There is a lock-out period
set in which the notional principal cannot amortize. The swap’s maturity date is the point at which
any remaining notional principal outstanding matures. And there is generally a clean-up feature: if
the notional principal falls below 5% of the initial amount, the swap amortizes completely.
Originally these instruments grew from the mortgage-backed securities market and the
amortization schedule was designed to correspond to the expected timetable of prepayments on a
pool of mortgages (by linking the amortization schedule with indices of mortgage prepayment
rates or of prepayments on a tranche of collateralized mortgage obligations) hence their name:
mortgage [replication] swaps or CMO swaps. Counterparties exchange a fixed-rate payment
stream for a stream of mortgage-related flows generated by a pool of mortgages or an index on
such a pool. Although the interest payments into this payment stream are fixed, the notional
principal can amortize as borrowers prepay mortgages if interest rates fall significantly. If this
happens, the notional principal on which the mortgage swap cashflows are calculated amortizes
accordingly. This kind of swap creates off-balance sheet investments that behave like a portfolio
of mortgages or like collateralized mortgage obligations without the need to take mortgage assets
onto their balance sheets.
The fixed-rate receive side of an IAR has negative convexity through the sale of the embedded
options. The fixed-pay side has positive convexity and can be used as a way of offsetting the
negative convexity of receiving mortgage-linked cashflows from mortgage-backed securities
which are affected by prepayment when rates decline. Most have had maturities of less than three
years to maximize the amount paid on the fixed leg. In addition to mortgage-related investment
and hedging, IARs can also be used for straightforward yield enhancement:
Example
Instead of investing in, say, vanilla one-year paper, a corporate treasurer can maintain three-month
rolling assets at Libor flat and pay that floating stream into an index amortizing swap with a oneyear lock-out and a final maturity that represents the maximum period for which he is comfortable
locking in his funds. As Libor decreases, the amortization speeds up. For the lock-out period, the
treasurer earns an above-market rate on his assets. In return for this, after the lock-out period he
accepts that, if Libor declines, instead of benefiting from paying less into the swap and receiving
fixed on the full notional amount, the swap will amortize, forcing him to reinvest the freed-up cash
at lower rates.
These swaps are usually structured so that as long as the amortization falls in a range between zero
(the swap matures on the full original notional amount) and 100% immediately after the lock-out
period, the treasurer achieves an above-market yield as well as a flexible medium-term investment
vehicle. Because amortization is expected, the swap performs rather like a money market
instrument after the lock-out period and provides cash liquidity as it amortizes. The swap works
best in a steep yield curve environment.
IARs can also be used for liability management in a steep yield curve environment:
Example
If a treasurer believes that short rates will not rise by more than, say, 100bp in the next two years,
an alternative to the vanilla swap is the index amortizing swap. For the two-year lock-out period,
the floating rate that the treasurer must pay into the swap can be up to 50bp less than he would
have to pay into a vanilla swap of the same maturity. Second, after the lock-out period, the
notional principal on the swap will amortize 50% as long as Libor does not rise more than 100bp,
so that the treasurer’s net position existing liability plus swap reverts gradually to a fixed-rate
liability. The transaction makes sense in yield-curve environments where the blended rate created
by the transaction is cheaper than the vanilla swap for the full term of the liability, a vanilla swap
for part of the remaining life of the liability or a cancellable swap. The danger is that the
treasurer’s prediction that Libor will rise no more than 100bp might be significantly mistaken. If
the rise is severe enough, no amortization will be triggered and the treasurer will have to remain a
floating payer for the remaining life of the liability. However, this floating rate will still be less
than that payable under the vanilla swap.
And these structures have now developed to the extent that now in an index amortizing swap
almost any amortization schedule is possible by agreement. So, for example, the currency-linked
index amortizing swap is an index amortizing swap whose notional principal is indexed to
currency value movements.
Example
A currency-linked index-amortizing swap could be used by a US exporter with such strong growth
in its Deutschmark earning exports that it needs more working capital. It wants to borrow fixedrate dollars while rates are low but wants protection against a strengthening dollar hitting its
export business and so reducing its working capital requirement. It could enter into a callable
interest rate swap but prefers an index amortizing swap under which it initially pays 4.30% fixed
and receives Libor on a notional amount equal to its borrowing. However, as the US dollar
appreciates against the Deutschmark, the notional principal of the swap reduces according to a
pre-set schedule, effectively shortening its life and facilitating early repayment of the underlying
loan and providing the required hedge profile, offsetting the reduced need for working capital as
the stronger dollar reduces Deutschmark revenue flows.
And reverse index-amortizing swaps can solve a number of problems faced by investors and
hedgers.
Example
A reverse index amortizing swap is an indexed principal swap in which the notional principal
amortizes faster as rates rise or which achieve the same effect by linking their floating-rate
payments to an index and increasing them if the index declines. Counterparties that wish to hedge
against instruments that amortize as rates decline can receive fixed in a reverse IAS. The
instrument is also used instead of a vanilla interest rate swap to transform a floating-rate asset to a
fixed-rate asset because it gets around the problem that an asset so swapped will lose its value if
rates rise and returns will be reduced if the investor is short-funded. An interest rate cap incurs an
upfront premium and may expire out of the money. The reverse IAS amortizes as rates rise thus
reducing the size of the fixed-rate asset. Higher cost funding can then be utilized to invest in
higher-yielding assets. In the same way, fixed-rate liabilities swapped into floating will incur
increasing interest expense when rates rise. The reverse IAS can be used to hedge against this.
Indexed principal swap (IPS)
Generic term for a (generally fixed-for-floating) swap whose notional principal can accrete or
amortize according to a predefined index, such as Libor, CMTs or a mortgage prepayment index
such as PSA rates. See clean index principal swaps, index amortizing (rate) swaps, reverse index
amortizing (rate) swap. Not to be confused with instruments whose interest payment streams alter
according to the level of an index (usually Libor but sometimes foreign exchange rates or
commodity prices) but whose notional principal remains constant, such as the incremental fixed
swap, Libor regulating swap, semi-fixed swap
Inverse floater swap
An interest-rate swap under which one counterparty pays fixed and receives a floating rate indexed
negatively to a reference index such as Libor. As Libor rises, the fixed payer would receive less;
as it falls, he would receive more.
Libor function swap
An interest rate swap to whose floating-rate leg a customized mathematical function or equation
has been applied to produce a payout profile tailored to a very specific view of rate movements.
Linear forex-linked swap
An interest rate swap one of whose legs is linked to movements in a foreign exchange rate.
Changes in the reference foreign exchange spot rate result in linear changes in the coupon rate
paid/received under the swap. This swap allows borrowers, for example, to swap their debt into an
interest rate that varies directly with a foreign exchange exposure they have. Adverse movements
in foreign exchange rates are offset by smaller interest rate payments on their debt.
Libor regulating swap
An interest rate swap under which one party receives Libor and pays a blended rate calculated as
the combination of a predetermined fixed rate and a predetermined floating rate. The blended rate
is capped at a maximum. It sits halfway between the incremental fixed {floating} swap and the
semi-fixed swap. The former links interest payments to a predefined ratchet table that dictates the
percentage of the notional principal on which fixed- and floating-rate payments are made. The
latter specifies just two rates payable a high and low rate determined by the level of Libor (or
some other underlying) on reset dates.
Example
A treasurer that could pay fixed at 6.71% in a three-year semi-annual swap could instead elect to
enter a US$100 million Libor regulating swap in which they receive six-month Libor and pay the
minimum of (6.90% + six-month Libor)/2 and 7.75%. So, if the average of the fixed and floating
rates stayed below 7.75%, then the treasurer would pay the blended rate. If that average were
above 7.75%, his fixed-rate payments would be capped at 7.75%. In this example, the blend of
fixed- and floating-rate is set at 50:50. This proportion can be customized according to the
hedger’s views.
The swap is constructed from a swap and cap, each for the requisite proportion of the original
notional amount. In this example, the swap can be imagined as two swaps, each on US$50 million
of notional principal. One is a 6.90% pay fixed receive six-month Libor swap, the other a pay sixmonth-Libor receive six-month Libor. The second swap clearly cancels itself and so the treasurer
has simply fixed US$50 million at the off-market rate of 6.90%. However, assuming he actually
has a liability of US$100 million on which he must pay Libor, that leaves US$50 million of the
original exposure unhedged. For the actual blended rate not to exceed 7.75%, a cap on that US$50
million floating portion is needed at 8.6% - ((7.75 x 2) - 6.9). The cap premium is the difference
between the swap rate (6.71%) and the fixed rate portion of the blended rate (6.90%) so that no
upfront premium is required.
In a positive yield curve environment the treasurer’s cost of funds will be lower than a regular
swap (but higher than Libor). Also, the maximum rate is known in advance, though it will be
higher than the current market swap rate. Like many other second generation swaps, this
instrument is for treasurers who wish to hedge against rate rises but who feel that the current yield
curve and implied forward curve overstate future rate rises.
Lookback swap
A swap in which, for example, the holder pays the highest Libor setting in the reset period and
receives Libor set at the beginning of the period plus a spread. In a three-year deal with six resets,
for example, the holder could receive six-month Libor plus 120bp and pay the highest daily sixmonth Libor rate in each six-month period.
Nearly-perfect swap
An interest rate swap in which a fixed rate is swapped into a low, off-market floating rate linked to
a reference index such as Libor but subject to the following type of formula: for every basis point
that Libor exceeds a pre-set cushion level between two reset dates, the spread over Libor increases
by a pre-set amount, say, one basis point. Libor is set at the end of each payment period. The
floating-rate payer is taking the view that the velocity of the increase in short rates will not exceed
the cushion level.
Example
A bond issuer issues a 7.0% fixed-rate bond and wants to swap it into floating. Under a nearly
perfect swap the swap counterparty pays 7.0% and the issuer pays Libor + 115bp subject to the
nearly perfect formula. Where Libor is set at the end of each payment period and where for each
basis point increase in excess of 25bp that Libor increases between two reset dates, the Libor
funding spread increases by one basis point. So if current six-month Libor is 3.85% then under the
swap the initial interest cost is 5.0%. But if rates rise rapidly, then funding costs will rise with
them.
Partial fill plus option
Commonest in the commodity derivatives markets, a partial fill plus option strategy is a swap
agreement in which one counterparty receives an off-market high fixed rate in exchange for the
market floating rate. In exchange for the off-market rate, the fixed-rate receiver grants the floatingrate receiver the option to double the amount of the swap if the price of the underlying exceeds the
swap rate.
Example
A company with a total hedge requirement of 100,000 barrels of crude oil per day could enter into
a swap under which it was paid US$1 more than the going swap rate for its oil on 50,000 barrels.
If oil prices rose substantially, then the floating-rate receiver would exercise the option and would
not only receive a floating rate higher than the fixed rate it was paying but would receive it on
twice the original notional principal of the swap. The swap can also be structured to be of use to
the floating receiver.
Participating swap
Any swap in which one of the counterparties participates in favourable movements in the
underlying price or rate while fixing a maximum cost. One interest rate version is an interest rate
swap in which the floating-rate payer caps his maximum payment but, by combining the swap
with a participating interest rate agreement, retains some participation in any falls in interest rates.
The commodity version works in much the same way.
Example
An oil consumer might elect to enter a participating swap under which he agrees to an off-market
fixed rate US$1 above the swap rate on a conventional fixed-for-floating commodity swap in
exchange for 50% participation in any downward movement in price. If the average of the index
price over the reference period is above the agreed fixed rate, then the consumer pays that rate and
receives the difference between it and the index rate capping its cost at the off-market swap rate. If
the index price of the commodity is less than the off-market swap rate, then, instead of paying
100% of the difference to the counterparty and receiving the index price as would be the case in a
normal fixed-for-floating swap, the consumer pays only 50% of the difference between the two,
benefiting from 50% of the price decline below the cap rate.
Performance swap
An interest rate swap in which the fixed-rate payer pays the at-market fixed rate and receives
Libor plus a margin unless Libor sets at a rate well above current implied forward rates. If Libor
does breach the trigger level, then the counterparty continues to pay the at-market fixed rate but
also receives the at-market fixed rate. If Libor does not breach the trigger level the fixed-rate payer
has fixed at below-market levels. If it does the swap effectively disappears for that period but a
rebate is paid in the form of a sub-Libor funding cost. The fixed-rate payer is long a standard
interest rate swap, short a call option on Libor (a cap) and short a binary option on Libor.
Periodic reset swap
An interest rate swap whose floating payments are reset according to a pre-agreed schedule or
index. Usually, the floating-rate payment is based on the average rate of the reference index over
the previous period rather than its level on the reset date. Variants include the window reset swap a
type of periodic reset swap in which the floating-rate payer is permitted to reset Libor at any time
within each reset period, as opposed to the beginning of each period as in a conventional swap, at
no additional cost. This embedded option allows the floating-rate payer immediately to take
advantage of windows of opportunity presented by declining rates or sudden dips in rates.
Polynomial swap
An interest rate swap in which polynomial equations (e.g., Ax2+bx+C) are applied to the Libor leg
creating payment profiles that can be tailored to outperform vanilla swaps within precisely defined
interest rate boundaries. The positions created give the precision of exotic options without the
associated all-or-nothing profiles.
Power Libor swap
Strictly speaking a swap that pays Libor squared or cubed (and so on) less a fixed amount/rate in
exchange for a floating rate. More generally, any leveraged swap that pays a multiple of Libor
usually in exchange for a greatly increased fixed rate if interest rates move against the end user.
Power Libor swaps often contain complex embedded options.
The most notorious example is the five-year/30-year swap entered into by Procter & Gamble
whose formula dictated that for every 1% increase in CMT yields above 5.78%, P&G’s payment
increases by more than 17% of notional principal per year and every 1% decline in long bond
prices costs P&G 1% of notional principal.
Semi-fixed swap
An interest rate swap in which there are not one but two fixed rates. Which of the two is
payable/receivable depends on whether Libor has reached a predetermined trigger point during
each periodic Libor setting. For example, a floating-rate borrower who believes that rates will not
rise as quickly as the implied forward curve predicts can receive Libor and pay a below market
fixed rate while Libor remains below the trigger point. If Libor exceeds the trigger, then the higher
fixed rate is payable. The trigger mechanism is created with an embedded binary option. There are
also commodity-linked semi-fixed swaps, particularly in the oil market. For example, an oil
consumer might pay a fixed rate of 4% if oil prices stay above US$12 but if prices go below that
level, he is swapped into 3.5%. That is, he has bought a swap plus a binary option on oil.
Trigger swap
A swap that pays a fixed-rate below the market rate. However, if rates rise above a certain trigger
level, the fixed-rate payer will pay a floating rate minus a spread determined by the then prevailing
floating rate. The result is a below market fixed swap that reverts to a below market floating rate
swap when the trigger is hit. The subsidized swap is the combination of a pay-fixed swap and the
sale of a cap. The cap premium is used to reduce the fixed rate paid under the swap. Also known
as a subsidized swap.
Example
The sale of a five-year sterling cap at 10.60% will earn the seller 50bp semi-annually. This amount
improves the five-year swap rate from 8.83% to 8.33%. If sterling Libor exceeds 10.60%, the
client will be put back into floating at a subsidized rate of Libor less 2.27%. The instrument is
ideal for borrowers who want to lock in their floating rate, but do not want to pay the market rate
as they believe the implied forward curve significantly overstates future rate rise. It generates an
attractive fixed rate as long as their rate ceiling is not breached. And even if it is, they still do
better than competitors paying vanilla floating rates.
Superfloater swap
A swap that imitates the characteristics of a superfloater bond in exchange for paying a fixed rate,
the counterparty receives a multiple of Libor minus a constant.
Example
A floating rate borrower wishes not just to protect itself against expected rate rises but actively to
benefit from them. It enters a superfloater swap in which it pays fixed and receives Libor as long
as floating rates stay between an upper and lower strike rate struck on either side of the fixed rate
payable. In a two-times multiplier superfloater for every basis point above the upper strike rate
that floating rates rise, the borrower receives two basis points of floating rate payment. If the
floating rate falls below the lower strike then the floating rate multiplier paid to the borrower falls
at a predetermined rate. So the borrower’s effective fixed-rate under the swap increases as rates
fall below the lower strike band but decreases as rates increase above the upper strike level. The
borrower has bought a cap and sold a floor.
Swap differential/difference agreement (SDA)
An interest rate basis swap contract to exchange or lock in the differential between a bond or note
yield and the swap rate of the same maturity. The contract moves with reference to the difference
between the same point on the two different yield curves. It allows an investor to profit from the
widening or narrowing between two yield curves. The SDA is customized with defined settlement
dates, a defined value per basis point move, and one defined point on two yield curves. All
payments are made in one currency so there is no currency exposure.
Example
An investor might believe that the differential between the two-year Euro swap rate and the twoyear Swiss franc swap rate will narrow over the next year. The investor enters a narrowing EuroSwiss franc SDA for one-year settlement. The value per point can be set at any value in either
currency, say Sfr10,000. The SDA price is given in terms of basis points. If at maturity the
difference between the two-year swap rates in the two currencies has fallen below the SDA entry
level, the investor will receive Sfr10,000 for every basis point lower. If the difference is higher
than the entry level the curves have widened the investor will lose this amount. The entry price is
calculated by taking the difference between the implied forward rates from the two yield curves. In
the example, the one-year forward two-year Euro and Swiss franc rates are calculated and the
difference is the SDA price. Investors who buy the SDA expect curves to widen; those who sell
expect curves to narrow.
Compound options
Key facts
A compound option is an option to buy or sell another option. There are relatively few uses for single
vanilla compound options. They can be used as a hedge for contingent exposures, such as the interest rate
and foreign exchange risks that will be incurred if a company wins a tender contract but that will not exist
if the company loses. They can also be used as a highly-leveraged way to gain exposure to the underlying
while limiting downside to the small initial premium.
However, in strips, or in combination with other options, compound options can be used to create less
specific and more useful products, notably pay-as-you-go options. These give holders tailored and
cancellable exposure to their chosen underlying asset. They resemble some forms of contingent premium
options but are created and behave very differently.
Definitions
Caption
Name sometimes given to an option on an interest rate cap. The option on a floor is sometimes
known as a floortion.
Chooser option
An option that is neither a call nor put until, at a predetermined date known as the choose or
choice date, or at any time during a preset chooser period, the holder of the chooser may trade it in
for either a call or put option. If the call and put have identical strikes and expiry dates the option
is called a standard chooser or regular chooser and can be priced via an analytical model. If they
differ in strike or expiry they are called complex choosers and can only be priced using numerical
models.
Choosers can be European-style or American-style in the sense that the holder is either given the
choice of a European put or call or an American put or call.
A chooser is not strictly speaking a compound option as in its basic variety the holder pays no
exercise premium for choosing call or put and cannot simply let the choice expire. It is more
similar to a European-style straddle (simultaneous purchase of a put and call) but, since the holder
must choose between one or the other at some point, it is cheaper. It suits aggressive investors who
wish to take a view on volatility.
The pricing relies on put call parity and the fact that the option writer knows that the option holder
will always choose the more valuable option on the choose date. So, if the call is more valuable,
the holder of the chooser will choose it, exercise it and create a synthetic put by shorting the
underlying and rolling the position forward at the strike price. Also known as a double option,
dual option or preference option.
Compound option
The right to buy or sell for a pre-agreed amount at a set future date a second option of
predetermined specification. This second option is known as the underlying option or back option
and the option to buy or sell the underlying options is known as the front option. The compound
option purchaser pays an initial premium (the front premium) and if they choose to exercise the
right to buy the underlying option they pay an exercise premium (the back premium). The sum of
these two premiums is greater than the premium that would have had to have been paid for the
underlying option at the outset. The higher the initial premium, the lower the exercise premium
and vice versa. A higher initial premium also results in a lower total premium. Compound options
can be used to lock in the forward volatility curve but are most often used to hedge contingent
exposures such as tender contracts.
Pay-as-you-go option
An option whose premium is payable in instalments at the beginning of each period at the
discretion of the holder. At each period start the purchaser can elect not to make a payment, in
which case the option is terminated. The initial upfront premium for the pay-as-you-go option is
below that for a conventional option but pay-as-you-go options are more expensive than
conventional options if all the premiums are paid. In this structure, the holder is long a strip of
compound options whose maturity matches the tenor of the payment periods. Also known as an
instalment option, though this is more usually applied to a type of contingent premium option,
instalment option. Also occasionally known as a rental option (since if the holder misses a
payment, the option is ‘repossessed’), mini-premium option and step payment option.
Example
A company might have sold a three-year floating-rate note that the buyer can put back under
certain circumstances. In return for this embedded option, the company receives a significant
discount on its coupon payments. The company is not very happy with the interest rate outlook
and thus wants to hedge this floating rate exposure. A normal three-year quarterly cap with a
7.35% strike would cost 174 bp. However, should the loan be called, the interest rate hedge will
no longer be required. They therefore decide to enter a pay-as-you-go (or instalment) cap which
would cost 23 bp per period (the rental payment). The company can simply terminate the cap
when desired by ceasing to make instalment payments. This scenario can be of use when the
underlying note gets called, or when the company decides it no longer requires the protection of
the cap. The price of the option will depend on the termination date of the option and so the
number of instalment payments made. If used for the whole original maturity it will be more
expensive than a vanilla option.
Wish collar
A risk reversal in which one of the two parties has the right to change the notional amount of one
side of the trade at a future date and time.
Contingent premium options
Key facts
One important difference between options and other derivative instruments such as forwards and swaps is
that, generally speaking, the buyer of an option has to pay an upfront premium to purchase protection or
express an investment view. This cash outlay is not always convenient and so structures have been
designed that defer premium payments so removing the need to make an upfront payment. In the same way
that some options’ strike price is only set on expiry, to more exactly match the holder’s risk profile or
investment views, the premium payment for these contingent premium options is dependent on the final
level of the underlying. In general, the structure ensures that holders pay for options they use and do not
pay for options that turn out to have been unnecessary.
Contingent premium options are constructed from a combination of standard options and digital options
and are usually more expensive than vanilla options that offer the same level of protection. The simplest
varieties combine the purchase of a vanilla option with the simultaneous sale of a digital option. The
incorporation of a digital option makes a very wide variety of structures possible. If the digital option is
struck on the same underlying as the vanilla option component of the contingent premium option, then the
option is known as a regular contingent premium option. If it is not, it is called a cross-contingent premium
option. The digital option can be European-style/at-the-money or American-style/one-touch. Its strike price
conditions can vary in as many ways as a naked digital. Strips of digital options can be sold to create
instalment-like premium payments. Because of this, contingent premium options are sometimes known as
instalment options, and so confused with pay-as-you-go options when the latter are referred to by the same
alternative name. The crucial distinction between contingent premium options and pay-as-you-go options is
that in the former, because of the embedded digital option(s), the payment or non-payment of the premium
is dictated entirely by the behaviour of the underlying. Because the latter are compound options, the holder
has ultimate control over the payment of the premium.
Definitions
Contingent premium option
A path-dependent option for which no upfront premium is payable. In one simple version the
premium is paid at expiration but only if the option expires in-the-money. Even if the option is inthe-money, but not deeply enough to recoup the premium, the option still has to be exercised and
the premium paid. If the option expires at-the-money or out-of-the money, no premium is paid.
For the option holder to benefit, the option either has to expire at- or out-of-the-money or it has to
expire sufficiently deep in-the-money to recoup the contingent premium. The premium is more
expensive than a conventional option premium because it is paid only if the option expires in-themoney, and this is not guaranteed. The premium can be approximated by dividing a conventional
premium by the probability of the option expiring in-the-money, i.e. its delta adjusted for the time
value of money. Also known as the paylater option.
Contingent premium options are constructed from the purchase of a vanilla option and the
simultaneous sale of a digital option. In the simplest case this digital option is struck at the same
level as the vanilla option strike with a payout equal to the premium the provider of the contingent
premium option calculates as sufficient compensation for the sale of the vanilla option. If the
vanilla option moves into the money, so does the digital creating the premium payment. If the
option remains out-of-the-money, so does the digital so no premium is payable. Contingency can
be applied to any derivative. So a contingent cap is a cap whose buyer pays a small upfront
premium and then has to pay a further premium instalment if the selected index fixes above the
preset contingency level. If the contingency level is never reached, then the premium is lower than
for a conventional cap. If the contingency level is breached then the total premium payable is
higher. In more tailored versions of the structure the premium is calculated and potentially paid
several times during the life of the option, not just at maturity. Structures that incorporate multiple
digital options, give this type of contingent premium options their other name, instalment options
(see over).
Cross-contingent premium option
A contingent premium option on one asset whose premium is contingent upon movements in
another.
Deferred premium option
A standard option except that the premium is payable at expiry rather than upfront. More
expensive than a conventional option by an amount equal to the cost of having effectively
borrowed the premium from the option provider. This is not a true contingent premium option
because the premium is payable regardless of the level of the underlying. It is the combination of a
conventional option and a loan.
instalment option
An option with zero upfront premium for which the buyer pays a pre-specified amount if the
underlying trades through one or more pre-determined levels at any time over the life of the
option. If the underlying fails to trade through any of these instalment levels, the buyer will have
acquired the option at no cost. See example 1.
The instalment structure can be applied to a most basic options and is behind many, seemingly
separate products. For example the self-funding cap has no up-front premium. Instead, a
predetermined premium is paid only at those strike resets where the cap is in-the-money. If the cap
expires out-of-the-money, the buyer makes no payment. In exchange for the guarantee that its
premium will not be wasted, the premium is higher than for a conventional cap.
Depending on the type of digital option sold, the structure can be altered so that the premium from
the sale of the embedded digital options is used to create not an all-or-nothing premium payment
but a high/low premium choice. In the reflex cap the premium is paid periodically, and each
instalment is dependent upon a trigger rate being reached. The total premium will be low if the
reference rate stays below the trigger, but will higher if the rate is above that trigger. See example
2.
Sometimes the name instalment option is applied to pay-as-you-go options as in both the premium
is paid for in stages. The former are strips of digital options, the latter are strips of compound
options. This difference is explained in the introduction to this chapter.
Example 1
An FX hedger is short EUR and long USD booked at 1.0450. A six-month instalment option (with
the forward at 1.0440 and volatility at 10%) would have no initial premium but the following
terms: premium trigger levels of 1.0250, 1.0200 and 1.0150 with the premium at each level 1.50%
EUR (110 USD pips). If any of the trigger levels are reached, then the premium for that instalment
is due. If all the levels are reached, a total of 330 USD pips will have been paid far more than the
2.72% EUR (280 USD pips) payable for the equivalent vanilla European option.
Example 2
A company wants to hedge a three-year floating-rate loan on three-month Euribor. It believes
Euribor will peak at 5%. A standard interest rate cap with a strike of 5% would cost them 4%
upfront. Instead they can enter a reflex cap at 5% with a trigger rate of 6%, just above the expected
peak. This structure would cost the company 19 bp for every period Euribor resets below 6% and
62 bp for every period Euribor resets above 6%. Euribor would have to stay above 6% for more
than 9 out of the 12 months for the standard cap to outperform the reflex cap on a present value
basis. The periodic premium is therefore low when Euribor is marginally above the strike and
higher when the cap is deep in-the-money the buyer is paying more for the cap when it is most
valuable. The reflex cap combines a normal interest rate cap with a series of digital options that
expire on every reset date. The normal cap is partially paid for by a preset amount per period
(which would be 19 bp in the example) and partially by the sale of the digitals (43 bp per period).
The reflex cap provides full cap protection without an immediate premium payment; costs less
than a vanilla cap if never used; and is advantageous where the view is that rates will not rise
dramatically above the strike, although if it does, the higher premium is only payable in those
periods where the cap is deep in-the-money.
Part-contingent knock-in option
An option that only knocks-in if spot moves sufficiently against the buyer’s underlying position.
The premium payable at origination is less than for a standard knock-in, but if the option is
knocked-in, then an additional premium payment is required that makes the option more
expensive than the standard would have been. Also known as a part contingent premium option.
Example
A hedger buys a GBP put/USD call with a strike of 1.65 and a knock-in at 1.57. The normal cost
of this option is 1.70% GBP. The part-contingent structure makes the upfront premium just 0.90%
but the buyer is obliged to pay an additional 2.40% GBP if the knock-in level is hit. So if sterling
appreciates the option will not knock-in. If sterling depreciates but does not hit 1.57 then the buyer
has no protection but has only spent 0.90%.
reverse contingent premium option
An option whose premium is only due if the option expires at- or out-of-the-money that is, the put
version of a contingent premium option. Like contingent premium options the premium can be
calculated on expiry or at set points during the life of the option. In the latter case no initial
premium is paid but if, subsequently, the underlying moves beyond preset trigger points set outof-the-money relative to the strike, certain fixed premiums are payable by the holder of the option
at maturity. For catastrophe insurance, the structure offers potentially zero premium protection.
However, if all the trigger levels are reached, then the holder pays more in premium than the
equivalent vanilla option (he pays the vanilla premium plus its financing costs). For this to have
happened though, the underlying must have moved in the hedger’s favour. This makes the option
useful where protection is required (put) but market view is bullish. The purchaser can elect not to
make an instalment payment, in which case the option is terminated.
Credit derivatives
Key concepts
Credit derivatives are instruments whose value is derived from that of an underlying bond, loan or other
credit agreement. They are used to assume or lay off credit risk in isolation from other types of risk. The
two main instruments are credit default swaps and total return swaps and they:
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.
enable holders of credit risk, such as banks, to lay off the credit risk in their loan, bond and
derivatives portfolios without having to sell the underlying credits upon which their relationships
may be based. This makes true credit portfolio management possible.
are an alternative to asset swaps for institutions unable to buy high-quality assets for funding cost
reasons, or unable to access the loan markets directly, to gain their desired credit exposure on an
unfunded basis (or, if embedded in credit-linked notes, on a funded basis.)
provide a mechanism for BIS capital management as they allow banks to remove assets from their
balance sheets particularly at key time regulatory capital requirements for banks are often
calculated on peak credit exposure at the end of reporting periods and can attract lower regulatory
capital than the cash equivalent.
allow synthetic assets to be created that do not exist in the markets.
allow hedging of contingent credit risks embedded in assets hedged with currency swaps.
allow corporates to hedge the credit exposures they have to other corporates via trade receivables
and long-term purchase contracts and to their banks via long term trade financings, loan facilities
such as revolving credits and swap and option portfolios.
allow corporates and banks to hedge against emerging market risk where guarantees are not
available or export agency cover is expensive.
create a link between the bank and insurance markets vastly increasing the capacity of both to
absorb credit risk. In sectors where corporate demand for insurance is so great that there is no
more reinsurance capacity, insurance and reinsurance companies can use a default swap to go
short the default risk of a basket of credits issued by entities in that sector so freeing up their lines
in that sector. See chapter 16 for: credit-linked notes credit-spread linked notes, first to default
bonds, repackaged notes, sovereign default notes
Definitions
asset swap
A transaction which transforms the cashflows of a security through the application of one or more
swaps. For example, bond coupons can be swapped from fixed to floating rate or vice versa;
interest and principal can be swapped into a different currency; the yield from a security can be
swapped for a cashflow based on an index in another asset class. An asset swap transaction
involves three distinct steps: an asset is purchased for cash; the cash flows are swapped into the
desired form; the package consisting of the asset and the swap(s) is held by the investor these are
on-balance sheet items.
The most common type of asset swap is the par asset swap. Here the notional amount of the swap
is equal to the face amount of the underlying asset. The asset is purchased for par and the investor
receives par at maturity. The other vanilla asset swap is the market value asset swap. Here the
notional amount of the swap is equal to the market value of the underlying asset at the time the
trade is executed. See repackaged securities for an explanation of the link between vanilla asset
swaps and true credit derivatives.
A number of important variations allow for more tailored transactions. Callable asset swaps and
step-up callable asset swaps are used where the underlying asset is either callable or step-up
callable. The swap matches the coupon and callable features of the underlying. They are also used
in convertible asset swaps (also known as stripped convertibles/converts).
Here the underlying asset is a convertible bond. Typically a bank purchases a convertible and sells
it to an investor for its fixed-income value. The investor then enters into a swap the asset swap
with the bank exchanging the fixed-rate coupon payments on the convert for a floating rate. The
swap agreement also contains an option giving the bank the right to call the bonds back at any
time and so terminate the agreement (hence the asset swap is callable). The investor effectively
holds a callable floating-rate note backed by the credit of the convertible issuer. He has given up
the equity option embedded in the convertible the option has been 'stripped' from the convert in
return for a higher spread on the swap. The bank now owns the equity component of the
convertible as it can call the bonds to extract any appreciation in the value of the equity option.
Callable and puttable asset swaps are also used to take views on changes in credit spreads they are
sometimes viewed as the first credit derivatives and in this context are often referred to as credit
spread options (see below). Other structures include cross-currency asset swaps in which the
coupon of the asset is swapped into a different currency; forward-[starting] asset swaps which can
be used by investors wishing to take advantage of a steep credit curve for a specific issuer. These
are also known as knock-out asset swaps as if the underlying asset defaults before the start date of
the swap, the forward swap is cancelled; maturity-shortened asset swaps where the maturity of the
swap is less than that of the underlying asset; and asset swaps can be combined with other options,
such as caps, floors and collars, to give more tailored payoff profiles. See also switch asset swap .
Contingent credit risk
Indirect credit risk such as that embedded in assets hedged with currency swaps.
Example
The arrangers of bond repackagings face contingent credit risks. Suppose a bank can buy Brazilian
Brady bonds cheaply in the US market but cannot sell them into Europe because of their unusual
floating coupon and amortization schedules. It places them into a special purpose vehicle (SPV)
and swaps them into fixed-rate Deutschmark bullet bonds. The underlying bonds are placed in a
trust and swapped using a cross-currency swap. Most SPV structures have triggers that force early
redemption of the new bonds and the unwind of the SPV in the event that collateral levels fall
below predetermined thresholds.So, if credit spreads rise, or in the extreme the bonds default, the
swap has to be unwound as the structure is dismantled. If the dollar strengthens against the
Deutschmark the bank will lose money on the swap unwind. In an extreme case, the residual value
of the bonds may not be enough to cover the shortfall, so there is a credit-related contingent
foreign exchange risk. To cover this risk, the arranger could take out default protection on the
underlying securities.
Similarly, banks run a quanto credit risk when cross currencies in an asset swap. For example: a
bank takes a dollar-denominated asset and swaps it into yen using a cross-currency swap. If the
underlying asset defaults and the swap is unwound there will be a windfall gain or loss on the
swap. Bankers recall deals when USD/JPY traded at 250 that, if they defaulted at today’s
exchange rate, would incur windfall losses as large as the bond principal. One way to hedge is to
take credit exposure in dollars and buy credit protection in yen. But the size of this hedge must be
managed dynamically in the same way as the risks embedded in quanto instruments.
[Credit] default swap
A swap agreement in which one counterparty (the protection buyer, risk seller, hedger) pays a
fixed periodic fee, usually expressed in basis points per annum on the notional amount, in return
for a payment by the other counterparty (the protection seller, risk buyer, investor) contingent
upon a specified default event relating to an underlying reference asset (a loan, bond, stream of
trade receivables or derivative contract cash flows). While there is no default event, the protection
buyer pays the swap counterparty the periodic coupon. If the specified credit event occurs there is
an exchange of cash flows and/or securities designed so that the net payment to the protection
buyer mirror the loss incurred by creditors of the reference credit in the event of its default. This
exchange of cashflows usually follows one of three patterns depending on the liquidity and
availability of the reference securities:
i.
ii.
iii.
Physical delivery: the protection seller pays full notional amount in exchange for
defaulted reference securities
Cash-settled recovery value-linked payment: the protection seller pays the counterparty
an amount calculated as the fall in the price of a reference security below par after default
as established by polling reference dealers. So a typical cash settlement formula would be
[(par - recovery value) x notional principal of swap]. The price of the reference asset can
be determined either as an average determined over a three-month period or via a single
poll.
Cash-settled binary payment: the protection seller pays the counterparty a predetermined
fixed amount or pre-determined percentage of notional is payable. This structure is
sometimes known as a digital credit swap. Because they are difficult to price and hedge
digital credit swaps command a premium most hedgers consider too expensive. The
economic effect of the transaction is to allow the holder of an exposure to a particular
counterparty to transfer the risk of that counterparty’s default to a third party without
transferring the underlying asset itself. The third party the protection seller assumes the
credit risk of the exposure in exchange for the return from it and so generates investment
income with no funding costs. Credit default swaps with the same maturity as the
underlying asset are equivalent to the underlying asset but with the interest rate risk
immunized. That in turn means that they are the same as total return swaps (see below)
with the interest rate risk immunized. Hence the pricing is directly comparable to asset
swaps.
The periodic coupon is generally derived from the Libor spread received by an investor in
the underlying obligation in the cash or asset swap market. The asset swap market spread
is the benchmark because the default swap is an unfunded transaction. To replicate it in
the cash market would require an investor to borrow the underlying bond through the
repo markets. On average, repo financing for high-grade corporate assets is available at
Libor flat, so the investor’s spread on the trade is a spread to Libor. This arbitrage, where
it is available, constrains the pricing of default swaps.
However, hedgers generally have to pay a premium to this Libor spread depending on the
tenor of the swap and the liquidity of the underlying bond. The swap is generally less
liquid than the underlying, which means investors demand more for taking the swap than
they would the bond. On the other hand, as the transaction is unfunded, it gives
institutions with high funding costs access to assets they would not normally be able to
purchase without incurring negative carry. Default swaps are also often structured with
maturities different to that of the underlying exposure. This creates synthetic investments
that are not available in the cash markets. Both these factors can make the swap more
desirable than a cash investment and drive the price down. In addition, in a credit default
swap the protection buyer has a credit exposure to the protection seller contingent on the
performance of the reference credit. If the seller defaults, the buyer must find alternative
protection and will be exposed to changes in credit spreads since the inception of the
original swap. So protection bought from higher-rated counterparties should command a
higher price. Also, the protection buyer must also take into account the correlation
between the risk of default of the seller and the hedged exposure. If there is a positive
correlation, then default by the underlying borrower may imply that the swap
counterparty’s ability to deliver the notional value of the reference security is impaired.
At worst this might means that the reference credit and protection seller default
simultaneously in which case the buyer is unlikely to recover the full default payment
due.
Credit default swap maturities usually run from three months to 10 years and transaction
sizes range from USD5 million to USD300 million. Bid/offer spreads range from 50bp to
150bp. The main participants in the market are banks. Also known as a credit swap, and
also, particularly if instead of paying a periodic fee the protection buyer pays a one-off
upfront payment, as a [credit] default option. Strictly speaking though, since the swap
contains no optionality, it is not a true option.
Credit default exchange swap
A type of default swap in which investors swap the default risk of one asset or basket of assets for
that of another. Each party serves simultaneously as a protection buyer and a protection seller.
These swaps are used mainly by commercial banks who want to hedge concentration risk in their
loan portfolios in a single unfunded transaction without buying and selling packages of
commercial loans.
Example
An investor seeks exposure to a loan yielding libor + 350 basis points that a double-A rated bank
has made to a Turkish corporation. The investor enters into a total return swap with the bank – the
total return payer under the terms of which it pays three-month Libor +100 basis points and
receives Libor + 350 basis points plus or minus any change in the market price of the loan. If the
loan’s market value remains unchanged over the life of the contract, then the investor earns 250 bp
on the transaction. However the investor bears both the price risk and any early repayment risk on
the loan over the life of the agreement.
Credit spread
The yield of a security or loan less the yield of a corresponding risk free security. Plotting credit
spreads the premium over the benchmark against bond maturity produces the credit curve for a
borrower. In theory this margin compensates investors for the risk of default. In practice it is also
affected by liquidity considerations and and number of other market factors. Credit spreads can be
traded with credit spread forwards and options. The key variables in trading are the term structure
and volatility of credit spreads. The forward credit spread is calculated as follows: the forward
prices of the security and the risk free benchmark are calculated and converted to the
corresponding forward yield. The forward credit spread is the forward security yield less the
forward risk free rate.
In theory the credit spread should increase in line with increasing default risk and maturity. In
practice, forward credit spreads increase in the case of a positively sloped yield curve but do not
seem to reflect investor expectations. Forward credit spreads appear to be a poor indicator of
future spot spreads and seem relatively more volatile than the underlying securities. This higher
spread volatility reflects the absolute lower level of spread and the imperfect correlation between
the yield on the security and the risk free rate.
Credit spread forward
Equivalent to a forward asset swap these are forward-starting transactions which enable credit
spread trading/hedging at a future date. Credit spread forwards allow counterparties to express
views on future credit spreads and benefit from a narrowing or widening of the credit spread
between debt instruments. One application is therefore for borrowers wishing to lock in future
borrowing costs without inflating their balance sheet today. In the case of a borrower hedging
issuing costs the reference credit would also be the protection buyer which significantly increases
the counterparty risk to the protection seller. Collateralization is being used to mitigate this risk.
Example
An investor wants to take advantage of steep forward credit curve by locking in wider spreads in
the future than are available today. He identifies an eight-year emerging market bond that
currently trades at Libor+70bp. Three-year bonds from the same issuer trade at 30bp over. At
current market prices, by agreeing to buy the eight-year asset in three years’ time an investor will
lock in Libor plus 95bp for the remaining five years of its maturity 25bp more than eight-year
assets today. The risk for the investor is that the issuer’s credit fundamentals deteriorate over the
the next three years and spreads widen by more than 25bp. However, the downside is limited by
the fact that if the issuer defaults in the next three years then the investor does not have to buy the
asset.
Credit spread option
Credit spread options are calls (puts) on a credit-sensitive asset with at a pre-determined strike
price expressed as a spread to an index typically Libor or to government securities. They allow
buyers to buy or sell the underlying asset at a pre-determined price for a pre-determined period of
time. The buyer of a credit spread call profits if spreads tighten. The buyer of a put profits if
spreads widen. The options are priced off the credit curve, expected credit spread volatility and
expected recovery rates.
The reference asset is usually either a floating rate asset or a fixed-rate asset swapped into floating
rate through an asset swap. Floating-rate reference assets are favoured because the floating coupon
immunizes interest rate risk any change in the principal value of a floating-rate asset is largely due
to changes in credit perception. This means that, in general, credit spread derivatives such as
forwards and options are equivalent to forwards and options on an asset swap.
So, for example, a put seller would be put into an asset swap paying par for the asset. The seller
would then pay the coupons on the asset and in return receive the pre-determined spread over the
Libor (the strike spread). Because of this, credit spread options are equivalent to callable or
puttable asset swaps and are sometimes known as asset swaptions.
Asset swap buyers can use these options to enhance yield by, for example, selling calls. This
exchanges upfront premium income against the loss of upside if spreads tighten. Investors can also
use combinations of calls and puts to lock in a current spread and transactions can be structured
that allow investors to go long or short the spread between two different assets say two countries’
sovereign debt or benchmark corporate bonds of different ratings or sectors. Institutional investors
are now using credit spread options to slice up a credit into different tenors. A sophisticated credit
risk player might buy a five-year bond and sell the first two years as a way of rolling down the
credit curve.
Basket/portfolio spread options are also available in which the option is written on a basket of
reference credits rather than just one. For example, in a basket credit spread put trade an investor
increases its premium income by selling a number of puts, each linked to a particular security at a
particular strike spread. The buyer has the right to exercise the seller into any one of the reference
assets at its strike spread, in the full notional amount.
Example
Improvements in economic fundamentals in Hungary have created new credit lines. In the Asian
credit crisis five-year paper went from 35bp over to as wide as 100bp. Spreads are now back at
Libor + 40bp but some expect a retreat in the short run. An investor with a price target of Libor +
50bp can utilize its new credit line while waiting for prices to move to its entry level by writing
one-year put options struck at the target spread. Typical terms would be that the investor receives
25bp running yield for giving the option counterparty the right to sell it bonds at the target spread.
The income requires no balance sheet and no funding requirements. If the spread remains below
Libor + 50 bp the investor has earned a premium of 25 bp multiplied by the notional principal of
the contract which it can use to buy the bond at a price below market (if it wants). If the spread has
risen above the target level then the option seller exercise their option to sell the bond at the strike
spread and the investor buys the bonds at its strike spread plus the premium which is the originally
targeted level. Alternatively the investor can settle the trade by paying the option seller the
notional principal of the contract multiplied by the difference in basis points between the final
spread and the strike spread. The higher the strike spread is above the current spread of the
reference asset, the more the credit spread put resembles a credit default swap in which the
investor pays a small periodic fee to buy protection against a large downward move in the price of
the reference asset.
credit spread exchange swap
A swap in which counterparties swap the credit spreads of two reference securities. This allows
them to take views on the relative spread between two comparable credit-sensitive securities.
Example
An investor wants to monetize the expectation that the credit spreads of Brazilian Brady bonds
will narrow relative to those of Argentine Brady bonds. The investor chooses a spread exchange
swap structure that produces a pay-off tied to the difference in the spreads of two Brady bonds.
The trade uses the exchange of two absolute spreads pegged to a US Treasury to to capture the
relative spread of two reference securities. the investor pays the dealer the notional amount
multiplied by the credit spread between the Brazilian Brady bond and the US Treasury. In
exchange, the dealer pays the investor the same notional amount multiplied by the credit spread
between the Argentine Brady and the US Treasury. If the credit spread of the Brazilian bonds falls
relative to that of the Argentine bonds , then the investor receives a payment from the dealer and
vice versa. This kind of trade is normally structured so that no payment is made unless a change in
the relative spread occurs.
First to default/basket default swap
A default swap in which there is not one but a basket of reference assets. The protection buyer
pays a fixed periodic payment, usually expressed in basis points per annum on a predetermined
notional principal.
The protection seller makes no payments unless some specified credit event relating to any one of
the reference securities in the agreed portfolio occurs in which case the protection seller is
obligated to make a payment based solely upon the defaulted security. In other words, the swap
only pays out on the first default which occurs in the basket. Other defaults are not hedged. The
investor receives a return approximately equal to the spread of the worst names in the basket plus
an additional correlation dependent spread for each of the other names in the basket. The
additional spread is based on the commitment fees on unfunded loans for those names.
These transactions started off with six or so reference assets and banks are now executing $1
billion-plus default swaps on a basket of between 30 and 50 credits. These larger deals are
essentially the same as a privately placed unfunded collateralized loan obligation. This type of
synthetic securitization has two key benefits over standard CLOs or credit-linked notes. First, it is
cheap: because it is unfunded it allows a strong bank to maintain its sub-Libor funding spread.
Second it is extremely flexible: as the assets remain on the banks balance sheet rather than being
having to be transferred from different branches or legal entities into a special purpose vehicle
which can cause internal, regulatory, legal and tax problems.
First to default swaps generate much higher returns than through a single investment in any of the
other names. As long as the basket is diversified, the hedging institution lays off exposure more
cheaply than hedging the names individually. Also known as basket/portfolio (credit) default
swaps.
Market contingent credit derivative
Credit derivatives whose payout depends on the mark-to-market value of an underlying credit
exposure and is contingent upon that value being negative. A hedger would determine its
maximum expected exposure on a credit or group of credits (these could be securities or
derivatives positions) if those credits were to default. It would then enter into a market contingent
credit derivative with a notional amount equal to this exposure. The hedger pays a commitment fee
say 15bp on the notional amount of the agreement unless the mark-to-market exposure on the
interest rate or currency position becomes negative. In that case the institution pays the full
hedging cost of the reference party but just on this negative amount. In the case of default, the
buyer of the hedge would have protection equal to the unwind value of the underlying position.
This is much cheaper than using a standard credit derivative because the hedger only pays for
credit protection when it is needed when the mark-to-market value falls below zero. If the markto-market value remains positive, then the commitment fee paid will typically be around half the
cost of fully hedging the underlying credit risk. See swap guarantee.
Mark-to-market cap
An interest rate hedge structure that puts an upper limit on the mark-to-market loss of a swap or a
swap portfolio. The portfolio version gives the client to enter into a portfolio of offsetting swaps at
any reset period over a chosen period, at strikes that ensure that the mark-to-market loss will not
exceed a predetermined amount. The premium depends on the underlying parameters of the swap
portfolio: tenors, notional amounts, strikes, correlations and embedded option features. Protection
on a portfolio basis is cheaper than buying caps on the individual swaps.
Example
A corporate treasurer may have a series of interest rate swaps on his books, hedging a variety of
underlying debt obligations. This treasurer was previously not required to mark his swaps to
market, but recent accounting changes force him to. In this case, he wants to limit any adverse
bottom line effects. Suppose he has a portfolio of five receive-fixed swaps maturing at different
dates between October 1996 and October 1998 which currently show a mark-to-market loss of $4
million. A mark-to-market cap would provide the company with an option to enter into pay-fixed
swaps at any rate reset date over the next 12 months exactly offsetting the existing swaps in the
portfolio and locking in a loss of $4.5 million. Alternatively the options could be cash-settled. The
cap premium can be paid upfront or on a periodic basis.
Recovery rate
The percentage of par paid to creditors after a default. This is an important concept in credit
derivatives because it can dictate the pricing and structure of a number of products.
Example
One bank is willing to take exposure to a credit at the senior unsecured level at a spread of 40bp.
Another bank with this existing exposure is willing to sell this exposure at the same spread.
However, the first bank considers the likely recovery rate for the credit to be 50% while the
second thinks that it is 70%.
The two institutions may enter into a credit swap in which the contingency payment is fixed at
50%, rather than at the floating recovery rate of senior unsecured debt as determined by dealer
poll. The second bank, the protection buyer, is prepared to pay up to 40 x 50/30 bp (67bp) for the
fixed recovery contract since it offers more protection than the floating recovery contract given the
bank’s recovery rate expectation of 70% (50/30 is the ratio of expected losses in the fixed versus
floating recovery transaction). Conversely, the first bank is happy to receive any premium over
40bp for the 50% fixed recovery transaction since in its view this is equivalent to the floating
recovery transaction. A transaction completed at, say, 55bp, would leave both institutions happy
given their divergent recovery rate views.
Reference asset
The underlying assets on whose price or status a credit derivative is based. Reference assets can
take almost any form including government bonds, Brady bonds, Eurobonds, MTNs, loans,.letters
of credit, trade receivables, receivables due under derivative agreements, CBOs, mortgages,
indices and funds.
Sovereign risk guarantee [option]
An option that gives the holder the right to demand delivery of a pre-determined amount of hard
currency to a bank account in a country with no exchange controls in exchange for the local
currency of a particular sovereign at the prevailing exchange rate in the event of a pre-determined
credit event. This event is usually defined in terms of foreign exchange controls or nonconvertibility.
Sovereign risk guarantees are used to hedge a counterparty’s business risk in a country rather than
investment risk in a security. Countries can suspend foreign-exchange convertibility without
defaulting on any specifiable reference asset. This makes credit swaps imperfect hedges against
many forms of sovereign risk. Even though the imposition of currency controls would likely lower
any sovereign borrower’s credit standing in investors’ eyes, which would increase the value of a
credit swap, the basis risk would be too large to make the hedge usable. The sovereign risk
guarantee lets the protection buyer transfer money out of a country even if it has declared nonconvertibility or non-transferability.
The protection seller may simply be an institution that wishes to be paid a premium for assuming
non-convertibility risk. However they can also be entities such as local banks or local or foreign
corporates who need local currency regardless of convertibility. In particular, corporations that
have entered into joint venture agreements that specify local investment at specified future dates
could write sovereign risk options against an underlying position to gain a premium.
The options are priced off the sovereign's outstanding hard currency obligations of the same tenor
as this is the investment alternative and carries equivalent risks. The option will be more
expensive because it is illiquid. Transaction sizes typically range between USD5 million and
USD20 million with tenors up to 12 months. These products are almost always embedded in shortdated credit-linked limited recourse notes. These are denominated in hard currency but pay out in
local currency if specified credit events occur (see chapter 16).
Structured repo
Flexible repo agreements that create financing and investment vehicles that are more efficient than
those available in the traditional repo market. Investors can use cancellable repos and convertible
repos to set initial financing rates well below those available on the plain vanilla market. They can
also achieve better risk-adjusted returns by using reverse structured repos to hedge credit risk on
emerging market bonds.
Example
An investor can achieve collateralized above-market, double-A rated returns using a reverse
structured repo to finance a single-B rated Ecuadorian sovereign bond for a double-A rated bank
counterparty. The investor pays cash for the bond at the trade date and at maturity it resells the
bond to the bank at par. In the interim the investor holds the bond on its balance sheet for a year
and earns superior risk-adjusted returns by exchanging the credit risk of the single-B rated bond
for the double-A rated counterparty risk of the bank counterparty.
Swap guarantee
A credit swap whose notional principal is adjusted in line with the mark-to-market value of a
reference swap (or other market sensitive instrument) and with a contingent payment triggered by
default on that swap. The main use of these products is to manage credit risk in cross-currency
swap portfolios.
The problem swap counterparties face is that the projected exposure on a swap can vary
enormously throughout the life of a swap. Take a 10-year $100 million notional yen/dollar
currency swap with principal exchange initiated in May 1990. At inception, prevailing rates
implied a maximum exposure at maturity of $125 million. five years later, as the yen strengthened
and interest rates dropped, that maximum exposure had risen to $220 million. Three years’ later
and it has fallen again by almost $100 million. This exposure is exacerbated if the counterparty’s
credit quality is correlated to the level of the yen and/or to interest rate movements. A lender in a
currency that has experienced depreciation and rising interest rates will be out of the money on the
swap and could be a weaker credit.
The swap guarantee hedges the exposure between margin calls on collateral posting. Alternatively
it can be structured to cover any loss beyond a pre-agreed amount. The buyer will only incur
default losses if the swap counterparty and the protection seller default. This means that the credit
quality of the buyer’s position is compounded to a level better than the quality of either of its
individual counterparties.
Switch[able] asset swap
An asset swap in which the buyer sells a counterparty (usually the dealer from whom the asset
swap was bought) the right to substitute the underlying asset with another asset (or any one of a
basket of pre-specified assets) on particular dates usually at a spread higher than the prevailing
market level. The embedded option is a spread option between the substitutable securities.
At the initiation of the transaction all the underlying assets will be of the same duration and credit
quality and so ought to move together in price with changes in interest rates. The main reason for
any spread change is therefore due to a change in perceptions of credit quality. The counterparty
exercises his option to substitute if one of the basket declines in credit standing versus the asset
that is asset swapped at the beginning of the transaction. These swaps are used to enhance the
yield of vanilla asset swaps. Also known as a basket asset swap. See basket credit spread option.
Example
An investor with a funding cost of Libor+10bp wants to buy an asset swap trading at Libor+10bp.
It sells an option under which it agrees to deliver the Libor+10bp asset to the counterparty in
return for another asset. This asset is either the cheapest to deliver from a basket or is a prespecified security. In the latter case, the counterparty can deliver the new asset at a spread 15bp
worse than the prevailing market level. The premium received for the sale of this option is 10bp.
The investor receives enhanced yield in return for the credit risk on the new asset. The option
buyer has hedged for less than the cost of a standard default swap.
Total return swap
An agreement that allows a counterparty to transfer the total economic risk both credit risk and
market risk of an underlying asset or portfolio of assets to another counterparty without
transferring the asset itself. (A default swap transfers only credit risk to the counterparty.) The
holder of an asset wishing to hedge exposure to that asset pays the total return (all interim cash
flows and any change in positive mark-to-market value) of a reference asset and (usually) receives
Libor plus a fixed spread and any capital depreciation from the other counterparty. Price
appreciation or depreciation may be calculated and exchanged at maturity or on an interim basis.
Total return swaps can be cash settled on the basis of a final mark-to-market value of through
physical delivery at maturity in which case the total return receiver pays the previous mark-tomarket price of the asset or portfolio in exchange for the actual securities. Eligible assets are those
with liquidity and observable prices.
Pricing depends on a number of factors: the cost of funds for the short counterparty; balance sheet
constraints the short counterparty may have the assets on its balance sheet with the associated
capital charges; credit risk both the riskiness of the asset as well as that of the long counterparty
are considered. Several methods can be employed to mitigate the risk of the transaction including
triggered collateral accounts, up front buffer collateral accounts and daily mark-to-market.
As with the credit default swap there is no exchange of principal and ownership and funding of the
underlying asset is unchanged. If the swap maturity matches that of a reference loan or bond asset
it is simply a synthetic version of the bond or loan that allows the holder to go long or short easily
and without funding. If it is not, then the swap is a synthetic asset that may not exist in the market.
Total return swaps can be structured on all the same asset classes as other derivatives. They are
used by investors to go long or short securities or instruments that are difficult to obtain, that they
cannot purchase in the form in which they are traded in the cash market or whose form is too
complex for them to analyze. They also offer unfunded so leveraged exposure to the underlying.
Total return receivers lock in term financing rates and total return payers effectively sell an asset
without ownership having changed hands. This makes them analogous to repurchase agreement
and some banks characterize certain types of repos for example those on emerging market debt as
credit-linked total return swaps. The main difference is that in a total return swap the underlying
securities are not exchanged whereas in a repo transaction they are. This transfer of economic
benefit without effecting a legal sale of the underlying asset can be advantageous for capital gains
tax reasons. (See structured repo.)
Total return swaps are also used by banks to keep assets off their balance sheets. Instead of
borrowing to fund the purchase of assets, a bank that wants exposure to a particular asset enters
into a total return swap with a counterparty (sometimes called the warehouse or balance sheet
provider) willing to buy the bonds. The balance sheet provider a bank or large corporate
effectively funds the position of the investing bank which then pays the funding costs to them and
receives a spread. If the originator of the transaction is itself balance sheet constrained, such as an
investment bank, then it will buy the underlying bonds and then repo them out to the final balance
sheet provider.
Example
An investor seeks exposure to a loan yielding libor + 350 basis points that a double-A rated bank
has made to a Turkish corporation. The investor enters into a total return swap with the bank the
total return payer under the terms of which it pays three-month Libor +100 basis points and
receives Libor + 350 basis points plus or minus any change in the market price of the loan. If the
loan’s market value remains unchanged over the life of the contract, then the investor earns 250 bp
on the transaction. However the investor bears both the price risk and any early repayment risk on
the loan over the life of the agreement.
Derivative-linked securities
Key Concepts
Any of the derivative instruments described in previous sections can be combined with a bond or note to
create a structured asset whose performance mimics that of the derivative itself. Although the derivatives
are sometimes said to have been 'embedded' (hence the alternative name 'embeddo' for these bonds) in the
fixed-income instrument it is more accurate to say that the buyer of the structured asset has purchased a
combination of derivatives in addition to a floating-rate, fixed-rate or zero coupon bond. The result is an
instrument whose coupon and/or principal payments create cashflows similar to those created by the
purchase of naked derivative instruments.
These securities are sometimes divided into two classes. First generation structured assets generally share
the following characteristics. They contain only one floating rate index; the maturity of the floating rate
index coincides with the reset and payment frequency - e.g. three-month Libor coupons must be reset and
paid quarterly; the floating-rate index is of the same country as the currency of denomination (no
quantization); and no exotic options are embedded. So in general, the index rate and discount rate of these
notes are equal to each other and to the to-maturity Treasury rate. Second generation structured assets
incorporate design complexity in addition to embedded options. These include notes containing index
maturity to reset frequency mismatch (such as a CMT FRN where coupons are linked to 10-year Treasury
rates but are reset and paid on a quarterly basis); notes that pay a coupon based on the differential or sum of
a number of indices; notes that include embedded exotic options; notes incorporating quantization; notes
where either the principal or coupons are linked to formulae and cross-category bonds whose performance
is linked to the relative performance of two or more different asset classes.
Structured assets can be created to give tailored exposure to any asset class on which there are derivative
instruments. Principal {coupon} Indexed Note is a generic phrase for any structured note whose redemption
{coupon} is linked to the performance of some asset or index. The examples below are just a few of the
combinations of fixed-income instrument, derivative instrument and asset class that can be tailored to meet
investors' requirements.
Definitions
CALLABLE FIXED FLOATER
An FRN that initially pays the investor an above market yield for a short non-call period and then,
if not called, steps-up to a higher coupon rate. If the bond is not called, the stepped-up coupon is
below prevailing market rates (if not the bond will have been called). The investor initially
receives a higher yield as he has implicitly sold a receiver swaption.
More complex versions, multi step-up callable bonds, have coupons that step up over their lives
and are callable on each step-up date. Investors have effectively sold the issuer a Bermudan call
option on the note for which they receive the higher coupon. The bonds are usually swapped into
floating rate Libor in which case the counterparty (paying the multi-step-up coupon in exchange
for Libor less a spread) holds a Bermudan receiver swaption (to receive the multi-step-up coupon)
which cancels the swap on exercise. The call on the note is triggered by the swap counterparty
calling the swap. Also known sometimes as a step-up bond.
CAPPED (floored, collared) FLOATING RATE NOTE
A capped FRN is a note whose maximum coupon is capped but paid at a premium over Libor
versus vanilla FRN coupons. Investors have bought an FRN and sold a strip of caps. The premium
from these sold options is monetized in the form of the higher interest rate spread over Libor.
Similarly, collared FRNs and floored FRNs are available.
The collared note has a minimum and maximum coupon and contains two embedded options. The
issuer {investor} is effectively long {short} a cap and short {long} a floor. In a falling rate
environment these notes outperform significantly since the cap is unlikely to be hit (so the investor
stands to keep the premium) and the floor is gaining in value. Conversely, it underperforms
significantly in a rising rate environment. This combination gives collared FRNs surprisingly high
duration compared with vanilla or floored FRNs.
The floored note has a minimum coupon giving investors protection against lower short-term rates
but, since they must pay for the floor, this is at the cost of a spread to Libor (or some other index)
below that on equivalent FRNs. The risk for investors is that rising rates push the floors out-ofthe-money, leaving them with a note that underperforms vanilla FRNs. They perform best in
environments in which the yield curve is steeply positive but in which rates fall. This makes the
floors cheap to buy at the outset and then means that they move handsomely into the money. More
complex variants can be constructed.
CLIQUET NOTE/BOND
A note or bond where the coupon each year is determined by the performance of some reference
asset or security in that year, subject to a minimum of zero. The bondholder therefore holds a strip
of options on which each of the strikes sets on a forward basis, most usually as a percentage of
spot on a given future date, rather than being fixed today. To determine the coupon, the spot rate at
the beginning of the coupon period would be compared with that prevailing at the end. The
investor would receive the option payout (or not) in the form of interest, and the option strike for
the subsequent period is fixed. Also known as forward starter/setting bond.
CORRIDOR [floating-rate] NOTE
Corridor options can be embedded in notes to create corridor FRNs (the investor is effectively
long an FRN and short the digital options). In any note or FRN of this genre, the principal and/or
coupons are linked to the number of or percentage of days during a specified period of time during
which some reference asset (e.g. Libor, a spread such as 10-year CMT rates less Libor or an
exchange rate) is above, below or in between a range.
There are two basic categories resurrecting and extinguishing (the latter is also called a [binary]
corridor or accrual note). With resurrecting notes, trading outside the range does not prejudice the
investor's ability to accrue further coupon if the reference asset subsequently trades back inside the
range. With extinguishing notes, no further coupon may be earned.
A common example of the former would be a note that in a market interest rate environment of
5%, paid a coupon of 10% x (n/N) where n is the number of days spot USD/DEM trades between
1.70 and 1.90, and N is the total number of days that it could potentially have done so (n/N is
therefore the percentage of days that it did.) With an extinguishing note, once the reference rate
trades at or outside the range, the day count is frozen and cannot subsequently increase.
The basic corridor FRN can then be further altered by overlaying strategies seen in other
structured assets. They can be principal guaranteed; instead of either accruing coupon or nothing,
notes can have two fixed coupons, one high coupon payable if the underlying trades within the
range and one low coupon payable if it does not (fixed accrual note); notes can be structured with
a re-settable range (re-settable corridor note), for example so that investors start each period in the
middle of the range; the range structure can be quit (quittable corridor note); investors themselves
can specify the range; the note can be structured so that accrual only occurs if both the upper and
lower range values are hit (limit range FRN) and so on. Investors in these notes are taking both a
directional view as well as a volatility view. They also face unusual price behaviour. For example,
if the underlying has traded through the lower boundary, the note will actually gain in value if spot
or the forward rate then re-approaches the lower boundary, while the same rise just below the
upper boundary will have the opposite effect. The note will begin to lose value even before the
forward price crosses this upper limit as the digital option, which the investor is short, has a non-
zero delta and so increases in value as it reaches the strike. Digital options also have larger vegas
than other options which increases the volatility risk of accrual notes. In high rate environments in
which the yield curve is steep, the notes rapidly approach the value and duration of a zero coupon
instrument.
The popularity of these structures has led to a proliferation of names including memory note,
accrual note, fairway note, range (floating-rate) note. Interest rate versions are sometimes known
as Libor enhancement accrual notes or LEANS.
Example 1
A US dollar investor might want to improve upon the current one-year dollar deposit rate of
5.90% by taking a view on EUR/USD exchange rates. For the right to enhance the yield if his rate
view is correct, he is willing to accept a lower minimum yield. So, part of the 5.90% depo rate is
used to buy an FX corridor option with a corridor range of 1.0250 to 1.0650 and a pay-out ratio of
1 to 2.75 (that is, the maximum payout of the option is 2.75 times the premium invested. The
option costs 4.00% of the amount invested, so if the spot rate traded outside the range for the
entire year, the investor would receive no additional payment and would receive a yield of just
1.90% (5.90% - 4.00%). The return on the note is calculated as yield = 1.90%($) + (no. of days
spot fixes in the range x fixed multiple of premium)/total business days in option period. In this
case, the minimum payout is 1.90% and the maximum is 12.90%. The binary version, given the
greater likelihood of a low payout is cheaper and has a higher maximum payout. A range binary
option with the same boundaries as the FX corridor option above would cost less (2.00% of the
amount invested) and its payout ratio would be much higher (1 to 10). This gives a minimum
3.90% return and a maximum 23.90%.
Example 2
A two-year Libor enhancement yield note might pay Libor + 100 bp with interest accruing only on
days when three-month Libor is between 3% and 4% in the first six months, 3.125% to 4.75% in
the next six months, 3.25% to 5.5% in the third six months and 3.5% to 6% in the last six months.
Assuming 360-day convention, so that each semi-annual period has 122 days, the investor has sold
the following package of binary or digital options for the first six month period 244 binary options
on three-month Libor - 122 calls with a strike of 4% and 122 puts with a strike of 3%; for the next
period, the same quantities of call and puts but with respective strikes of 4.75% and 3.125%, and
so on. Every day, one call/put combination is either exercised or expires. If on any day threemonth Libor is high enough for the calls to be exercised, then the purchaser of the range note - the
seller of the binary option - effectively pays the holder of the binary call ((3-month Libor + 100
bp)/360) x principal amount of bond. Likewise, on any day, if three-month Libor is low enough
that the binary put is exercised, then the purchaser of the note effectively pays the buyer of the put
the same amount. Hence the spread to Libor payable by such notes - in this case 100 bp - is
determined by the level of premium obtained for the options. This will be determined by the width
of the range (the broader it is, the less likely the options are to be exercised and so the less
premium they will command) and the volatility of Libor (the higher it is the higher the premium
for the options.)
Credit-linked note (CLN)
A bond whose coupon payment and/or principal redemption is linked to the credit status of an
underlying reference asset. In the simplest form, a CLN consists of a bond issued by a highly-rated
borrower packaged with a credit default swap on a less creditworthy risk. Usually the issuers are
banks that wish to sell exposure to a particular emerging country or a particular borrower by
issuing a bond linked to that country's or borrower's default. If default occurs, the investor receives
a educed payout (and in some cases nothing). Because the embedded credit default swap can
create synthetic maturities of the underlying reference asset, and because it can be combined with
currency-swaps to alter the currency of the credit-linked note's coupon and principal payments, the
buyer of the note gains access to an investment that is either difficult to find or that does not exist
in the cash market.
The structure of the CLN depends on the the underlying credit default swap. Usually the swap is
cash-settled according to the recovery value of the relevant reference securities. So, if the
reference asset defaults and its recovery value is 60%, then the CLN will pay accrued interest from
the last payment date until (say) five days before the credit event notice day plus 60% of par (or
instead of cash settlement may deliver the physical). The more aggressive versions of the CLN
incorporate a digital credit swap which pays a predetermined fixed amount or pre-determined
percentage of notional principal. This creates CLNs with fixed payouts in the event of default in
the reference asset. The most aggressive variety, known as the zero/one CLN (because it is
digital), pays out zero in the event of default. Many other coupon/principal combinations are
possible such as example 1.
Example
An Italian investor wishes to increase exposure to Brazil but is not allowed to purchase Brady
bonds. A bank issues a Euro-denominated note linked to the Republic of Brazil's $4,331,319,000
EI floating-rate notes due April 15 2006. If there is a credit event, the Euro notes pay a 5.95% per
annum coupon accrued until the credit event with redemption at par. If there is no credit event
before maturity then the interest paid is zero but redemption is at 156.9% of par. The Euro notes
maturity matches that of the reference asset but is rated single-A against the reference asset's
rating of B1/BB-.
Example
An investor is looking to add to his Czech Republic exposure and would like to purchase SPT
Telecom's five-year 5.125% Euro note. However no paper is available. At the same time, a
number of western banks are looking to buy default protection on the Czech Republic and default
swap levels look attractive to cash. The investor must buy funded exposures and cannot enter
directly into derivative contracts. A four-year 5.125% CLN linked to default on the Czech
Republic, collateralized by World Bank Eurobonds can be structured that is 10bp cheap to SPT
Telecom despite being one-year shorter and a better credit.
The basic structure is sometimes known as a two-name basket credit note, but CLNs can be
constructed with more than one reference asset. Basket CLNs'. principal and/or coupon payments
depend on there being sufficient non-defaulted assets in a portfolio of reference assets to pay them
and so offer investors risk mitigation through diversification. First to default basket notes increase
risk (and return) as it only takes a credit event in one of the reference securities/counterparties to
trigger redemption of the CLNs. CLNs can also be structured in which the issuer has the right to
substitute different securities for the reference assets effectively securitizing a switch asset swap.
Basket CLNs are usually issued by banks optimizing regulatory capital usage. They are complex
in structure because to obtain ratings the underlying exposures have either to be individually
identified, which is not optimal for those borrowers or counterparties, or they have to be
aggregated and tailored in a way that suits both the buyer of credit protection and the ratings
agency.
Example 3
A bank wishes to free up credit lines to European bank counterparties. It issues a note that pays
six-month Libor plus a margin, where the margin is 0.875% minus 20bp for each merger between
two reference entities subject to a minimum margin of 0.40%. Interest payments will stop if a
credit event takes place. Redemption is 100% of the nominal amount unless there is a credit event,
in which case notes are redeemed 15 business days after the credit event takes place. The
redemption amount in principal x a reference price which is the best available bid, 10 business
days after the credit event, for a predetermined amount of the reference securities that triggered the
credit event. A credit event is triggered if any of five reference entities fails to make timely
payments in respect of any of their obligations. The investor is taking the risk of default of any of
the institutions in return for an attractive margin over Libor. This kind of note can also be
structured with reference to specific securities.
Leveraged CLNs have also been popular. These are created simply by increasing the notional
principal of the underlying default swap above the principal value of the CLN. Some of these
notes (and all CLNs and repackagings can incorporate this feature) incorporate a trigger spread. If
the reference assets fall below a certain price - usually at the point at which the investor will lose
more than the initial investment - the whole trade is unwound and the investor receives the
recovery value of the note.
Example
An investor believes that Russian debt will recover from current levels. He buys a two-year
leveraged CLN callable after one year that, for an initial investment of $30 million gives him
exposure to $60 million of Russian risk. Yields are high, but if the reference assets default, and the
recovery rate is lower than 50%, the investor will lose more than the initial investment.
And credit-linked notes can be constructed that securitize sovereign risk guarantees rather than
credit default swaps. These notes offer investors increased yield in return for taking on the risk
that there will be a non-convertibility event in the currency in which the reference assets are
denominated.
Example
A Portuguese investor believes that the market's perception of Brazilian risk is exaggerated. He
buys a convertibility-linked note whose underlying assets are real-denominated assets of the
Republic of Brazil. If the real remains freely convertible, the investor receives three-month Lisbor
+ 130bp and 100% of the notes' principal in escudos. If there is a convertibility event as defined in
the term sheets, then the notes will redeem and the issuer will have the right to deliver any of the
reference obligations in a face value equivalent to the notional amount multiplied by the last
exchange rate, or the same amount of the now unconvertible reals. The investor also accepts the
risk of tax and regulatory changes.
See chapter 15 for descriptions of the derivatives that underlie these structures. Also see total
return [index] notes.
CURRENCY PROTECTEd NOTE
A bond whose coupon and/or principal redemption is denominated in one currency and indexed to
a reference index in another. For example a US dollar FRN paying an investor Deutschmark Libor
in dollars plus a fixed spread. Such structures are said to be quantized and are also known as
differential notes and diff floaters because they incorporate differential swaps/options (not to be
confused with notes such as the CMT-Libor differential note, so-called because they offer a play
on the spread between two reference indices).
Quantization can be applied to almost any structured asset. So a quantized inverse floater might
pay 9.11% minus six-month sterling Libor paid in US dollars to provide a currency risk averse US
investor with a bullish interest rate play on sterling. A leveraged diff floater is just the combination
of a leveraged FRN with a differential swap and might pay 2 x (three-month Euribor minus threemonth dollar Libor) minus a spread.
DE-LEVERAGED FRNS
Floating-rate notes whose coupon is determined by formulae in which the reference index is
multiplied by less than one. They underperform when the reference index rises and investors can
be paid for taking this risk with high floors or high initial coupons. An example is the step up
recovery FRN (SURF), a de-leveraged CMT FRN with a coupon floor that steps up over time. A
five-year dollar Surf might pay 0.5 x (10-year CMT) + 1.50% subject to a coupon floor of 4.50%.
The floor is higher than yields on benchmark Libor FRNs. The investor is short T-bonds and long
an in-the-money T-bond call option.
DUAL CURRENCY NOTE/BOND
The combination of a fixed-rate bullet-repayment bond and a long-dated forward or option
contract to create bonds with the principal denominated in one currency and interest payments in
another. Either the investor or issuer could have the option to repay a fixed amount of either the
first or second currency, based upon the interest payments, the principal, or both. If the investor
has the option, the note will tend to have a below-market coupon and vice-versa.
An example is a USD note issued at par in either USD or EUR at the issuer's choice, and where
the coupons are in fixed USD or EUR at the investor's choice. The exchange rate for converting
USD amounts into EUR amounts is set at, say spot on the day the note is launched. There are a
number of variants the tri-currency note/bond (also known as tri-colour note/bond) in which the
principal or coupon can be paid in any one of three pre-specified currencies instead of only two
e.g. a USD bond with issue and redemption at par and an annual coupon of 3% in JPY or 7% in
USD or 8% in GBP at the issuer's choice; and the reverse dual currency bond - a dual currency
bond in which the currencies that can be chosen for principal and interest are different e.g. a USD
bond where coupons can be in USD or EUR and the principal can be in USD or JPY.
EXTENDIBLE FLOATER
An FRN, usually one-year, extendible to two or three years at the issuer's option. For example, if
the two-year swap rate was trading at 5.65% and the forward curve was implying rates of 7.5%, an
investor who wished to take a view against the pessimistic forward curve could buy a one-year
extendible FRN under which he receives six-month Libor plus 50 bp for the first year. Then, if the
two-year swap rate at the end of that first year is higher than 7.5%, the note will be extended a
further two years and the investor's coupon would be fixed at 7.5%. Effectively the investor is
selling a one-year option on the two-year swap rate at the forward rate (7.5%). If the investor had
instead bought one-year paper paying Libor flat, the fixed reinvestment rate at the end of the year
would have to be higher than 8% to outperform the extendible. See also index amortizing rate
note.
INDEX AMORTIZING NOTE
A combination of a fixed-income instrument with a swap that mimics the performance of
mortgage-backed products by amortizing according to a pre-set quarterly schedule that is linked to
the level of a specific index, usually Libor or the PSA. As interest rates increase (or prepayment
rates decrease) the slower the notes amortize, and so the longer their average life.
In this respect they behave like collateralized mortgage obligations (CMOs).
In some cases the amortization is all or nothing. So after the first year the notes will be called in
their entirety if Libor has not risen by, say, more than 100 bp from the current level. The notes'
coupon is fixed, and for the first year there can be no amortization. The coupon is set significantly
higher than the prevailing yield on one-year notes. The notes are attractive if the yield curve is
steeply positive and if the future sharp rate rises predicted by the curve do not occur. The note will
then amortize more quickly than their initial pricing took into account (or they will be called after
the lockout period) and the yield will be higher than on a vanilla FRN of the same life. The risk is
that rates do rise quickly and the notes' average life extends, leaving the investor with a coupon
fixed at levels that become more unattractive with every rate rise.
Interest-rate DIFFERENTIAL NOTES
An interest-rate differential note pays the investor the difference between two different interest
rate indices. So, a CMT-Libor differential note pays the difference between the CMT rate and a
short-term Libor index. A note might have a three-year maturity and pay 5.00% for the first year
and thereafter pay the 10-year CMT rate less three-month Libor plus 1.60% reset quarterly with a
minimum coupon of zero. This could give investors higher spot floating-rate yields than are
possible with either the CMT FRN or vanilla FRNs. As with other CMT-linked notes, the main
risk is that yield curve flattening will erode this advantage as the CMT-Libor spread decreases.
The investor is effectively long a CMT FRN and long Eurodollar futures. Any indices can be used
in place of CMT rates and Libor.
INVERSE/REVERSE PRINCIPAL {COUPON} INDEXED NOTE
Notes whose principal redemption {coupon payments} fall if a reference index/asset/spread rises
and vice versa. The commonest example is the Inverse/reverse floating rate note, an FRN whose
coupon rises (falls) as a floating reference rate falls (rises). A typical coupon is calculated as a
fixed coupon minus the floating reference index e.g. 7.5% minus three-month Euribor. Such notes
combine an FRN and an interest rate swap of twice the notional size. Libor resets in a rising rate
environment will cause the bond to fall in value, not to reset its value to par and so investors
usually also purchase a cap whose strike is set at the level of Libor that will produce a zero coupon
that is, it is struck at the fixed rate element. If rates rise beyond this strike, producing negative
coupons, then the long cap makes up the difference back to zero so coupons cannot become
negative. The notes suit investors who want a high initial yield in an upwardly sloped yield curve
and to benefit if rates fall.
There are many variants of the structure. The step-up inverse floater features a step-up constant
(7.5% minus three-month Libor for the first six months, 8.5% minus three-month Libor for the
second six months and so on) and also a fixed above-market first coupon followed by the inverse
formula.
The principle of inverse performance can be applied to any asset class. For example, investors
have used reverse principal exchange rate linked securities to take views on foreign exchange
rates. These are currency-indexed notes whose principal repayment varies inversely with the value
of a currency versus the repayment currency for example, a note repaid in dollars that varies in
value according to the yen/dollar rate. Spread versions, known as dual index inverse floating-rate
notes, are also available. These are FRNs whose coupon or principal redemption rises as the
spread between (or the average of) two rates falls. For example, an investor might believe that
Swiss and Euro short-rates are unsustainably high. If so, they could purchase a note that redeemed
at par + 10 x (8.70% - Average CHF/EUR 3-year swap rate). This version is leveraged 10 times:
every 1 bp change in the average results in a 10 bp change in redemption value. A variant, the
stepped dual index floater, pays a fixed first coupon before reverting to the dual index formula.
LEVERAGED FLOATER/SUPERFLOATER
A floating-rate note whose coupon rises as the reference index which can be in any asset class
rises but in a ratio greater than 1. For example, a leveraged FRN might pay a coupon of 2 x (sixmonth Libor) - 7.09%. This would be the mirror image of an inverse floater. The coupon is floored
at zero. The investor is effectively long an FRN of twice the notional principal of the leveraged
floater and short a fixed-rate bond whose coupon is the fixed-rate element of the formula. The
investor is also usually long floors to prevent negative coupons.
Superfloater characteristics are useful to corporations when structured as superfloater liabilities the combination of a floating rate liability (that is, borrowing floating) and receiving floating in an
interest rate swap of the same notional principal. This structure creates a net coupon of two times
Libor less the fixed swap coupon. It might be used by a corporate with an existing liability whose
profits fall when interest rates fall, but whose profit losses are greater than any benefits of lower
debt funding costs, perhaps because it has little debt.
One extreme example of a leveraged FRN is the power note. This bond or note pays a coupon that
is linked to a power of the underlying index. For example, a coupon might be equal to 25.00%
minus (3-month Libor)2 with a floor at zero. Investors in these notes want extremely high returns
over a short period of time and in return accept extremely large duration and negative convexity.
The equivalent investor position is long a fixed-rate note, short a highly leveraged (and changing)
amount of FRN, long a highly leveraged (and changing) amount of out-of-the-money interest rate
caps.
Leverage can be applied to any structure. Leveraged inverse floaters incorporate swaps of more
than twice the notional size of the bond to create coupons such as 12.5% - [2 x (Libor-6%)]. In this
case, for every 1% fall in Libor, the coupon of the bond rises by 2% and for every 1% rise in Libor
the coupon falls 2%. Since a standard inverse floater is created from an FRN combined with a
swap of twice its notional principal, this structure uses a swap of three times the notional principal.
The leveraged capped floater behaves like a normal FRN when Libor (or another index) is below a
predetermined strike. Once the index rises above that strike, the note behaves like a leveraged
inverse floating rate note. So the schematic coupon (where m is the leverage factor) is Libor + a if
Libor < strike; B - m x Libor if Libor > strike. The equivalent investor position is long an FRN,
short (m + 1) Libor caps at the initial strike, and long (m) Libor cap options at the higher strike. A
typical actual formula might be the minimum of Libor + 30bp and 28.45% less (3 x Libor).
These instruments work best in a steeply positive yield curve environment in that case the caps are
priced off the even steeper implied forward curve (and so raise a large amount in premium) and
Libor has to move significantly above implied forward rates before the investor's guaranteed
pickup is threatened by the formula.
Leveraged structured assets are also known as duration enhanced notes as leveraged formulae tend
to create high synthetic positive or negative duration.
PRINCIPAL {coupon} INDEXED NOTE
The generic name for any structured asset whose redemption {coupon} is linked to the
performance of one or more assets or reference indices. For example, a currency indexed note's
performance is linked to the performance of one or more foreign exchange rates. A typical note
might pay no coupon but have redemption formula of 100% x [1 + 150% x (FX - 1.0250)/FX]
where FX is the EUR/USD exchange rate at maturity, perhaps with some minimum and maximum
redemption levels. Some notes have a further barrier-like condition that would trigger a
predetermined redemption level (say 110%) if a particular level were breached at any time by the
underlying spot FX rate. Any other asset class may be substituted for foreign exchange and the
formula for principal repayment can reflect a long, short or more complex forward or optionrelated position in the index. The performance of the reference asset(s) can also be linked to
coupon payments, leaving principal redemption unchanged/guaranteed (see below) or can affect
both coupon and principal.
PRINCIPAL PROTECTED/GUARANTEED NOTES
A fixed-income instrument that offers investors the guaranteed redemption of 100% of the
principal plus some, all or a multiple of the rise in value of a particular underlying asset, e.g. a
stock index. They are most easily constructed by the purchase of a zero coupon bond whose
maturity and nominal value matches that of the capital guaranteed instrument. The difference
between the price of the zero and its nominal amount is used to buy options on the desired
underlying. The amount of participation in the underlying asset depends on the cash available to
buy calls. Falling interest rates push up the cost of the zero coupon bond. This affects the level of
gearing that can be offered. Each bank has a name for its own offerings. For example: equitylinked notes (ELNs), protected equity notes (PENs), index growth-linked units (IGLUs), protected
equity participations (PEPs), protected index participations (PIPs), equity participation notes
(EPNs). Non-equity names include guaranteed return index participations (GRIPs) and guaranteed
return on investment units (GROIs).
RATCHET FLOATER
An FRN combined with a ratchet option that pays a high floating coupon subject to a condition
that the coupon cannot rise by more than a fixed amount from the previous coupon level nor fall
below the previous coupon.The note has a high first coupon of, say, 4.655%. The investor is
effectively long an FRN, short a path-dependent or periodic cap and long a path-dependent or
periodic floor.
In a steep forward curve environment in which it is implied that rates will rise 2% or 3% in the
next year, a cap that protects its holder against any rate rise of more than 25 bp in a quarterly
period will be expensive (though not as expensive as a vanilla cap) and so the investors in this
ratchet floater will be well compensated for selling it. Equally, the floors the investor is buying
will be cheap. This combination allows the note to offer a high coupon floor when although the
forward curve is implying sharp rate rises, the investor believes rates may fall or rise much more
slowly. The note will severely underperform if rates do rise (particularly for variants of this
structure where the coupons are made more generous by the inclusion of a knock-out feature so
that if rates rise above the knock-out level the note only pays a nominal coupon). Also known as a
one-way floater, one-way collared floater and sticky floater.
REpackaged securities/repackagings
The generic name for securitized asset swaps. The simplest versions are securities issued by a
special purpose vehicle that has purchased the underlying bonds and swaps. The asset swaps
generate the coupon and principal payments that are passed to investors via the newly issued
securities. The use of swaps and other derivatives allows the cashflows of complex securities to be
transformed into currencies and structures acceptable to mainstream investors.
Overcollateralization of the SPV means that repackaged securities can obtain much higher ratings
than the original securities.
Repackagings are used instead of asset swaps for a number of reasons. First, any investors cannot
hold derivative instruments directly. Second, asset swaps tie up swap credit lines, expose the buyer
directly to the credit of the counterparty, are not easily transferable and can cause accounting
difficulties if the asset is trading above par. Because the special purpose vehicle is issuing new
securities, it can use a combination of derivatives and structural mechanisms to transform loans
into bonds, unrated notes into rated notes, premium into par or discount notes and so on without
utilizing investors' swap lines.
Most repackagings incorporate price triggers: if the value of the collateral in the SPV falls below a
certain point, the vehicle is unwound and the investors repaid according to the remaining value of
the assets and derivatives. In these cases, as with asset swap terminations, the derivative
counterparty has a senior claim on the collateral. As with credit derivatives, the investor is
exposed both to the credit risk of the underlying assets and the swap counterparty. The
incorporation of the unwind trigger creates a similarity between the basic repackaged note and the
credit-linked note since an asset swap with an unwind trigger is almost a default swap and a creditlinked note is simply a securitized default swap. The key difference is that the default swap
requires an explicit credit event to trigger a credit-related loss to one counterparty while the asset
swap requires a price level that may or may not be associated with an actual credit event.
Credit-linked repackaged note combine both a securitized asset swap and a default swap is a . In
one structure, a special purpose vehicle purchases a portfolio of assets and any required interest
rate and currency swaps, and enters into a credit default swap with a swap counterparty. If a predetermined credit event occurs in the SPV's portfolio, the trust settles the swap and investors in the
notes suffer to the extent that the assets of the portfolio are not sufficient to repay principal and
coupon in full. Investors obtain leverage to the extent that the face amount of the portfolio they
have exposure to exceeds the face amount of SPV notes. The amount of leverage that can be
obtained increases with the credit quality of the portfolio. The difference between this structure
and the standard repackaging is simply that the unwind event is explicitly a credit event.
Example
Italian investors are looking for higher-yielding assets than currently available EUR Eurobonds. A
bank purchases a portfolio of Argentine Brady bonds and asset swaps the floating USD coupon
payment into fixed-rate EUR. The principal exchange on the swap eliminates FX volatility on the
currency swap. At the same time the bank enters into a credit default swap with Argentine Brady
bonds as the reference assets. In return for an annual fee, the bank undertakes to accept the
recovery value of the bonds if a credit event takes place. This is equivalent to undertaking to make
a contingent payment to the swap counterparty of the principal value of the credit-linked notes
issued against the swap less the recovery value of the Brady bonds. Investors in the CLNs receive
a yield around 70bp higher than current Eurobonds.
RE-SETTABLE FRN
The generic name for the combination of an FRN and options to create a note on which the
coupons can be reset depending on the performance of some underlying reference asset. For
example, an investor's choice FRN is a note plus digital options that pay a conditional coupon.
Investors are asked to guess the level of Libor in the upcoming period. If their guess falls within a
predetermined range, then they receive an above market coupon. If it falls outside the range, they
receive nothing.
RAINBOW NOTE/BOND
A note incorporating a multi-factor option so that the performance of either the principal or
coupons is linked to the best performing of a number of assets. So an investor could buy a oneyear bond with redemption of 100% plus the best performance out of the S&P 500, 30-year
Treasury futures or EUR/USD, subject to a minimum redemption of par. Also known as best-of
bond or chooser bond.
TOTAL RETURN [INDEX] NOTES
A bond incorporating a total return swap so that, for example, its coupon consists of the total
return from a bond or equity index plus or minus a spread. As most investors cannot incur
negative coupons, any negative returns are usually rolled over and, if necessary, deducted from
principal at the end. Leveraged structures have been popular with investors - the diagram below
illustrates a two-year note embedded with a collateralized five-time leveraged total return swap.
The swap generates 200bp plus the price movement on a $50 million loan portfolio. The return to
the investor is 6.5% from the collateral plus 10% from the swap assuming no price movement.
However, the investor runs the risk of losing principal if the loans fall in value or default.
Digital options
Key concepts
A digital option, also known as a binary option or an all-or-nothing option, either pays out zero or a pre-set
amount. Unlike standard options therefore its payout is discontinuous: if the strike price is reached, a predetermined amount is payable no matter by how much the option is in the money. (In the following pages
‘binary’ can be substituted for ‘digital’ in any of the definitions.
The payout is expressed either as a multiple of the premium a payout of 1:2.5 would mean that the option
would pay 2.5 times the premium invested or as a percentage of premium to payout, in this example the
premium would be 40%. The conditions that determine a digital option’s payout are many: for example, a
spot rate trading or not trading at a certain level, a range maintained or broken, a level trading only after
another level trades or if that level has never traded. Like barrier options, digital options are difficult to
value and hedge because, around the one particular spot price, small moves in the underlying can have very
large effects on the value of the option because of the discontinuity in payout around that point.
Digital options can be combined to create products that enable the holders to express extremely precise
views on market movements. They are often embedded in fixed-income instruments to create innovative
structured assets. And basic digital puts and calls are a fundamental building block for many structured
products including installment options, range and timer options and structured forwards.
The three basic structures are the European digital option, the American digital option and the range digital
option. The other products are variations on these three themes.
Definitions
american digital option
An option that pays a fixed amount at expiry providing that the underlying has traded above (call)
or below (put) a pre-determined level at any time over the life of the option. If the underlying fails
to achieve this strike level then the option pays nothing. Also known as one-touch options (as it
only needs the underlying to reach the strike price/trade within the one-sided range once within the
life of the option for there to be a payout) and no-touch options (as if there is no-touch there is no
payout).
Asset-or-nothing digital option
Not a true digital option, these are options that pay out the value of the underlying at maturity if
spot trades above (call) or below (put) the strike levels. A slight variation is the digital gap option
that pays out an amount defined by the underlying asset price minus a constant providing that the
underlying has traded above or below the strike. The payout profile is very similar to that of
standard knock-in options whose knock-in level is in-the-money.
Digital barrier option
A digital option whose payout depends not only upon the digital strike level but also on whether
the spot price has reached a knock-out or knock-in barrier level. A knock-out digital call {put}
pays a fixed amount at expiry providing that the underlying is trading above {below} the strike at
maturity and that it has never hit a predetermined knock-out strike level at any time over the life of
the option.
There are two basic types:
i.
ii.
in one the strike is set at-the-money and the knock-out is set out-of-the-money relative to
the strike. If the underlying moves away from the strike to the knock-out the option is
extinguished. If it does not hit the knock-out and trades above the strike at expiry
(European) or has ever traded above it (American) then the option pays out. This is
known as an out-of-the-money knock-out digital option.
in the other, the strike and knock-out are set at the same level which is below the current
spot price. As long as the underlying never trades through the knock-out level the option
pays out a set amount. If it hits the knock-out at any time it is extinguished. This second
variety is essentially a range option with only one boundary. This is known as an in-themoney knock-out digital option and is (like the American digital) known as a no-touch
option though in this case because the underlying must not touch the knock-out level. A
knock-in digital option is an American-style digital whose payout is not made at expiry
but is automatically triggered as soon as the underlying asset price hits the barrier levels.
Some varieties require that the barrier has been hit more than once.
European/At-maturity digital option
An option that pays a fixed amount at expiry providing that the underlying is trading above (call)
or below (put) a pre-determined level at expiry. If the underlying is trading outside the level, then
the option pays nothing even if it has previously traded through the barrier. For example, a digital
cap {floor} pays a fixed amount if the interest rate moves above {below} the strike. These are
useful where rates are expected to move just far enough to trigger the option but not much further
and so where a conventional cap or floor would represent the purchase of unwanted protection.
Mandarin cylinder/collar
A collar that is restructured using an overlay of digital options. The underlying collar remains in
place but the holder buys a range binary option whose range matches that of the collar. As long as
the underlying trades within the boundaries of the collar the range binary option produces an
additional payout. If spot trades outside the range, the holder loses the premium paid for the
option. In the example illustrated, the holder has bought a GBP put struck at 1.45, has sold a GBP
call struck at 1.54 and has bought a GBP 1.45/1.54 range binary option. If the barriers are not hit,
then at maturity the holder receives 0.0620 GBP plus the value of the cylinder.
Range [digital] option
A digital option with more than one boundary condition. The basic version comes in European and
American varieties. The European-style range option, often known simply as a range option, pays
a fixed amount providing that spot is trading within a predetermined range at expiry. If spot is
trading outside the range then the option pays out nothing. The American-style or knock-out range
option only pays a fixed amount at expiry providing that spot has never traded outside the range
during the life of the option.
Options can be structured that pay out only if both boundaries have been touched (limit digital
option); if either boundary has been touched (double-one-touch option); if neither boundary has
been touched (double-no-touch option); or if the underlying either stays inside a predetermined
range, or hits both extremes of the range. The latter is sometimes known as a boundary [digital]
option and is the combination of a range option with a limit digital option.
All these options take views on trading ranges and thus volatility rather than any directional trend.
For example, the boundary binary option takes the view that volatility will either be low, or high
against no strong directional trend. They are embedded in structured assets to create various types
of accrual or range note.
Timing/timer option
An option whose payout is based on the amount of time the underlying has traded within a pre-set
range or above or below a specified barrier. The resurrecting/standard timing option accrues a
fixed proportion of a total amount for every period (typically one day) that the underlying
continues to trade within a predetermined range or band over the life of the option. The buyer
specifies the range and pays a premium upfront. The option’s maximum payout is specified in
advance as a multiple of the premium. Then, for every period in which the underlying trades
within the range, a portion of that maximum payout is locked in. No daily payment is made for the
time that the underlying trades outside the range. The final payout is calculated on a pro rata basis
and is payable on expiry. Also known as corridor options, memory options or accrual options.
The extinguishing/knock-out timing option is a standard timer option but with the additional
condition that if the underlying trades outside the range during any given period then the option
will stop accruing any further payment from that point on the option is knocked-out. The total
amount accrued prior to knock-out is payable at expiry.
Forwards
Key concepts
Vanilla forwards and forward rate agreements are over-the-counter derivative instruments that lock in a
guaranteed price for an underlying asset, such as a foreign exchange rate or an interest rate, for a prespecified amount at a pre-specified time in the future. They are typically executed with zero upfront cost
and for this reason are preferred by some counterparties, particularly corporate hedgers, to options.
However, zero upfront cost does not mean zero cost. Because they guarantee a price or rate, forward
transactions remove not only the risk that the underlying will move against the holder over the life of the
forward, they also remove any potential to benefit from advantageous moves in the underlying. They
therefore trade certainty for potential opportunity costs. They also remove the ability for a hedger to take
views on support/resistance levels or the timing and strength of any market moves. Options, though they
incur an upfront premium, allow the holder both to hedge and to benefit from any upside.
The flexibility of options has led to the development of hybrid combinations of forwards and options to
create structured forwards. These can be tailored to meet a wide variety of client needs and are used both
for hedging and for outright view-taking. The examples in this chapter illustrate the most important classes
of forward and structured forward – but any of these basic templates can be altered with additional strike
levels, barriers and reset features.
Definitions
Cancellation Forward
A standard forward which is automatically cancelled if a predetermined level is breached over the
life of the forward. Typically cancellation forwards enable the client to buy or sell forward at a
rate which is better than the forward outright for the same maturity, at the risk of losing the
forward if the cancellation level is touched, in which instance the client will suddenly find his
position unhedged.
Example
A client short EUR/USD wants to buy Euros over a six-month horizon. Spot is 1.0310 and the
forward is 1.0378. The client can buy at 1.0278 as long as the 1.1200 cancellation level is never
touched.
Conditional Forward
A structured forward which gives the client the right, but not the obligation, to buy or sell forward
on the maturity date of the conditional forward, providing a pre-determined trigger level is not
breached at any time during a pre-specified part of the life of the forward. If the trigger level is
breached then the client will be obliged to buy or sell forward at a rate which is worse than that
which would have been incurred if a normal forward outright was executed instead. Also known
as a forward extra.
Example
An investor is short EUR/USD over six months with spot at 1.0310 and the forward at 1.0378. The
investor could buy EUR at 1.0400 but this is a right which becomes an obligation only if the
trigger level of 1.0100 is ever breached, locking the investor into a rate of 1.0400. Exchange Rate
Agreement (ERA) A type of synthetic agreement for forward exchange (SAFE) settled on the
spread between two forward foreign exchange rates instead of with reference to the spot rate. See
non-deliverable forward.
FADING forward
Also known as an accrual forward, this is a synthetic forward where for each period that a predetermined fixing condition is established, a portion of the contract is locked in. It provides an
opportunity to deal at a rate significantly better than the forward outright rate but only for a
portion of the amount corresponding to the frequency that spot has fixed above {below} the
trigger level. The product is an alternative for those with cash flows spread over a period of time
or for balance sheet hedgers. One version is also known as a weekly reset forward.
Forward contract
An agreement to buy or sell a given quantity of a particular asset (such as a currency) at a
specified future date at a pre-agreed forward price. A forward is the over-the-counter equivalent of
a future. The difference between the spot price and the forward price is largely influenced by the
cost of carry, that is, for financial assets, interest rates. For example, for currencies the forward
rate for a given future point in time is determined from the interest rate differential between the
two currencies. The theoretical forward price of a carryable asset like a currency contains no
expectations of the future spot price since the seller of the contract can hedge by holding the
underlying.
Forward-Forward
An FX swap in which both of the two value dates occur after spot value. This is simply the
forward sale {purchase} of a currency against a forward purchase {sale} with two different dates.
Forward Point Agreement (FPA)
A forward agreement to trade a forward at some point in the future at fixed foreign exchange or
interest rate forward points, fixed at the outset of the contract. An FPA is the FX swap equivalent
of an FRA. The agreements can be cash settled on a net payment basis or physically settled by
entering into the forward at the fixed points directly on the FPA maturity date. The FPA contract
enables users to separate the timing of the spot exchange rate and forward points used in a forward
foreign exchange contract by fixing the level of forward points used in a forward foreign exchange
rate and leaving the spot exchange rate to be determined at any time up to the forward exchange
date. These are much less frequently used than forward-forwards.
Forward Rate Agreement (FRA)
An interest rate contract in which buyer and seller agree to exchange the difference between the
current interest rate and a pre-agreed fixed rate, struck on the date of execution of the FRA
contract. If rates have risen, then at maturity the purchaser of the FRA receives the difference in
rates from the seller. If they have fallen, the seller receives the difference from the buyer. The
buyer of an FRA fixes a future borrowing cost; the seller fixes the rate of return on a future
deposit.
FRA prices are quoted as interest rates on the basis of the bid and offer yield levels for the period
of the FRA. The FRA rate itself is the implied forward rate for the relevant date. FRAs are labelled
on the basis of the number of months to the start and end of the FRA. So a three-month FRA
starting one-month forward is a 1 x 4 FRA or 1 v 4 FRA and a 3 v 9 FRA is trading the implied
six-month rate in three months’ time. So if the 3 v 9 were trading at 6.90% and a hedger or
speculator believed that in three months’ time six-month Libor would be above 6.90%, then they
would buy the FRA on their desired notional principal. Unlike interest rate futures, there are no
up-front margin payments. FRAs are the building blocks from which swaps are constructed. See
forward-forward [interest] rate, implied forward.
Knock-in cancellation forward
A structured forward transaction which is automatically cancelled if a predetermined cancellation
level is breached over the life of the forward. In addition the holder of the position incurs no
downside liability providing that spot has failed to breach a preset knock-in level over the life of
the forward. This is effectively a combination of the cancellation forward and knock-in forward
strategies described above.
Long-term foreign exchange (LTFX)
The outright forward purchase or sale of a currency for a future date at a price agreed at the
inception of the agreement with no spot exchange at the time of closing. They enable the holder to
lock away forward foreign exchange points for periods over 18 months and are used primarily to
hedge existing or anticipated exposures such as long-term borrowings or future receivables. LTFX
agreements usually entail a single exchange at a future date or a series of exchanges spread evenly
over a number of years. Equivalent to zero-coupon currency swaps they can be used to replicate
fixed-to-fixed currency swaps.
Non-deliverable forward (NDF)
A form of synthetic agreement for forward foreign exchange (SAFE) whose value at maturity is
based on the difference between the forward rate on the start date and the spot rate at settlement
and may be settled with a different asset from those that related to the forward. Generally used as a
forward when one of the two currencies is not freely convertible for net value and so which is
cash-settled in the freely convertible currency. Also known as a forward exchange rate agreement
or FXA.
Out-performance forward
The name given to a wide range of structures whose central aim is to construct a synthetic forward
using barrier options, which give the holder the potential to out-perform the forward outright if his
market view, around which the out-performance forward is tailored, is correct. These are
constructed from any combination of knock-in or knock-out options. Variations include barrier
windows and strips of out-performance forwards to cover a series of foreign exchange
transactions.
Participating forward
An adaptation of the range forward in which fewer options must be sold than are purchased. Also
known as profit-sharing forwards they are a type of ratio forward and are usually structured to be
zero premium. So in-the-money put {call} options are sold to finance the purchase of out-of-themoney call {put} options. In the FX version of the trade the holder might have a long call position
twice as large as the short put. This gives the holder participation in 50% of the weakening of the
underlying currency whilst retaining full protection on the upside if used as a hedge for an
underlying short position.
Prepaid forward sale
The sale of the underlying for the future with the present value of the forward sale paid to the
seller at the outset of the transaction. This is common as a loan substitute in the commodities
markets. Oil producers sell oil on a prepaid basis to a lender/counterparty who pays the producer
and then hedges his forward oil price risk through the sale of physical crude or using a commodity
swap. Producers use such transactions because it enables them to pay off debt today with
tomorrow’s revenue. In other markets also known as an off-market forward.
Range forward
The combination of an anticipated position in the underlying such as a foreign currency
receivables payment in three months time with a risk reversal (see chapter 8). Unlike a standard
forward, which locks in a fixed exchange rate for a forward exchange of currencies effectively the
buyer pays away all upside potential to the seller in exchange for an equivalent payment if rates
move the other way the range forward gives the holder exposure to spot rates but only within the
range set by the short put and long call position. These floor the downside risk at the cost of
capping the potential upside. The range forward is usually structured so that no premium is
payable upfront. This is the name given by currency markets to what in the interest rate markets is
called a collar (the interest version entails the purchase of a cap and sale of a floor). The position is
also called a cylinder. See corridor.
Example
A EU based company will receive USD 100 million in 3 months. It wants to hedge against USD
depreciation while maintaining some exposure to USD appreciation against the EUR but does not
want to pay the upfront premium associated with a naked USD put/EUR call option. Nor does he
want to lock in the forward rate because he is hopeful that the USD will move in his favour. He
chooses to purchases a EUR call/USD put struck at 1.0475 and sells a EUR call/USD put at
1.0275. The forward is at 1.0377 and so the range forward has zero upfront premium. If the USD
depreciates beyond 1.0475, then the company exercises its put at 1.0475 and fixes a minimum
value of EUR 94.56 million for its USD 100 million receivables. If spot remains between the two
option strikes then both expire worthless and the USD position is exchanged at the spot rate. If the
USD appreciates beyond 1.0275 and the holder of the sold call option will exercise it, capping the
maximum value of the USD 100 million at EUR 97.32 million. The three diagrams opposite
represent the same transaction. Strictly speaking the third shows only the pay-off from the option
position which without the underlying short forward position is a risk reversal. However it is often
used to represent a range forward and risk reversal and range forward are sometimes used
interchangeably.
[Range] bonus forward
A forward transaction struck at a rate that is better than the forward outright the bonus forward
rate. If spot ever trades outside a pre-specified range around the initial spot rate at any time over a
pre-specified period over the life of the forward, then the bonus forward resets to a rate which is
worse than the forward outright the reset forward rate. So an investor who could sell EUR against
USD in three months at 1.0377 could instead execute a range bonus forward where the bonus rate
of 1.0577 is achieved if EUR/USD trades inside a 1.0000 to 1.0950 range and the reset range of
1.0277 if not.
Ratio forward
In general any position similar to the range forward but where the call and put positions are
unequal. Often used specifically of a strategy in which the long call position is combined with a
short put position with a larger notional amount. This enables both a more conservative strike on
the put and a more aggressive strike on the call. It is a bullish strategy with stronger views that
spot will not go below the put strike. See participating forward.
Rebate forward
A forward in which the buyer enters into a structured forward transaction which is automatically
cancelled or knocks out if a predetermined trigger level is breached over its life. However, unlike
the standard cancellation forward, the client will permanently lock in a rebate should the forward
be cancelled..
Synthetic agreement for forward exchange (SAFE)
The generic term for exchange rate agreements (ERAs) and non-deliverable forwards (NDFs also
known as forward exchange agreements or FXAs). While forwards involve the actual sale and
purchase of the underlying, SAFEs are notional principal contracts like FRAs and are cash settled,
NDFs with reference to both the spot rate and forward premium/discounts, ERAs with reference
only to the latter. They were created to overcome capital adequacy requirements which
constrained banks in the forward market rather than as a result of demand for an alternative to
forwards. They are now used as a substitute for forwards in markets where currencies are not
freely or easily convertible.
Synthetic forward
The combination of a long European-style call and short European-style put or vice versa with the
same expiration and at-the-money-forward strike prices. A long forward position is a long call
position combined with a short put position. A short forward position is a long put position
combined with a short call position. The volatility used for both must be the same to avoid
conversion arbitrage (see chapter 8), reflecting put-call parity (see chapter 1)
Trigger Forward
The combination of a standard forward and a short knock-out option position. The holder can enter into a
forward to buy or sell at a rate which is better than the forward outright, but is exposed to the risk that if
spot hits a predetermined trigger level the structure knocks out. However, this risk is less than is the case
with a cancellation forward, as the client’s protective put or call is transformed into a put or call spread
rather than being totally cancelled out.
For example, in order to hedge a long position the client would buy a put and sell a call to create a synthetic
forward, and simultaneously sell a put struck below the put that is being purchased, that is only activated if
spot trades below a pre-specified level at any time during a pre-specified period of the life of the position.
More structured variants of this strategy exist which attach an out-of-the-money knock-out on the put or
call spread that the client is purchasing and a knock-in feature on the call or put that the client is selling.
Trigger forwards which use more than one option are known as double trigger forwards.
Multi-factor options
Key facts
The value of a multi-factor option is determined by the behaviour of two or more underlying assets and
therefore by the correlation between those assets. The main classes are rainbow and basket options. These
options are useful in managing complex combinations of risks although the details of their pricing and
mark-to-market value are complex. The assets involved need not all be from the same asset class. So, for
instance, an option may have a payout which is linked to the performance of the S&P index, the price of
Brent crude and the dollar-yen exchange rate. There is an additional special case where payments from a
single asset option are made in a different currency to the denominating currency of the underlying asset.
These are generally called currency translated , quantized or just quanto options. The values of many of
these options depend upon correlation and as such are described here.
Quantization can also be applied to most types of derivative instrument (see the entries for currency
protected swap, quanto FRA and the descriptions of cross-currency swaptions and quanto caps under the
entries for swaption and cap). Options on all asset classes can be quantized. This is just a small selection of
the more common uses.
Definitions
Asset-linked foreign exchange option
A quanto option on a foreign asset in which the exchange rate used on maturity is the greater of
zero, and the exchange rate at maturity less a preset strike. So the payout for a FTSE-100 equitylinked foreign exchange option would be:
$[FTSE-100 x max($/GBPmaturity – $/GBPstrike], 0)] where the maximum function determines
the exchange rate to be used. This strategy combines a currency option with an equity forward to
create a variable quantity forex option. It gives the holder of the option the ability to gain from any
dollar strengthening but places a floor (the strike) on the exchange rate component of the
investment since if at maturity sterling has weakened from the strike level then the option payout
is zero. An alternative way to look at this is that it is a foreign exchange option which is quantoed
into FTSE units.
Basket option
An option that pays out on the basis of the aggregate value of a specified ‘basket’ of financial
assets rather than on the value of the individual assets. The premium of the option will reflect the
correlations of the basket components: if they are negatively correlated, then moves in the value of
one component will be neutralized by opposite movements of another. Unless all the components
are perfectly correlated, the option will be cheaper than a series of individual options on each of
the assets in the basket. Basket options are used in all asset markets and can be cross-asset i.e.
contain more than one asset type and can be constructed with digital payoff characteristics. For
example:
Better off (worse off)
An option that pays the holder a return based on the price level/percentage price change achieved
by the better (worse) performing of two or more underlying assets. So its payout is: Max(asset1,
asset2,asset3...assetn) where asseti is either the price level or percentage change in asset i. For
example, an investor could buy a better-of option on two equity indices and receive the returns
from the better-performing of the two. If all the underlying assets fall in value, the holder must pay
the performance of the asset with the smallest decrease in value.
Composite option
An option where the strike is denominated in the payout currency of the option and fixed at
inception. The asset price is then translated at spot at expiry of the option. So in the example
described if the FTSE option is struck at the money spot (i.e. the strike is 4800*1.6 = $7680 per
FTSE unit) then the payout in the two cases is max(5000*1.5 - 7680,0) = max(7500 - 7680,0) = $0
and max(5000*1.7 - 7680,0) = max(8500-7680,0) = $820.
In this case the investor is still exposed to currency movements but the hedger has significant
correlation risk. He will be funding his position in dollars and every hedge re-balance will involve
a sale or purchase of sterling and a sale or purchase of the sterling asset. Thus his cost of hedging
will be affected by the frequency with which the exchange rate and the FTSE index move.
Currency basket option
An option that gives the holder the right to exchange a portfolio of predetermined amounts of
currency for a fixed amount of a base currency.
Example
A US multinational is expecting flows of Euros, Canadian dollars, Japanese Yen and Swiss Francs
in three months and has budgeted rates for all flows. Unwilling to lose upside potential by using
forwards, the company must choose between buying currency puts for each currency struck at the
budgeted rate or buy a basket option struck at the total US dollar value of all the currency amounts
at the budgeted rates. If on the option expiry date, the dollar value of the currencies at the final
spot rates is less than the basket strike, the corporation will exercise the option. If the total dollar
value of the portfolio is higher than the basket put strike then the option expires worthless and the
currencies are converted at spot.
Currency-translated/Quanto options
Also known as quantos or quantized options, these options payout in a currency different from the
natural denominating currency of the asset. For instance an option on the Nikkei index that pays
out in dollars or an option on dollar-mark that pays out in Italian lire. There are five main types.
To illustrate them we will take the example of a dollar-based investor who wishes to buy a call
option on the FTSE-100 but receive the proceeds in dollars. Let us suppose that at inception of the
trade the FTSE-100 is trading at 4800 and the dollar-sterling exchange rate is 1.60. At expiry let us
suppose that the FTSE-100 is trading at 5000 and the dollar-sterling exchange rate is either 1.50 or
1.70.
Flexi-quanto/flexo
An option where the payout of an option is translated at spot from the denominating currency of
the underlying asset into the denominating currency of the option. So for instance in the example
described if the FTSE option is struck at the money spot then the payout to the investor in the two
cases is (5000-4800)*1.50 = $300 and (5000-4800)*1.70 = $340. In this case the investor is fully
exposed to currency movements and the hedger has no correlation risk since he funds his position
in sterling, transacts his delta hedges in a sterling asset and simple does a spot foreign exchange
transaction at expiry.
Hybrid barrier option
An option on one asset knocked-in or out by movements in another. These are also called twofactor or outside barrier options and also dual trigger options. The commonest types are hybrid
barrier caps and floors linking exchange rates (example 1), commodity prices (example 2) or
equity indexes with interest rates (example 3) though hybridization has also been applied to other
types of derivative. For example semi-fixed swaps (and other resettable or contingent structures)
have been constructed with the rate reset trigger dependent on the price of oil (a swap plus a
binary oil option) and other assets.
Example 1
A Japanese exporter with large floating-rate debt outstandings might be very profitable when
dollar-yen exceeds 105 but below 95 cash flow becomes critical and he requires interest rate
protection. A normal five-year 7% cap might cost 364 bp. Instead they buy a five-year 7% cap
which knocks-in when the dollar-yen rate hits 95. This reduces the cost of the cap by 160 bp.
Example 2
Or take a gas producer whose profits usually rise with rising gas prices but fall with rising interest
rates. The company fears the combination of rising rates and falling prices. It could buy a standard
interest rate cap, but is unwilling to pay so much premium as it is only the combination of factors
that pose a threat. Instead it can buy an interest rate cap that is knocked-out if the gas price
exceeds a specified barrier in any quarter. The strike and knock-out levels are set at the company’s
combination pain or breakeven threshold. The company pays floating interest rates only when it
has profits with which to pay.
Example 3
A UK-based company might wish to buy interest rate cover for some debt. However, it is
contemplating floating off a large subsidiary in the next two years in which event it will not
require the cap. Instead of buying a three-year cap at a cost of 339 bp, they buy a knock-out cap
that knocks-out when the FTSE midcap index rises by 15%. This cap costs 140 bp less. As well as
the lower premium, the cap will disappear at exactly the right time: when the company will be able
to float its company or sell it at an attractive price and pay down its liabilities. The knock-out can
either be permanent, as in these examples, or the cap can be structured so that it is only knockedout for the period in which the knock-out trigger is breached. If the underlying moves back
through the knock-out trigger, then the cap is reactivated, making it resemble a range transaction.
Interest rate basket option An option on a basket of interest rates designed to reduce overall interest costs
across a number of different markets.
Example
A borrower may believe that his European interest rate bill would rise more than is implied in the market
but is unwilling to fix in case his view is incorrect. Instead of purchasing a series of options on the
individual markets, he wants a basket because the mix of EU and non-EU currencies exhibit some negative
correlations that will reduce the premium cost. He could buy a one year 8% strike basket option
denominated in his base currency with the underlying the average two-year swap rate in the chosen
currencies. If the average rate rose above 8% this hedger would be protected. The sensitivity of the basket
would be similar to that of a basket of payer swaptions. (An investor would buy the product if he wanted a
customized, balanced exposure to a region and is prepared to accept a degree of upside limit.)
Joint quanto option
An option where the payout is translated from the denominating currency of the underlying asset
into the denominating currency at a rate which is at least as good as a predetermined exchange
rate. So in the example described if the FTSE option and fixed dollar/sterling exchange rate are at
the money spot then the payout in the two cases is max(5000-4800,0)*max(1.6,1.5) = $320 and
max(5000-4800,0)*max(1.6,1.7) = $340.
In this case the investor is fully protected from adverse movements in the foreign exchange rate
and participates fully in favourable movements. The hedger clearly has significant correlation risk.
In fact he has two rainbow options since he has a contingent flexo option and a contingent quanto
option where the contingency is based on the exchange rate: he will have the flexo option only if
the exchange rate is above the predetermined foreign exchange rate and he will have the quanto
option only if the exchange rate is below the predetermined foreign exchange rate.
Max/min option
An option that pays the holder a return based on the asset that performs best against its strike out
of a basket of assets each with its own pre-set strike price. Unlike the better (worse) of option, the
strikes are above zero and the payout cannot be negative. So a Max call option pays out:
Max(max(asset1-strike1, asset2-strike2,... assetN-strikeN), 0) and a Min call pays out:
max(min(asset1-strike1,asset2-strike2,... assetN-strikeN),0). Outperformance option An option
that pays the holder the difference in the performance of two assets – that is: max(asset2-asset1, 0)
An investor with a cash position in the FTSE-100 but worried that the S&P500 might outperform
it could buy a rainbow outperformance option that paid a return based on the difference if positive
between the two indices’ performance. So if the S&P500 did outperform the FTSE-100 by 8%,
then the holder would receive a payout based on that 8% difference. If the S&P500 did not
outperform, then there would be no payout. Also known as difference options.
Quanto option
An option where the payout is translated from the denominating currency of the underlying asset
into the denominating currency of the option at a predetermined exchange rate. So in the example
described if the FTSE option and fixed dollar/sterling exchange rate are struck at the money spot
then the payout in the both cases is max(5000 - 4800,0)*1.6 = $320.
In this case the investor is fully protected from movements in the foreign exchange rate and the
hedger has correlation risk. He will have a funding position in both dollars and sterling always
maintaining the net value of sterling assets and liabilities at 0. This means that correlation impacts
him as the size of his FTSE position varies linearly with the dollar/sterling exchange rate.
Rainbow option
An option whose payout is based on the relationship between multiple assets as opposed to the
price or performance of a single asset. A rainbow option whose payout depends on two assets is
said to be a two-colour rainbow, on three assets a three-colour rainbow and so on. There are five
basic types of rainbow option.
Spread option
An option on the spread between two asset prices or indices. They differ from outperformance
options as they are struck not at zero but at some level of the spread. So the payout is: max(asset2asset1-strike, 0).
Example 1
A yield curve option is an option on the spread between interest rates at two different points on the
same yield curve. They are usually struck on the yield of a longer maturity bond/index less the
yield of a shorter maturity bond/index. So an at-the-money call on the US dollar two-year CMS
versus the 10-year CMS would have a strike equal to the implied spread between the expected
levels of the two rates on the exercise date. Yield curve calls profit if the spread widens (yield
curve steepens), puts if it narrows (yield curve flattens). They allow investors to take a view on the
shape of the yield curve without taking a directional view on the underlying bond market. A yield
curve option costs less than the series of calls and puts on the underlying securities/indices used to
construct the yield curves because it only pays off on the change in spread whereas one of a pair of
separate options might be in-the-money as the result of a parallel shift in the yield curve. For the
option writer a key consideration, as in all multi-factor options, is the correlation between the two
points on the yield curve. An investor holding a three-year floating rate asset yielding Libor
believes that the spread between the three-year swap rate and six-month Libor will be higher than
implied by the forward curve. So he buys a digital cap (i.e. series of digital calls) on the three year
swap rate minus six month Libor spread struck at 60bp with an immediate payoff of 120bp as soon
as the spread hits the strike. The premium is 65bp a year semi-annually. If the spread is lower than
60bp the investor receives Libor less 65bp. If the spread is higher he receives Libor plus 55bp. He
could bet on a narrowing of the spread by buying a digital floor.
Example 2
Acrack-spread option is an option on the spread between the price of crude oil and one or more of
its refined products. Simple versions are traded on Nymex – one on the spread between heating oil
and crude oil prices and one on the spread between crude and unleaded gasoline. They are quoted
in terms of the price of one barrel of the refined product less the price of one barrel of crude.
Vanilla options
Key Concepts
An option is a contract, traded over-the-counter or on an exchange, that grants the buyer the right to buy
(call option) or sell (put option) a pre-agreed amount (the notional principal) of a specified asset or index
(the underlying) at a pre-determined price or rate (the strike price) and time specified in the contract. The
option holder is said to exercise the option if he exercises his right to buy or sell the underlying. If exercise
is allowed only at the option’s maturity then the option is said to be European-style. If exercise is allowed
at any point during the life of the option it is said to be American-style. If it can be exercised on a number
of pre-determined dates it is said to be a Bermudan or Atlantic option.
The seller of the option is also known as the option writer. The underlying asset can be a foreign exchange
rate, an interest rate or swap rate, a commodity or commodity index, an individual equity or an equity
index, baskets of any of these assets or spreads between them. In exchange for this right, the buyer of most
types of option makes an upfront payment, the premium. The level of this premium is not straightforward
to calculate but depends on, among other things, the option’s payoff structure and maturity, the level of
interest rates and the volatility of the underlying.
What distinguishes options from other financial instruments is the asymmetric payments they generate.
Options allow their holders to profit when the price of the underlying moves in their favour while limiting
downside to the premium paid. This is unlike the symmetrical payoffs associated with forwards and futures.
This chapter deals with vanilla options. These are options whose payoffs are continuous, whose strike
prices are fixed, which exist in the same form throughout their maturity and for which a standard upfront
premium is payable. It also includes a number of combinations of standard options (known as spreads)
which are commonly used to create more complex payoff profiles tailored to more specific views on the
part of the buyers and sellers. More complex options fall into a number of different types and are dealt with
in the following chapters.
Definitions
Back {forward} spread
(i) Any option spread where more options are purchased {sold} than sold {purchased}. (ii) An
option spread whose value will rise {fall} given a sharp movement up or down in the price of the
underlying.
Box [spread]
Generally the term box position refers to any offsetting spread positions; for example, the
combination of bull and bear spreads. Another example is the combination of a horizontal or
calendar call spread and a calendar put spread with both spreads having the same expiration dates
on their long and short positions. These types of spread positions are used to capture the value in
mispriced options while hedging against market risk or, alternatively, are used to tie up or free up
cash.
Butterfly [spread]
A combination of four options. Used to describe either the combination of a bull with a
bearspread, or of an at-the-money straddle with an out-of-the-money strangle. A long butterfly
might be long an option (put or call) at 40, short two at 60 and long one at 80. For a limited
premium upfront the strategy will achieve a maximum payoff of 20 if spot at expiry is 60. A short
butterfly is short two options and long two options. The characteristic shared by all the
combinations is that the holder benefits from stable prices in the underlying while remaining
protected against large movements in prices.
Call {put} option
An option that grants the holder the right but not the obligation to buy a pre-agreed amount of a
specified underlying at a pre-determined price or rate. The buyer of a call is expressing a bullish
view on the underlying and also implicitly, since he is long an option, believes either that volatility
will rise or at least that it will not fall. The right to sell the underlying and so express a bearish
view is called a put option. In a foreign exchange option, since the option is to exchange one
currency for another, all options are call options on one currency and, by definition, put options on
the other currency. For example, in EUR/USD, a EUR call is a USD put and vice versa. See putcall parity.
Call {put} spread
An option spread involving the simultaneous purchase and sale of call {put} options on the same
underlying either with different strike prices or maturities. These positions can be bullish or
bearish depending on the strike prices of the options. So a bull {bear} call spread is the purchase
of a call option with one exercise price and the sale of a call with a higher {lower} exercise price
both generally with the same expiration. This reduces the cost of the position at the expense of
limiting participation in the appreciation of the underlying. And a bull {bear} put spread is the
purchase of a put option with one exercise price and the sale of a put with a higher {lower}
exercise price, both generally with the same expiration. These two combinations produce identical
payoff profiles.
Cap {floor}
A call {put} option on an interest rate index. Above {below} the strike the holder of a cap {floor}
with a notional principal equal to an underlying liability is hedged against rises {falls} in interest
rates. Caps are therefore used as hedges against rate rise by borrowers and floors as hedges against
rate falls by lenders or investors.
In practice, caps and floors are medium-term agreements under which, in exchange for a one-time
upfront premium payment, the seller agrees to pay the buyer the difference (if positive) between
the strike rate and the current rate at preset times over the life of the cap {floor} thus establishing a
maximum {minimum} interest rate for the holder. The buyer selects the maturity, interest rate
strike level, reference floating rate, reset period and notional principal amount. The maturity of
standard caps {floors} means that they are actually made up of a series of caplets/single period
caps {floorlets/single period floors}. A caplet {floorlet} can be viewed either as a call {put} on an
interest rate index or a put {call} on an interest futures contract or zero coupon bond.
Vanilla caps and floors are not a continuous rate guarantee; claims can only be made on specified
settlement dates. This makes them best suited to hedging the interest rate on floating-rate
instruments that are reset periodically. Caps and floors are priced off the implied forward curve the
relevant implied forwards being either the swap rate for the period of the cap/floor or the FRA rate
for a caplet/floorlet. The simplest approach to pricing caps/floors assumes that these forward
interest rates are lognormally distributed. Caps are sometimes also known as ceiling rate
agreements.
Interest rate caps can be quantized (see chapter on ‘Multi-factor options’). A quanto cap is one
option or a series of options whose payout is based upon a reference (foreign) Libor exceeding
(cap) or falling below (floor) an absolute strike rate or spread with respect to the base (domestic)
Libor. The payout is in arrears on a money market basis, as in a normal cap/floor, but is
denominated in the base (domestic) currency. The ability to cap or floor the differential between
two currencies’ Libors in one currency can be more efficient and economical than doing vanilla
caps and floors in each currency individually. Variants include: The rate differential option is a
quanto cap or floor on the foreign interest rate or domestic currency payment stream in a
differential swap. For example, a borrower of Swiss francs paying three-month dollar Libor plus a
spread in exchange for three-month Swiss franc Libor under a differential swap could cap his
absolute rate payment.
The spread differential option is a quanto cap whose buyer receives the spread between two
interest rates in different currencies minus a strike spread, with the payment denominated in his
required currency. Payments are made in the domestic currency when the spread exceeds the strike
level. It can be viewed as a strip of options on forward spread agreements. The floor version is
used to ensure that coupons in leveraged currency protected notes do not become negative.
Capped call {floored put}
An option with both a strike price and an in-the-money cap {floor} strike. If the underlying hits
the cap {floor} strike then the option is automatically exercised for its intrinsic value. It is
different from a call {put} spread in that the cap {floor} is locked in if ever the trigger is hit
regardless of subsequent movements in the underlying. Payment can either be immediate or made
on the original expiry date of the option. These instruments can be used as one element of a collar
or risk reversal strategy in which, as soon as the underlying trades through the cap strike, the short
option explodes (expires) and the long option pays out. Because the automatic exercise locks in
the intrinsic value of the option, these options have a similar risk profile to vanilla options.
Also known as cliquet options (‘cliqueter’ is French for ‘to knock’; the automatic exercise became
known as the cliquet clause),exploding options, lock-in options, trigger options (not to be
confused with true barrier options also sometimes called trigger options). See knock-out trigger
option, switchback option.
Condor [spread]
An options (or futures) spread position similar to the butterfly. The holder is long and short two
spread positions strangles on the same market. A long condor can be constructed by, for example,
buying a call struck at 40, selling a call struck at 60, selling another call struck at 70, and buying a
call struck at 80.
The maximum payoff on this strategy occurs when the underlying trades between 60 and 70 a
wider range than the butterfly but with the compromise of a smaller maximum potential profit.
The position is limited on both the up- and downside and is directionally neutral. The short condor
sells the lowest strike call, buys the two higher strike price calls and sells the highest strike price
call.
Conversion [arbitrage]
An arbitrage trade so called because it can be used by the holder of a put to alter his position to a
call or vice versa. A conversion, also known as a long option box, is the purchase of the
underlying or future, purchase of a put and sale of a call with the same exercise price and
expiration date. This converts a put to a call and creates a short synthetic futures position hedged
by a position in the underlying or future.
The opposite is known as a reversal or short option box or reverse conversion and is the purchase
of a call, sale of a put with the same exercise price and expiration and sale of the underlying or
future. This is the reverse of a conversion as it converts a call position into a put. The position is a
synthetic long futures position hedged by the sale of the futures contract. These arbitrages
maintain and rely on put-call parity. If a put is overvalued (or if the put is fairly valued but the call
is undervalued), a riskless profit can be made by executing the reversal. If the call is overvalued
(or the call is fairly valued but the put is undervalued), the riskless profit is generated by selling
the call, buying the put and buying the underlying or a future. The actual arbitrage return depends
on the additional borrowing costs {investment returns} from the money market transactions which
fund (result from} these trades.
Corridor
A call spread constructed from the purchase of an interest rate cap at one level and sale of another
at a higher level. The holder of the corridor is protected against rate rises between the strikes of the
two calls. Unlike the holder of a collar though, the holder benefits fully from any downward
movement in rates. Also sometimes applied to a collar on a swap created by using two swaptions.
Diagonal spread
An option spread in which the holder is short the same type (call or put) of options of one maturity
and strike price and long options of a different maturity and different strike price. A diagonal bull
spread is the sale of a shorter maturity option and purchase of a longer maturity, lower strike price
option. A diagonal bear spread is the purchase of a longer maturity option and sale of a shorter
maturity, lower strike price option. ‘Diagonal’ because it is a cross between a horizontal and
vertical spread.
Fraption
An option on an FRA giving the holder the right but not the obligation to purchase an FRA at a
predetermined strike. A cap can be thought of as being constructed from a string of interest rate
guarantees. Also known as an interest rate guarantee. Participating option An option which
changes the rate of participation in a price or rate movement once the strike price has been
reached.
Example
A participating call option on the FTSE-100 stock index might give 100% participation from a
strike at-the-money up to the point at which the index has moved up 10%. Then further
participation is limited to 50%. Effectively the option holder has sold a call at that level on half the
notional principal of the original call. Because of this, the participating option is cheaper than the
standard variety. The cap version is known as a partial cap. It reduces exposure to an upward
move in the price of the underlying rather than eliminating it completely. Either the hedger simply
buys a cap with a smaller notional principal than the underlying exposure, giving both
counterparties participation on an average basis. Or, if a zero premium structure is required, the
hedger simultaneously buys an out-of-the-money cap and sells an in-the-money floor with a lower
notional amount. Since the floor is in the money, it needs to be struck on less notional principal to
create a zero premium. The structure limits participation in downward rates to the portion of the
underlying exposure not covered by the floor sale.
Horizontal spread
A generic name for the simultaneous purchase of one type of option (call or put) and sale of the
same type of option with the same strike price but a different maturity. Usually used specifically
of the simultaneous sale of an option with a nearby expiry date and the purchase of an option with
a later expiry date, both with the same strike price. This trade will profit if the time decay on the
short position is faster than that of the long. Also known as a calendar spread, money spread. See
vertical spread.
Ratio/variable spread
Any option spread in which the number or notional amount of options purchased is not the same
as the number or notional amount sold. For example, a ratio bull spread is the simultaneous
purchase of in- or at-the-money options and sale of a larger quantity of out-of-the-money options.
And a call ratio forward spread is the simultaneous purchase of at- or in-the-money calls and sale
of a larger number of out-of-the-money calls. The position will make money from the long calls as
long as the underlying rises. If however it rises beyond the strike of the short calls so far that the
intrinsic value from the long position is overwhelmed, the position can lose value, The potential
losses on the position are unlimited. The strategy is cheaper to purchase than the plain bull spread
and is suitable when spot is expected to be stable or if it moves is more likely to rise. The position
then benefits from the time decay of the options sold.
Risk reversal
The simultaneous purchase of an out-of-the-money call {put} and sale of an out-of-the-money put
{call} usually with zero upfront premium. To a trader, the term means more specifically the
purchase of a less than 50% delta (³) call {put} financed by the sale of a similar delta put {call} for
zero upfront cost. The options being bought and sold will typically have the same notional size
and pre-specified maturity and the deltas will typically be set to 25%. According to Black-Scholes
the purchase and sale of options with similar deltas (and so out-of-the-money forward to the same
extent) should be zero cost. In practice the market favours one side versus the other in the simplest
case the implied volatilities of out-of-the-money puts and calls of the same strike and maturity are
different and the extra cost of the favoured side is commonly known as the risk reversal spread.
This reflects the market’s perception that the relevant probability distribution is not symmetrical
around the forward but skewed in the direction of the favoured side. Another way of interpreting
this is to say that implied volatility is correlated with spot, an impossibility in a Black-Scholes
world.
The one-month 25³ risk-reversal is the market’s benchmark indicator of skew in a particular asset
class and is commonly used to trade the skew. When positive, it indicates that calls are more
favoured by the market than puts and that the market is more likely to rally significantly than fall
(and vice versa when it is negative). The 15³ risk-reversal is also gaining in popularity as a
benchmark trade as it is used to hedge the second order volatility risk (ðvega/ðspot) in some
barrier options. See strangle. When combined with a short {long} forward position, a risk reversal
becomes a cylinder or range forward. Confusingly, because this latter position is known as a collar
in interest rate markets, the naked risk reversal is also sometimes referred to as a collar.,
Example
A short EUR cash position could be hedged with the purchase of a EUR call/USD put struck at
1.0477 and the sale of a EUR put/USD call struck at 1.0290. Assuming a forward rate of 1.0377
this three month collar would be zero premium. At expiry if the spot is above 1.0477 then further
losses on the underlying position are hedged by the purchased option. If spot is between the two
strikes the underlying is exchanged at the prevailing spot rate as with a normal forward. If it is
below 1.0290 then the profits on the underlying are capped by the sold option.
Seagull
A ratio spread comprising the purchase of a call spread and the sale of a put with a strike below
that of the calls, or vice versa. This produces a schematic payoff profile that resembles a tilted and
elongated ‘M’ – like a schematic representation of a bird or seagull.
Straddle
A long straddle is the purchase of a put option and a call option on the same underlying with the
same strike price and the same time to expiry. The position is usually initially delta neutral since it
is typically struck at the forward rate. The position (which can also be constructed from two long
puts and a long position in the underlying or two long calls and a short position in the underlying)
will make money if volatility is high. A short straddle is the sale of a put option and a call option
on the same underlying with the same strike price and the same maturity. It exposes the holder to
unlimited downside but will make money if volatility is low. Since the strategy, when struck atthe-money-forward, has a large vega position and zero delta, it is the commonest way to trade
volatility used by interbank and other sophisticated players.
Strangle
Similar to a straddle except that the strike of the call lies above the strike of the put. A long
{short} strangle is the purchase {sale} of a put option and a call option on the same underlying
with the same expiry date but with different strike prices. The short strangle generates less
premium than the straddle because options sold are out-of-the-money and so cheaper. This is
compensated for by the wider break-even range of the position. Straddles and strangles involve
combinations of two options, which differentiates them from, say, butterflies, which involve
combinations of four options, and can in fact be constructed by combining a strangle and short
straddle and vice versa. The one month 25³ strangle is the standard strangle trade and a common
market indicator of the amount of leptokurtosis or “extreme event probability” in that asset class
and is sometimes used to hedge second-order vega risk in some barrier options. See risk reversal.
Swaption
The option to enter into a swap contract. The simplest swaption is an option to pay or receive fixed
rate in an interest rate swap. This can be considered an option to buy or sell a fixed-rate bond
versus selling or buying a Libor flat floating-rate note. A payer(’s) swaption is an option that gives
the buyer the right but not the obligation to enter into an interest rate swap paying fixed and
receiving floating. It is also called a put swaption as it is analogous to a put on a fixed-rate
instrument (that is, an option to issue a bond). The buyer benefits if rates rise as the option will
become more valuable. If rates rise above the fixed rate payable under the swaption, then the
holder can exercise it and swap an existing floating rate liability into an advantageous fixed
rate.The payer swaption is similar to a cap in that it provides an interest rate ceiling, but it has to
be exercised to provide the fixed rate, and once exercised, the holder is locked into paying a fixed
rate, unlike the cap holder who can still benefit if rates fall. Also, while caps tend to reference the
short end of the yield curve, the payer swaption tends to reference the two- to 10-year part of the
curve.
A receiver(’s) swaption is an option giving the holder the right to receive fixed rate under an
interest rate swap. As it is analogous to having a call option on a fixed-rate bond, it is also known
as a call swaption. A receiver swaption behaves like a floor.
Typically, the option period is for a year or less on swap maturities of between three and ten years.
So a typical transaction might be to buy a three-month payer swaption with a strike price of 7.50%
for cash settlement on a notional principal of $50 million. If swap rates rise to 8.00%, the option
would be exercised and a cash payment made to the swaption buyer. (Most swaptions are cashsettled.) Swaptions are usually European style, although American-style swaptions, allowing the
buyer of the option to enter into a swap at any time after the exercise date, typically on a payment
date, are available. Swaptions are available on most vanilla and exotic swaps on most underlying
assets.
Swaptions can be combined with swap positions to create extendible or cancellable swaps. See
callable swap, extendible swap, reversible swap. They are also used, like forward start swaps, to
monetize call and put options embedded in callable and puttable bonds. See monetization. A
swaption on a cross-currency swap is sometimes known as a cross-currency swaption. Here one
counterparty sells/buys the right to enter into a currency swap with another counterparty on a predetermined date under which the first counterparty pays a pre-set fixed or floating rate in one
currency in exchange for a pre-set fixed or floating rate in another currency. The principal amount
for final exchange is set for both currencies. Initial exchange of principal amounts is not
necessary. A borrower who wished to reduce his funding costs by issuing a note denominated in
one currency but convertible into one denominated in another could use this instrument to hedge
against investor exercise.
Example
An investor seeks the better performing of a EUR fixed-rate asset and a USD fixed-rate asset.
Both currency and interest rate exposure are sought. For the option the investor pays the premium
in the form of reduced yield. A EUR 100 million 5% five-year note convertible once in one year
into a $103.10 million 6.75% note with the same maturity is issued by a European agency. This
borrower buys a fixed EUR/fixed USD cross-currency swaption to hedge potential investor
exercise. This gives him the right, once at the end of one year, to enter into the following currency
swap: USD notional: $103.10 million payable at maturity by the swaption writer; EUR notional:
EUR 100 million payable at maturity by the borrower; Maturity: four years from exercise; Payer
of 6.75% semi-annual USD: writer; Payer of fixed 5% annual EUR: agency; Initial currency
exchange: none; Premium: 2.53%, EUR 2.53 million paid by the agency. The borrower gets
cheaper funding regardless of option exercise, the investor gains the required exposure. The
swaption is more economical than the purchase of an FX option and two interest rate options,
unless the implied correlation between them is zero.
Table-top
A ratio spread in which the purchase of an option is paid for by sales of the same option at two
different strike prices. So called because of the representation of its payoff profile. For example,
the purchase of one call option with a strike of 40 and the sale of one call struck at 60 and another
struck at 80.
Vertical spread
A generic term used to describe the simultaneous sale of one type of option (call or put) and
purchase of the same type of option with the same maturity but a different strike price. Contrast
with horizontal spread.
Warrant
An option in the form of a listed security rather than an over-the-counter contract. Warrants are
available on all the asset classes used as the underlying in option contracts. Warrants are also
available on a variety of government debt instruments and both debt and equity warrants can be
attached to public bond issues by corporations and financial institutions. See cover.
Vanilla swaps
Key facts
Swaps are the archetypal over-the-counter derivative instruments. Although they are highly liquid, traded
instruments, they remain privately negotiated contracts between counterparties – though these contracts
now take the form of standardized master agreements. They involve the exchange of fixed and/or floating
payment streams based on interest rate, currency, equity, bond, commodity and real-estate indices. They
enable counterparties to exploit different markets’ perceptions of their credit to raise cheaper capital, to
access new markets by creating synthetic instruments and to manage currency and interest rate risks. The
main uses are:
i.
ii.
iii.
iv.
v.
To modify existing or future cash flows from an asset or liability for risk management purposes.
To alter an existing cash flow so that it matches a changed set of circumstances.
To decrease borrowing costs or increase investment yields.
To access markets synthetically that would otherwise be closed or uneconomical.
To modify cash flows for tax and accounting purposes.
Swaps are now used by every kind of user of the financial markets – banks, insurance companies, nonfinancial corporations and institutional investors. This chapter focuses on standard swaps characterized by
the following features:
i.
ii.
iii.
iv.
v.
The term of the swap is a whole number, commonly one, two, three, five, seven and 10 years.
The fixed and floating coupon payments take place at regular intervals, for example every six or
12 months.
The notional principal of the swap remains constant for the term of the swap.
The fixed rate remains constant for the term of the swap.
The floating rate is set at the beginning of each interest period and paid in arrears at the end of the
interest period.
The basic swap structures are the commonest. However, because swaps are privately negotiated contracts
the terms and conditions of any swap can be altered to suit the exact circumstances of the user. This
flexibility has led to the creation of a family of swaps still regarded as vanilla – that is they do not have
other derivative instruments combined with them and in particular which do not contain any option-like
characteristics – whose basic form is the same as the standard interest rate and currency swaps but whose
notional principal, coupon payments, period, fixing or final settlement terms are non-standard.
This chapter also contains examples of swaps that do not conform to the standard structures in the
following respects:
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.
ix.
Payment frequency mismatches when the floating-rate payment frequency differs from the term of
the floating-rate index – example a swap in which a counterparty receives three-month Libor reset
quarterly but paid semi-annually.
Day-count mismatches where the fixed or floating payments of a swap are based on a convention
that differs from the market convention of the currency involved.
Irregular coupons such as short or long fixed or floating interest periods at the beginning or end of
the swap or zero coupons.
Non-standard tenors such as 18 months.
Variable notional principal.
Forward or deferred starts vii Off-market pricing where the fixed-rate payable is above or below
the market rate resulting in additional payments by one counterparty.
Floating margins – that is structures incorporating indices such as six-month Libor plus or minus a
margin. While the market convention is based on flat Libor-based payments, a margin causes a
discrepancy if the payment frequency and day count conventions for fixed and floating payments
differ.
Alterations to the standard rate set in advance, payment in arrears structure.
Standarde swaps but on different asset classes.
Definitions
Accreting [principal] swap
A currency or interest rate swap whose notional principal increases over its maturity. Created to
fix the interest costs of projects funded by a series of predictable future drawdowns on a loan
facility. Also known as a [staged] drawdown swap, escalating [principal] swap, step up swap.
Amortizing swap
A swap whose notional principal decreases over the life of the instrument. In the simple versions
the notional principal decreases according to a fixed schedule that is determined at the outset of
the swap. This allows, for example, users to convert amortizing fixed-interest securities (i.e. bonds
with sinking funds or early redemption provisions) into floating-rate securities. This type of
amortizing swap can be seen as a series of separate swaps with the total swap price reflecting a
weighted average of the individual swap rates or it can be seen as one swap of a particular duration
and priced as a swap with this maturity. An interest rate swap that converts the cash flows from an
amortizing debt instrument or index into a fixed-swap payment is also known as a level payment
swap.
In the more complex versions the amortization is linked to an underlying index, for example
interest rates, a foreign exchange rate or mortgage prepayment rates, but the timing of the
amortization is not known at the outset of the swap and depends on the path of the underlying
index. These are known as index amortizing swaps. In these instruments the fixed-rate receiver has
sold a series of complex swaptions to the payer. They are described in the chapter on complex
swaps and see also under index amortizing note in the chapter on derivative-linked securities.)
Annuity swap
A cross-currency swap in which there is no exchange of principal, just interest payments. Also
known as coupon-only swaps they are usually used to swap dual currency bonds into just one of
their currencies. So a bond whose principal is denominated in dollars with coupons paid in yen can
be swapped into dollars by an issuer willing to pay dollars and receive yen to cover the interest on
the bond. Since the bond principal is already in dollars there is no need to include it in the swap
agreement.
The term annuity swap is also used of a type of amortizing swap in which an irregular payment
stream is exchanged for a regular payment stream of the same present value. It is usually found in
currency swap markets where it is used to exchange a set of even cashflows in one currency for an
equivalent (in value and evenness) cash flow in a second currency. A series of long term forward
foreign exchange contracts would create an uneven series of payments. So if a company wishes to
swap an annuity stream of USD10 million over five years into yen, it can either treat the dollar
annuity as an amortizing loan and execute an amortizing USD/JPY currency swap or it can use a
currency annuity swap whose swap rate depends on how much of the early JPY cash flows have to
be borrowed from (or lent to depending on interest rate differentials) the later cash flows.
asset swap See chapter 16.
Back-to-back swap
A swap agreement with the same terms and opposite counterparties to an existing swap such that,
if entered into, it will cancel out the obligations of the original swap. These swaps are more
complex than simply cancelling an existing agreement and so will only be used if there are
specific tax or accounting benefits. Also known as a reverse swap.
Basis swap
A floating-floating interest-rate or cross-currency swap under which two counterparties exchange
interest flows on two different floating rate reference indices. They arose from banks’ needs to
hedge the spread exposure between different short-rates, for example lending at prime and funding
in Libor. So some of the commonest basis swaps are, CP for Libor, T-bill for Libor, six-month
Libor for six-month Libor reset monthly or three-month USD Libor for three-month JPY Libor
and hence the alternative name for a basis swap, money market swap. They are also used to hedge
the interest rate risk in a currency swap. Basis swaps can be created from two interest rate swaps
in which the two fixed legs cancel out between counterparties and the two floating legs are as
required. They are also common in commodity markets where they are used to hedge fluctuations
on spreads between different products.
CMT/CMS swap
A yield curve swap in which one leg is linked to CMT rates. One counterparty pays the CMT or
CMS rate at one part of the curve, say the two-year CMS or CMT rate, and receives it at a
different part of the curve, say 10 years. Interest rate swaps can be indexed to many indices. So, a
COFI swap is a swap one of whose legs is referenced to the 11th District Cost of Funds Index, a
US interest rate index important to savings and loan institutions. Sometimes used as a reference
rate in swaps and bonds, particularly when short rates are expected to fall, because movements in
COFI tend to lag short-term rates. Other indices commonly used are Prime, Fed Funds, Libor and
the many domestic rates found in local fixed-income markets.
Collateralized swap
A swap agreement in which one or both counterparties puts up collateral to guarantee its ability to
meet its obligations under the agreement.
Commodity swap
In its vanilla form an agreement identical to a fixed-for-floating interest rate swap except that the
payment streams are based on the price of a commodity such as crude oil or its distillates, nonferrous metals and bullion.
Example
An oil producer wishing to lock in the price of his production of 600,000 barrels a year can pay a
floating rate equal to the pre-agreed price index times 50,000 barrels a month and receive a preagreed fixed amount per barrel on the same notional 50,000 bbl/month. The fixed price is set
upfront by reference to the prevailing swap or forward market. An oil consumer would enter such
a swap as a fixed payer. Typically no oil changes hands. The producer continues to sell 50,000
barrels a month to the market and channels that floating payment stream into the swap in exchange
for the fixed rate. Physical delivery can be accommodated.
Concertina swap
An interest-rate swap whose notional principal varies according to the present value of an existing
fixed-rate paying swap and which is used to increase near-term protection from high floating rates.
While the notional principal is normally adjusted within a concertina, rate and tenor can also
adjusted. Also known as an accordion swap and a net present value (NPV) swap.
[Cross]-currency swap
The spot sale of one currency for another combined with a simultaneous forward agreement to
repurchase the agreed currency amounts at a pre-set date and an agreement by the counterparty in
the lower interest rate currency to make periodic payments to the counterparty in the higher
interest rate currency in that currency. This payment is approximately equal to the interest rate
differential between the two currencies. Also known as a cross-currency [interest] rate swap. As
with interest rate swaps they can be fixed-fixed, fixed-floating or floating-floating.
Example The issuer of a five-year $100 million 6.0% fixed-rate Eurobond is funding a project
which will generate a five-year floating-rate yen return. It wants to hedge the mismatch between
the floating-rate yen asset and the fixed-rate dollar liability (i.e. against yen depreciation and
falling yen rate) to lock in the spread on the project. Currency swap rates are at 6.20% against yen
Libor. The issuer pays yen Libor and receives 6.20% in USD. The notional amount is $100 million
on the dollar side and ¥12.5 billion on the yen side. As well as the periodic payments under the
swap there are two principal exchanges: at the initiation of the swap the issuer pays $100 million
(the bond proceeds) to the swap counterparty and receives ¥12.5 billion. On maturity this
exchange reverses: the issuer pays ¥12.5 billion to the counterparty and receives $100 million with
which it redeems the bond. The combination of the swap and bond synthesizes a five-year floating
yen liability and the issuer retains any spread earned.
Cross-currency basis swap
A basis swap in which the reference indices are in different currencies. Principal amounts can be
exchanged or the basis swap can be structured as a coupon swap. They are functionally just a
rolling series of short-dated forward foreign exchange transactions and are used to hedge a wide
variety of cross-currency products such as cross-currency interest rate swaps and cross-currency
equity swaps. They are also used by borrowers to transfer liquidity in one currency into their
desired funding currency.
Deferred [coupon] swap
A swap in which some or all of the payments are deferred for a pre-set period after they have been
calculated and come due. These are tax or accounting driven and payments tend to be deferred
across fiscal year ends and other key balance sheet dates.
Equity swap
A type of total return swap (see chapter 16) in which both capital appreciation and any dividend or
coupon income from one index or individual equity are exchanged for a floating-rate payment
usually based on a Libor plus or minus a spread. The instrument swaps all the economic risks
associated with the underlying without actually transferring the underlying, often with tax
advantage. They are useful for gaining unfunded (leveraged) exposure to equity markets and can
also be used for quick and relatively cheap asset allocation alterations.
So, for example, an investor long US dollar floating rate assets wishes to exchange that exposure
for exposure to the S&P500 equity index. He enters a swap paying US dollar Libor and receiving
the total returns where positive from the S&P500 plus or minus a margin. If the index returns are
negative he pays the difference between zero and the index performance to the counterparty in
addition to the floating rate payment. Typically both the index-return payments and the floatingrate payments occur monthly or quarterly. The payments are calculated on a notional principal
amount that is not exchanged. Payment streams can be denominated in the same or different
currencies. Equity swaps can be structured to have variable or fixed notional principal; they can be
single- or cross-currency; and the cross-currency versions can be currency hedged (quantized) or
unhedged.
An equity swap is essentially a long-term equity future and so the cost of carry is crucial in pricing
(and determines the margin paid out with or deducted from the index returns. The payer of the
index return is short the index. To hedge this position he borrows floating rate, using the Libor
payment stream he receives from the swap counterparty to service the loan, and buys the index. To
fulfil his obligation to pay the total returns from the index, he pays out the dividends and capital
appreciation he receives from his position in the index.
Fixed [-for-] fixed swap
Currency swaps in which both counterparties pay a fixed rate.
Fixed-rate payer
The swap counterparty that undertakes to pay fixed in a swap. Also said to be the buyer of or long
the swap. So the floating-rate payer is the swap counterparty that undertakes to pay floating in a
swap. Also said to be the seller of or short the swap.
Forward/deferred [start] swap
A swap that begins at an agreed date in the future. Forward or deferred start swaps are used to
lock-in funding costs commencing at a specified time in the future when the borrower believes
that funding costs will rise significantly in the intervening period. They can also be used to extend
existing swaps or liabilities to suit a changing asset or liability profile. A forward start swap whose
start date coincides with the termination date of an existing swap and which will automatically
extend the original transaction is also known as an extension swap. Forward start swaps can be
used to re-finance capital markets transactions and monetize embedded call options in bonds in
much the same way as swaptions.
Interest rate swap
An agreement between two counterparties to exchange interest rate payments on a notional
principal sum which is not exchanged. The commonest structure is the fixed-for-floating swap in
which one counterparty agrees to pay a fixed rate over the term of the swap in exchange for a
floating-rate payment payable by the other counterparty.
The vanilla fixed-for-floating interest rate swap is also sometimes called a coupon swap since it
can be viewed as swapping the coupons from two bonds with the same principal. A swap, viewed
from the pay fixed side, can be considered either as a portfolio of FRAs all with the same strike or
as a portfolio which is short a coupon bond and long an FRN or, alternatively, as the combination
of a cap and floor with the same strike.
Swaps are actively traded and are generally quoted on a yield basis, that yield being the yield to
maturity that equates the present value of the fixed side to that of the floating side. Quotes
generally refer to the fixed leg or coupon. A five-year dollar swap quoted at 60 bid 65 offer means
that a counterparty wishing to pay fixed and receive Libor flat would have to pay the marketmaker a fixed rate which is 65 basis points over the yield to maturity of the five-year US treasury
at the time the swap is initiated. If he wanted to pay Libor and receive fixed, the counterparty
would receive a fixed-rate of 60 bp over. So, the swap bid is the price at which a counterparty will
buy a stream of Libor-linked cash-flows and the offer is the price at which they would sell a
stream of Libor-linked cashflows.
Swap pricing depends on the term structure of interest rates, the swap spread, transaction costs and
credit risk. There is generally no upfront premium for a swap, as at the outset of the swap both
parties are theoretically indifferent as to whether they are in fixed or floating: the net present value
of the two payment streams is zero. Since the price of an interest rate swap is the level at which
the market is indifferent between paying a fixed rate or interest and a stream of Libor, it depends
entirely on implied forward Libor rates. This means that a hedger must, before he decides to fix,
determine whether he believes rates will rise as far as the implied forward curve implies. In steep
yield curve environments, where the implied forward curve is even steeper, fixing incurs negative
carry.
Example
A typical hedging application would be a corporate treasurer with US$1 billion of US dollar
outstanding floating rate debt who believed that dollar interest rates were set to rise. To increase
his level of fixed-rate debt and protect himself against rate rises, this treasurer could enter into a
fixed-floating semi-annual swap on US$500 million notional principal under which he would pay
a fixed rate (the swap rate) and receive a floating rate linked to an index such as Libor. Every six
months, a net interest payment is made between swap counterparties. If the prevailing level of
dollar Libor is higher than the fixed rate (the swap rate) then the swap counterparty pays the
treasurer the difference. If the swap rate is higher than Libor, the treasurer pays the counterparty
the difference. This netting fixes the treasurer’s interest rate. The swap can be reversed at any
time. The unwind valuation is the difference between the present values of two sets of cashflows:
that of the future cash flows payable/receivable under the swap and that of the cash flows for a
matching but offsetting swap. The market is effectively buying the right to continue the swap on
its original terms. If these are better than the current terms then the swap has positive value. On a
five-year swap that had run for one year, the comparison would be with a current four year swap.
If the implied forward curve had shifted up sufficiently for the current four-year swap rate to
exceed the original five-year swap rate, then the swap would have positive value as the market
would be able to buy the (now higher) stream of Libors for the old (lower) price.
Leveraged swap
A swap in which the fixed-rate receiver receives an above-market fixed rate and pays a multiple of
the floating rate index.
Example An investor who believes that the future spot rate will be lower than the rate implied by
the forward curve can simply receive fixed and pay floating under a swap. To increase returns, he
can transact the swap on twice the notional principal of his liabilities. If he has a limit on notional
principal, he can substitute a leveraged swap: he pays twice (or more) the floating rate but on the
same notional principal as the original swap. At the extreme he can receive a very high fixed rate
and pay Libor-squared. Since Libor-squared rises faster the higher Libor is, this is extremely
speculative.
Libor-in-advance swap
An interest rate swap in which the Libor rate is reset at the beginning of the previous period except
for the first period where Libor is set at the beginning of the corresponding period as in a
conventional swap. This effectively shifts the floating Libor periods back by one period except for
the first. The Libor-in-advance swap allows the fixed-rate payer to pay a lower fixed rate in
exchange for receiving Libor in advance in the same type of interest rate environment if the yield
curve is positive.
Libor-in-arrears swap
In a conventional swap, floating interest payments are reset in advance, at the beginning of each
(usually semi-annual) period and paid in arrears. So the six-month Libor rate payable in six
months’ time is determined by the Libor rate in effect at contract origination and paid at the end of
the six-month period. At the 12-month settlement, the coupon payment is determined by the sixmonth Libor rate prevailing at month six and so on.
In a Libor-in-arrears swap, interest payments are both set and paid in arrears. That is, the first
Libor fixing is after six months, just two days before the payment date, and is determined by the
six-month Libor rate in effect at month six (not at contract origination) and subsequent rates are
set at the end of each period. So, with a standard swap both parties know the amount of the
floating-rate payment six months in advance. With the Libor-in-arrears swap, neither party knows
what the payment will be until it is due. This effectively extends the floating-rate payer’s exposure
to Libor by one additional interest period and means that the forward rates that are used to
determine the fixed-rate payment in the swap are one period further out than on a standard swap. If
the yield curve is steeply positive, this means that the fixed-rate for the Libor-in-arrears swap will
be higher than for the standard swap because the forward rates are higher.
Another way of looking at it is that the market is implying that short-term rates will rise. Therefore
the market expects that setting Libor in arrears will result in a higher Libor being set and therefore
a higher payment than if Libor is set normally. Therefore the market will pay an incentive to any
counterparty that wishes to pay Libor in arrears. So, if the market is expecting Libor to be on
average higher at the end of each six-month period by 50 bp, then in a floating-floating Libor-inarrears swap a counterparty could receive Libor and pay Libor-in-arrears less 50 bp. The swap
would be advantageous if Libor falls over the period or rises by less than 50 bp. This shows how
the swap is priced: the market expects Libor to rise 50bp over each floating period and so is
willing to receive Libor-set-in-arrears less 35 bp. The price adjustment is therefore the present
value of the average expected increase in Libor over the period, calculated from implied forward
Libors for that period. Also known as an arrears [rate] reset swap, delayed Libor reset swap.
Libor-in-arrears swaps are a way of taking a view that future spot rates will be lower than those
implied by the forward curve, though the buyer’s view on absolute rates may not be much
different from that expressed by a conventional swap. If interest rates do not rise as sharply as the
yield curve suggests, the Libor payments will be less than those on a conventional swap. Most
commonly they are used by fixed-rate receivers (for example, treasurers swapping fixed-rate bond
issues into floating) who benefit from the steepness of the yield curve by paying Libor-in-arrears
in exchange for a higher fixed rate. Floating-floating versions are sometimes used by investors
who would receive Libor and pay Libor-in-arrears if they believed rates will not rise as fast as the
implied forward curve suggests.
There are a number of more recent variants of the structure. In a less aggressive version of the
Libor-in-arrears swap, counterparties can choose to receive a fixed-rate and pay floating with the
other counterparty having the option to pay Libor-in-arrears (and receive a higher fixed rate). The
fixed-rates payable will be lower than that in the full Libor-in-arrears swap to take into account the
cost of this option. Alternatively, if the counterparty wants to take a more aggressive view on the
forward curve than in the standard Libor-in-arrears swap, he can choose to receive an even higher
fixed rate than in the Libor-in-arrears swap in exchange for agreeing to pay the greater of sixmonth Libor and six-month Libor in arrears. This floating rate liability could be capped at a
catastrophe level.
Mark-to-market swap
A standard swap (of any kind) except that, periodically, it is marked to market. The counterparty
on whose side the mark-to-market value is positive pays that value to the counterparty showing a
mark-to-market loss. This greatly reduces the credit exposure the counterparties have to each other
through the swap. Options on this varying exposure are also available see mark-to-market cap.
Credit derivatives hedge this credit risk in a different way by guaranteeing to make good any
mark-to-market loss realized in the event of default. See swap guarantee.
Mismatched payment swap
A swap in which payment streams are not exchanged on the same date. For example, the floating
amounts are payable semi-annually but the fixed amounts are payable quarterly.
Multi-rate reset swap
A swap in which the reset and payment periods are unusually frequent. For example, an interest
rate swap in which the floating-rate payer pays one-month Libor on a monthly basis.
Off-market swap
A swap in which the fixed-rate payments are below or above the market rate. Where it is below
the swap is known as a discount swap. At maturity the discount is repaid with one payment. The
structure is useful in financing projects which will not generate income to pay under the swap until
they are completed. When the fixed rate is above the market rate the structure is known as a highcoupon swap. The floating-rate payer may compensate the fixed-rate payer either by higher
periodic payments or by payment of an upfront fee.
Overnight indexed swap (OIS)
A fixed-to-floating interest rate swap whose floating rate is based on a weighted average rate for
overnight transactions. The two counterparties exchange at maturity the difference between
interest accrued at the fixed rate and the compounded daily overnight rate. These swaps are used
to hedge cash deposits.
Real-estate swap
A swap involving the exchange of the returns from a pre-agreed property index, such as the US
Russell NCREIF Property Index, a benchmark index which takes into account the yield on 1,800
properties throughout the US, for a financial index such as Libor. Such swaps are used by
institutional investors who wish to re-allocate assets away from property at times of low yields,
but who do not want to take the capital loss of selling the property in a bear market.
Roller-coaster swap
A generic name applied to swaps whose notional principal is different in different payment
periods. Such swaps’ notional principal generally increases and decreases periodically to
accommodate cashflows that differ predictably on a seasonal basis or to accommodate debt
obligations scheduled to rise and fall. Hence the alternative name, the seasonal swap.
Rolling reset swap
A swap where one counterparty pays the lower of the arranged swap rate or the prevailing market
rate on the roll date for the same tenor.
Roll-lock swap
A swap used to hedge roll risk. This is the risk that long-term hedgers face when using short-term
contracts. As each expiration approaches, hedgers sell futures contracts they own and re-enter the
position in a more distant month. The cost differences can be expensive and can also create
tracking error. Under a roll-lock swap the roll-lock payer pays the average of the cost of the roll
(defined as the difference between the near and next futures contract) measured at pre-agreed
times before expiration. The roll-lock receiver pays a Libor-based rate set at a pre-agreed time
after the expiration of the near contract. Also known as a roll-over lock.
step up [coupon] swap
An interest rate swap whose fixed-rate payments rise over time according to a schedule
determined at the outset of the swap. Also known as an escalating rate swap.
[Swap] spreadlock
A contract that locks in a predetermined swap spread for a deferred start swap. So a future fixedrate payer {receiver} is guaranteed a maximum {minimum} spread over a specified benchmark
index (usually a government bond rate) in a forward swap. The swap provider agrees to provide a
borrower/hedger with a swap deferred over a defined period (usually less than six months) at a
preset spread over a reference (usually Treasury) rate comparable to the maturity of the swap into
which the hedger is obliged to enter. This guarantees the issuer of a bond, for example, a swap at a
known credit margin over the relevant reference rate while enabling that borrower to take
advantage of any absolute movement in rates. A payer {receiver} spreadlock allows the holder to
enter into a swap paying {receiving} the reference rate plus the agreed spread. Also known as a
deferred rate setting swap.
Tax-exempt swap
An interest rate swap with one or both payment streams based on tax-exempt US municipal bond
yields or a tax-exempt index such as the JJ Kenney. Also known as municipal swaps.
Variable maturity swap
A swap whose maturity is uncertain but whose range is predefined. For example a swap whose
maturity is between two and three years contingent on Libor reset dates.
Yield curve [arbitrage] swap
A swap in which the counterparty moves up or down the yield curve twice in the same swap, yield
curve swaps are a type of basis swap in which a shorter-term floating-rate index is swapped for a
longer-term floating-rate index.
In the commodity markets such swaps are known as contango swaps and backwardation swaps
depending on whether the forward curve is positively or negatively sloped. In both, counterparties
exchange a payment stream based on the nearby futures contract for one based on a more distant
futures contract. A yield curve swap in which the returns from more than one market are swapped
for the returns from one market is known as a combination yield curve swap. For example, a
counterparty might pay the two-year CMS Deutschmark rate and receive 50% of the two-year
CMS Deutschmark rate plus 50% of the two-year CMS yen rate.
A yield curve swap can be viewed as a series of forward swaps each of which starts on the yield
curve swap’s reset dates. (This is also one way to hedge them but in practice it is expensive and
hedging is done on a portfolio basis).
Example 1
An investor believes that the US dollar yield curve will steepen. He enters into a US$50 million
notional principal yield curve swap under which he pays three-month Libor and receives the 10year Constant Maturity Swap (CMS) rate less 200 basis points. If the spread between short and
long rates widens, then the investor can reverse the swap or close it out to take profits. The swap is
priced by comparing the forward curves for the two indices, valuing the implied cash flows
separately and then calculating a swap spread such that the net present value of the two implied
cashflows is zero.
Example 2 A commodity consumer could use a backwardation swap to fix the spread differential
between spot and forward prices to offset the costs they would incur if the spread relationship
reversed, for example if absolute prices fell. Under this type of swap the consumer might pay the
average daily price of the nearby futures contract and receives the six-month or 12-month contract
plus a spread. If the curve flattens, the profit on the swap offsets the higher cost of hedging new
forward purchases. In a contango swap the user locks in a favourable contango, or positive spread,
between forward and nearby prices. So an oil producer might pay the monthly average of the daily
difference between the nearby and 12-month WTI futures contract on a pre-agreed notional
principal amount of oil and receive a fixed spread of 30 cents per barrel. Another way of looking
at the structure is that the producer pays a floating amount equal to the average of the 12-month
futures contract and receives a floating payment equal to the average nearby contract plus the 30
cent spread. This enables the commodity producer to lock in the positive spread between forward
and nearby prices and also to hedge against anticipated backwardation.
Zero-coupon swap
An interest rate swap in which the floating payment streams are usually conventional but the
fixed-rate payments are made through a single lump sum payment calculated on the basis of the
present value, discounted to that payment date, of the stream of fixed payments that would have
been payable over the term of a conventional swap. The present value is usually adjusted to take
account of the greater credit risk involved in this kind of mismatched structure. The lump sum
payment can be made at any time during the life of the swap.
The term zero coupon swap is usually applied to a swap in which the fixed-rate payments are
deferred to maturity. If the lump sum payment is made at the outset of the transaction the swap is
known as a reverse zero-coupon swap or pre-paid swap. In both cases the mismatched structure
makes the swap is functionally equivalent to a loan and entails similar credit risks, though it is
usually off-balance sheet for accounting purposes. Zero-coupon swaps can also be used to hedge
the payment stream on a zero coupon bond.