A derivative is a (financial) contract, whose value depends on the future value of an underlying
asset (in our examples, car prices).
 Forward contract: the buyer and seller agree today on the delivery of a specified quantity and
quality of an asset at a future date, for a given price. Futures are similar to forwards, except they
have a standardized specification and are typically traded on organized exchanges. In addition,
while in a forward any profits and losses are realized at maturity, futures P&L are realized daily,
in a process known as mark–to–market. Another difference is that typically, forward contracts
are kept open until maturity, and the underlying is delivered. In contrast, futures are typically
closed before maturity, and no delivery of the underlying asset takes place.
 Options: the buyer of a call option contract obtains the right, but not the obligation, to buy a
given asset at a given strike price. Likewise, the buyer of a put option obtains the right, but not
the obligation, to sell a given asset at a given strike price. American options give their holders
the right to exercise (buy, financial derivatives 5 if a call; sell, if a put) at any date until maturity.
European options provide the right to exercise only at maturity.
 Swaps: involve the periodic exchange of cash flows between two parties.
The principal use of derivatives is as hedging instruments. Hedgers, speculators, and arbitrageurs
are the main users of derivatives. When an investor is exposed to a risk they do not want to bear,
they may enter a derivative contract to transfer it to another party who is willing to accept it.
Speculators, on the other hand, are happy to bear risk. They buy or sell derivative contracts in the
hope to profit from future price changes. Finally, arbitrageurs trade derivatives with the objective
of exploiting mispricing in the market, to make a profit.
Derivatives can generate leverage: the ability to take large speculative positions with little initial
capital.
Forwards
Payoff to the buyer: 𝑆𝑡 − 𝐹0
Payoff to the seller: 𝐹0 − 𝑆𝑡
Because it costs nothing to enter a forward contract, the payoff from the contract is also the trader
total gain loss from the contract.
Currency: If you take a long position in the US dollar forward, and the dollar appreciates (𝑆𝑡 > 𝐹0),
you make a gain. For the short position, it’s the opposite.
Futures
Suppose that the futures price (of a commodity) is above the spot price during the delivery period.
Traders then have a clear arbitrage opportunity:
1. sell (short) futures contract
2. Buy the asset
3. Make delivery
Strategy: buy the cheaper one (asset in this case), and sell the expensive one (future). So: the
arbitrager sells (shorts) the future contract and buys the asset (commodity), and makes delivery
for an immediate profit.
If the future price is less than the spot price, there exists an arbitrage opportunity but no similar
perfect strategy. Companies interested in acquiring the asset will find it attractive to enter into a
long futures contract and then wait for delivery to be made. An arbitrager can take a long futures
position but cannot force immediate delivery of the asset. The decision on when the delivery will
be made is made by the party with the short position.
Under the forward contract, the whole gain/loss is realized at the end of the life of the contract,
while under the futures, it is realized day by day.
2.4 Suppose that in September 2012 a company takes a long position in a contract on May 2013
crude oil futures. It closes out its position in March 2013. The futures price (per barrel) is $68.30
when it enters into the contract, $70.50 when it closes out its position, and $69.10 at the end of
December 2012. One contract is for the delivery of 1,000 barrels. What is the company’s total profit?
Total profit: 1000($70,50 - $68,30) =2200
Margins
Entering a future contract requires a margin, i.e deposit of cash or highly liquid securities (e.g.
Treasuries) with a broker. This means that entering a futures contract is not “free” as for forwards.
The buyer (long position) earns ΔFt = Ft − Ft−1; the seller (short position) earns −ΔFt. If any losses
accumulate, they are charged to the margin account of the trader who makes them. This is
equivalent to closing the contract and opening a new one every day.
Margin accounts have a maintenance level, i.e. a minimum level M∗ such that, if the margin account
goes below M*, the trader must either close her position, or replenish the margin account up to
restoring it to the initial level.
Another important feature of futures trading is the fact that a clearinghouse stands ready to take a
position in all contracts, at any point in time.
2.3 Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. The
size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is
$3,000. What change in the futures price will lead to a margin call? What happens if you do not meet
the margin call?
There will be a margin call when $1000 has been lost from the margin account (until we reach the
maintenance margin.
5000 ∗ (17,20 − 𝐹𝑡) = −1000
𝐹𝑡 = 17,40
A 0,20 change in the future price will lead to a margin call. If the margin call is not met, the broker
closes out my position.
CHAPTER 3: HEDGING
When an individual or company choses to use futures markets to hedge a risk, the objective is
usually to take a position that neutralizes the risk as far as possible. Consider a company that knows
it will gain $10,000 for each 1 cent increase in the price of a commodity over the next 3 months and
lose $1, for each 1 cent decrease in the price during the same period. To hedge, the company's
treasurer should take a Short futures position that is designed to offset this risk. The futures
position should lead to a loss of $l, for each 1 cent increase in the price of the commodity over the 3
months and a gain of $1,00 for each 1 cent decrease in the price during this period. If the price of
the commodity goes down, the gain on the futures position offsets the loss on the rest of the
company's business. If the price of the commodity goes up, the loss on the futures position is offset
by the gain on the rest of the company’s business.
Short hedges: A short hedge is a hedge that involves a short position in futures contracts. A short
hedge is appropriate when the hedger already owns an asset and expects to sell it at some time in
the future. Ex: a US exporter who knows that he will receive euros in 3 months. The exporter will
realize a gain if the euro increases in value relative to the US dollar and will sustain a loss if the euro
decreases in value relative to the US dollar. A short futures position leads to a loss if the euro
increases in value and a gain if it decreases in value.
Long hedges: Hedges that involve taking a long position in a futures contract. It is appropriate when
a company knows it will have to purchase a certain asset in the future and wants to lock in a price
now.
Example
It is January 15. A copper fabricator knows he will need 100000 pounds of copper on May 15. The
spot price of copper is 340 cents per pound, and the futures price for may delivery is 320 cents. The
fabricator can hedge its position by taking a long position in 4 futures contracts (with delivery of
25,000 pounds) and closing its position on May 15. On may 15:
1. St = 325 cents. Gain: 100,000 x ($3.25-$3.20) = $5,000 on the futures contracts
Pays 100,000x$3.25 = $325,000 for the copper. Net cost = $320,000
2. St = 305. Loss: 100,000 x ($3.20-$3.05) = $15,000 on the futures
Pays $305,000 for the copper. Net cost = $320,000
BASIS RISK
In practice, hedging is often not quite as straight forward. Why?
1. The asset whose price is to be hedged may not be exactly the same as the asset underlying
the futures contract.
2. The hedger pay be uncertain as to the exact date when the asset will be bought or sold.
3. The hedge may require the futures contract to be closed out before its delivery month.
These problems give rise to the basis risk.
Basis = Spot price of asset to be hedged – Futures price of contract used = ST - FT
Basis risk can lead to an improvement or a worsening of a hedger's position.
 Short hedge. If the basis strengthens (increases) unexpectedly, the hedger's position improves.
If the basis weakens (decreases) unexpectedly, it worsens.
 Long hedge: If the basis strengthens (increases) unexpectedly, the hedger's position worsens.
If the basis weakens (decreases) unexpectedly, it improves.
The asset that gives rise to the hedger's exposure is sometimes different from the asset underlying
the futures contract that is used for hedging. This increases the basis risk.
Suppose it is March, your home currency is the U.S. dollar, and you have a 375, 000€ receivable in
July. Suppose you enter your futures with a short position on the Euro, at a futures price F0 =
$1.20/€. Let’s consider two scenarios:
(a) In July, ST = $1.10/€, and the September futures contract price is FT = $1.125/€. The difference
ST − FT = −$0.025/€ is known as the contract’s basis. Because the basis is unknown at the time we
enter the contract (March), you face basis risk. Your Euro receivable is worth €375, 000 × $1.10/€
= $412, 500; the payoff on your futures hedge is €375, 000 × (FT – F0) = $28, 125; thus your net
payoff is $440, 625.
(b) In July, ST = $1.25/€, and the September futures contract price is FT = $1.27/e. The basis is thus
ST − FT = −$0.02/€. Your Euro receivable is worth $468,750; the payoff from your futures hedge is
€375, 000 × (FT − F0) = −$26, 250; thus your net payoff is $442, 500.
The net payoffs in the two scenarios differed because of basis risk. Your unhedged position (payable
or receivable) is worth ST at maturity. Your futures hedge gives you a payoff equal to −(FT − F0). As
a result, your net payoff is ST − (FT − F0) = F0 + Basis. Intuitively: If there were no basis (and no basis
risk), you would lock in a payoff equal to F0. The basis generates some volatility around that payoff.
To design an optimal hedging policy, we need to minimize basis risk.
ln general, basis risk increases as the time difference between the hedge expiration and the
delivery month increases. Because of this, you choose a delivery month that is as close as possible
to, but later than, the expiration of the hedge. Suppose delivery months are March, June, September
and December for a futures contract on a particular asset. For hedge expirations in December,
January and February, the March contract will be chosen; for hedge expirations in March, April, and
May, the June contract.
Example 1: taking a short position
Gain on the futures contract: F0 – Ft = 0.78 – 0.7250 = 0.0550 cents per yen.
Basis: St – FT = 0.7200 – 0.7250 = -0.0050 cents per yen when contract is closed out.
The effective price obtained in cents per yen is the final spot price PLUS the gain on the futures:
𝑆𝑇 + (𝐹𝑂 − 𝐹𝑇 ) = 𝑂. 7200 + 0.0550 = 0.7750
Or we can write it as F0 + Basis = 0.7800 – 0.0050 = 0.7750
The total amount received by the company for the 50 million yen is 50m x 0.00775 = $387,500
Example 2: Long position
Gain on the futures contract: FT – F0 = 69.10 – 68 = $1.1O per barrel.
Basis: ST – FT = 70 – 69.10 = $0.90 per barrel when contract is closed out.
The effective price paid (in dollars per barrel) is the final spot price LESS the gain on the futures:
𝑆𝑇 − (𝐹𝑇 − 𝐹0 ) = 70 − 1.10 = 68.90
Or we can write it as F0 + Basis = 68 + 0.90
Total price paid is 68.90 x 20,000 = $1,378,000
Cross hedging and optimal hedge ratio
The hedge ratio is the ratio of the size of the position taken in futures contracts to the size of the
exposure. When the asset underlying the futures contract is the same as the asset being hedged; it
is natural to use a hedge ratio of 1.
𝜎
The hedge ratio that minimizes the variance of the hedger’s position is: ℎ∗ = 𝜌 𝜎𝑆 where 𝜎𝑆 is the
𝐹
standard deviation of S, i.e the asset price volatility.
Intuitively, basis risk is minimized if the price of the asset we would like to hedge S and the futures
price F move closely together. We thus want to pick the optimal hedge ratio h∗ such that movements
in the value of our hedging instrument ΔF track changes in the asset’s price ΔS.
𝑐𝑜𝑣(∆𝑆, ∆𝐹)
ℎ∗ =
𝑣𝑎𝑟(∆𝐹)
𝜎 2
Hedge effectiveness is the proportion of the variance that is eliminated by hedging, ℎ∗ 2 𝜎𝑆 2
𝐹
Estimate the relation between change in the asset price S and futures price 𝐹: ΔS = 𝑎 + hΔ𝐹 + 𝑒
 h = 0 ⇒ No hedging possible
 h > 0 ⇒ Hedge a drop in S by shorting 𝐹
 h < 0 ⇒ Hedge a drop in S by going long 𝐹
3.4 Under what circumstances does a minimum variance hedge portfolio lead to no hedging at all?
A minimum variance hedge leads to no hedging when the coefficient of correlation between the
futures price changes and changes in the price of the asset being hedged is 0.
𝜌 = 𝐶𝑜𝑒𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 ∆𝑆 𝑎𝑛𝑑 ∆𝐹
Relation between change in the asset price 𝑃 and related futures price 𝐹: Δ𝑃 = 𝑎 + hΔ𝐹 + 𝑒
If h = 0 ⇒ No hedging possible: there is no relation between Δ𝑃 and ΔF.
𝑐𝑜𝑣(∆𝑃, ∆𝐹)
ℎ∗ =
𝑣𝑎𝑟(∆𝐹)
Since 𝑐𝑜𝑣(∆𝑃, ∆𝐹) = 𝜌𝜎𝑆 𝜎𝐹 , ℎ = 0 only if 𝜌 = 0.
3.6 Suppose that the standard deviation of quarterly changes in the prices of a commodity is $0.65,
the standard deviation of quarterly changes in a futures price on the commodity is $0.81, and the
coefficient of correlation between the two changes is 0.8. What is the optimal hedge ratio for a 3month contract? What does it mean?
The hedge ratio is the ratio of the size of the position taken in futures contracts to the size of the
𝜎
0,65
exposure. The optimal hedge ratio is: ℎ∗ = 𝜌 𝑆 = 0,8 ∗
= 0.642
𝜎𝐹
0,81
This means that the size of the futures position should be 64.2% of the size of the company’s
exposure in a three-month hedge.
Optimal number of contracts
QA: size of position being hedged (units)
QF: size of one futures contract (units)
N*: optimal number of futures contracts hedging
ℎ∗ 𝑄𝐴
∗
𝑁 =
𝑄𝐹
When futures are used for hedging, the equation becomes: 𝑁 ∗ =
ℎ ∗ 𝑉𝐴
𝑉𝐹
where VA is the value of the
position being hedged and VF is the value of one futures contract (the futures price times QF)
Stock index futures
Hedging an equity portfolio
P: current value of the portfolio
F: current value of one futures contract (futures price x contract size)
If the portfolio mirrors the index: h* = 1 and N* = P/F
But if your portfolio does not perfectly track the index, h* = β. A portfolio with β=2 is twice as
sensitive to market movements as a portfolio with β = 1. It is necessary to use twice as many
𝑃
contracts to hedge the portfolio. 𝑁 ∗ = 𝛽
𝐹
3.7 A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts
on a stock index to hedge its risk. The index futures is currently standing at 1080, and each contract
is for delivery of $250 times the index. What is the hedge that minimizes risk? What should the
company do if it wants to reduce the beta of the portfolio to 0.6?
ℎ∗ = 𝛽 = 1.2
The number of contracts that should be shorted is:
𝑃
20000000
𝑁 ∗ = 𝛽 = 1.2 ∗
= 88.88
𝐹
1080 ∗ 250
Where P is the current value of the portfolio (20,000,000) and F the current value of one futures
contract (futures price times the contract size)= 1080*250.
To reduce the beta to 0.6, half of this portion is required: short position in 44 contracts.
3.18 On July 1, an investor holds 50,000 shares of a certain stock. The market price is $30 per share.
The investor is interested in hedging against movements in the market over the next month and
decides to use the September Mini S&P 500 futures contract. The index futures price is currently
1,500 and one contract is for delivery of $50 times the index. The beta of the stock is 1.3. What
strategy should the investor follow? Under what circumstances will it be profitable?
50,000 ∗ 30
𝑁 ∗ = 1.3 ∗
= 26
1500 ∗ 50
The investor should short 26 contracts to hedge the portfolio. It will be profitable if the stock in the
portfolio outperform the market in the sense that its return is greater than the predicted by the
capital asset pricing model.
Changing the Beta: To change the beta of the portfolio from 𝛽 𝑡𝑜 𝛽∗ , where 𝛽 > 𝛽∗ , we need to short
𝑃
𝑃
(𝛽 − 𝛽∗ ) 𝐹 contracts. When 𝛽 < 𝛽∗ , we need to long (𝛽∗ − 𝛽) 𝐹 contracts.
Reasons for hedging an equity portfolio: the hedger feels that the stocks in the portfolio will
outperform the market. A hedge using index futures removes the risk arising from market moves
and leaves the hedger exposed only to the performance of the portfolio relative to the market.
Another reason may be that the hedger is planning to hold a portfolio for a long period of time and
requires short-term protection in an uncertain market situation.
Summary: a company can take a position in futures contracts to offset an exposure to the Price of
an asset. If the exposure is such that the company gains when the price of the asset increase, and
loses when the price of the asset decreases, a short hedge is appropriate. If the exposure is the other
way round (i.e., that company gains when the price of the asset decreases and loses when the price
of the asset increases), a long hedge is appropriate.
Case study: Jet Blue
JetBlue has been hedging its exposure to jet fuel prices using WTI derivatives. Would it make sense
to switch to Brent? 𝛥𝐽𝑒𝑡𝐹𝑢𝑒𝑙𝑃𝑟𝑖𝑐𝑒 = 𝑎 + ℎ𝛥𝐹 + 𝑒
 Δ𝐹 : change in the price of the hedging instrument
CHAPTER 5: FORWARD AND FUTURES PRICES
Derivatives are priced by no–arbitrage: market forces align prices such that no investor can obtain
an “abnormal” profit (or return). Forward and futures prices are usually very close when the
maturities of the two contracts are the same.
Forward prices
Law of one price: If two assets, or portfolios of assets, give you identical payoffs tomorrow, today
they must have the same price.
From the short position: you face the risk that the underlying price appreciates (St > F0) and you
make a loss of F0 – ST.
The value of the portfolio at T must also be zero. 𝐹0 = 𝑆0 (1 + 𝑟)𝑇
Known Income
Known Yield
Valuing forward contracts
Valuing forward contracts: K is the delivery price for a contract that was negotiated some time ago,
the delivery date is T years from today, and r is the T-year risk-free interest rate. The variable F is
the forward price that would be applicable if we negotiated the contract today. We also define
ℱ = 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡 𝑡𝑜𝑑𝑎𝑦
At the beginning of the life of the forward contract, the delivery price, K, is set equal to the forward
price, 1$, and the value of the contract, 𝑓, is 0. As time passes, K stays the same, but the forward
price changes and the value of the contract becomes either positive or negative.
 Value of a long forward contract: ℱ = (𝐹0 − 𝐾)𝑒 −𝑟𝑇
 Value of a short forward contract: ℱ = (𝐾 − 𝐹0 )𝑒 −𝑟𝑇
The value of a forward contract on an investment asset that provides no income:
𝓕 = 𝑺𝟎 − 𝑲𝒆−𝒓𝑻
5.9 A 1-year long forward contract on a non-dividend-paying stock is entered into when the stock
price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.
a) What are the forward price and the initial value of the forward contract?
𝐹0 = 40𝑒 0,1∗1 = $44.21
The initial value of the forward contract is 0.
b) Six months later,the price of the stock is $45 and the risk-free interest rate is still 10%. What are
the forward price and the value of the forward contract?
The delivery price K = F0 = $44,21. The value of the forward contract:
𝓕 = 𝑺𝟎 − 𝑲𝒆−𝒓𝑻 = 45 − 44,21𝑒 −0,1.0,5 = $2,946
The forward price is: 𝐹0 = 45𝑒 0,1∗0,5 = $47.31
Forwards vs futures: When the r is constant and the same for all maturities, the forward price for
a contract with a certain delivery date is in theory the same as the futures price for a contract with
same date. But
 If S is strongly correlated with r, a long futures contract is more attractive, and futures prices
will tend to be slightly higher than forwards’.
 If S is negatively correlated with r  forward prices are higher.
Future prices of stock indexes:
Forwards and futures contracts on currencies
From the perspective of a US investor, where r is the US dollar rate:
𝐹0 = 𝑆0 𝑒 (𝑟−𝑟𝑓)𝑇
If this isn’t true, there would be arbitrages opportunities:
Suppose rf = 7% and r=5%, S0 = 0.62 USD per AUD. The 2-year forward exchange rate should be:
0.62𝑒 (0.07−0.05)2 = 0.6453
1. If F = 0.63 < F*, AUD is undervalued. An arbitrageur can:
a. Borrow 1000 AUD at 5% per annum for 2 years, convert to 620 USD and invest the USD
at 7% (continuous compounding)
b. Enter into a forward contract to BUY 1000𝑒 0.05𝑥2 = 1,105.17 (to repay 1000 aud) for
1105.17 ∗ 0.63 = 696.26 𝑈𝑆𝐷
The 620 usd invested grow to 620𝑒 0.07𝑥2 = 713.17 𝑈𝑆𝐷 in 2 years. Of this, 696.26 usd are used to
purchase 1105.17 aud under the terms of the forward. Riskless profit: 713.17 – 696.26 = 16.91
2. If F = 0.66 >F*, AUD is overvalued. An arbitrageur can:
a. Borrow 1000 USD at 7% per annum for 2 years, convert to 1000/0.62 = 1612.90 AUD,
and invest AUD at 5%
b. Enter into a forward to SELL 1612.90e0.05x2 = 1782.53 AUD for 1782.53x0.66 = 1176.47
USD
The 1612.90 AUD invested grow to 1782.53 AUD in 2 years. The forward contract has the effect of
converting this to 1176.47 USD. The amount need to payoff the USD borrowingis is
1000e0.07x2=1150.27 USD. Riskless profit: 1176.47-1150.27 = 26.20 USD.
Storage costs
Let’s denote the present value (value at t = 0) of the storage costs by C:
F0 = (S0 + C)(1 + r)T or 𝐹0 = (𝑆0 + 𝐶)𝑒 𝑟𝑇
Ex: If it costs $2 per unit to store the asset, with payment being made at the end of the year (T=1)
and r=7%, then C = 2e-0.07x1 = 1.865
If the storage costs net of income incurred at any time are proportional to the price of the
commodity, they can be treated as negative yield. F0 = S0e(r+u)T when u denotes the storage costs
per annum as a proportion of the spot price net of any yield earned on the asset.
Income yield
When the underlying asset generates income: just as storage costs increases the value of the
forward, income makes it cheaper.
F0 = (S0 - I)(1 + r)T
A slightly separate case can be made for forwards (or futures) on stock indexes, such as the S&P500.
Within an index, each stock pays a dividend at a different point in time in general. Because of that,
it can be convenient to move to continuous time, and assume that the index earns an infinitesimal
dividend q at each instant. q is then known as the income (in this case, dividend) yield: 𝐹0 =
𝑆𝑜 𝑒 (𝑟−𝑞)𝑇
Cost of carry
The cost of carry is defined as storage cost, plus the interest paid to finance the asset, minus the
income earned on the asset. For a non-dividend paying stock with no storage cost, the cost of carry
is simply r; with an income yield q (stock index) , it is r – q. For a currency r – rf, and for an asset
that provides income at rate q and has storage costs at rate u, it is r – q + u. If c is the cost of carry:
𝐹0 = 𝑆𝑜 𝑒 𝑐𝑇
5.3 Suppose that you enter into a 6-month forward contract on a non-dividend-paying stock when
the stock price is $30 and the risk-free interest rate (with continuous compounding) is 12% per
annum. What is the forward price?
𝑭𝟎 = 𝑺𝟎 𝒆𝒓𝑻 → 𝐹0 = 30𝑒 0,12∗0,5 = $31.86
5.12 Suppose that the risk-free interest rate is 10% per annum with continuous compounding and
that the dividend yield on a stock index is 4% per annum. The index is standing at 400, and the
futures price for a contract deliverable in four months is 405. What arbitrage opportunities does
this create?
𝑭𝟎 = 𝑺𝟎 𝒆(𝒓−𝒒)𝑻 = 400. 𝑒 (0.1−0.04)∗0.33 = $408.08
Since 405 < 408.08, profits can be made by shorting/selling the stocks underlying the index and
taking long position in/buying future contracts. Strategy: BUY LOW (F0), SELL HIGH: buy future
contracts; sell the shares underlying the index.
5.16 Suppose that 𝐹1 and 𝐹2 are two futures contracts on the same commodity with times to
(𝑡 − 𝑡1 )
maturity 𝑡1 and 𝑡2, where 𝑡1 < 𝑡2. Prove that:
𝐹2 ≤ 𝐹1𝑒𝑟 2
where 𝑟 is the interest rate (assumed constant) and there are no storage costs. For the purposes of
this problem, assume that a futures contract is the same as a forward contract.
If 𝐹2 > 𝐹1 . 𝑒 𝑟( 𝑡2 − 𝑡1 ), an investor could make a riskless profit by:
a. Buying futures contract with maturity t1
b. Selling futures contract with maturity t2
When the first futures contract matures, the asset is purchased for F1 using funds borrowed at rate
r. It is then held until time t2, at which point it is exchanged for F 2 under the second contract. The
costs of the funds borrowed and accumulated interest at time t2 is 𝐹1 . 𝑒 𝑟( 𝑡2 − 𝑡1 ). A positive profit
of 𝐹2 − 𝐹1 . 𝑒 𝑟( 𝑡2 − 𝑡1 ) is then realized at time t2. This type of arbitrage opportunity cannot exist for
long.
CHAPTER 4: INTEREST RATES – BRIDGE TO SWAPS
If the interest rate is measured with annual compounding, the bank's statement that the interest
rate is l0% means that $1 grows to $1x1.1= $1.1 at the end of 1year.
With semiannual compounding, it means that 5% is earned every 6 months, with interest rate being
reinvested: $1 x 1.05x 1.05= $1.125
Generalization: If an amount A is invested for n years at an interest rate of r per annum:
 Annual compounding: A(1+r)n
𝑟
 Rate compounded m times per annum: 𝐴(1 + 𝑚)𝑚𝑛
Continuous compounding  Aern
Compounding a sum of money at a continuously compounded rate r for n years involves multiplying
it by ern. Discounting it at a continuously compounded rate involves multiplying by e-rn
Zero rates
The n-year zero-coupon rate is the rate of interest earned on an investment that starts today and
lasts for n years. All the interest and principal is realized at the end of n years.
Bond pricing
Most bonds pay coupons to the holder periodically, and the bond’s principal is paid at the end of its
life. Suppose a 2-year treasury bond, measured with continuos compounding, with a principal of
100 provides coupons at 6% per annum semiannualy. The price of the bond is 3𝑒 −0.05𝑥0.5 +
3𝑒 −0.058𝑥1 + 3𝑒 −0.064𝑥1.5 + 103𝑒 −0.068𝑥2 = 98.39
if d is the present value of $1 received at the maturity of the bond, A is the value of an annuity that
pays one dollar on each coupon payment date, and m is the nº of coupon payments per year, then:
Determining treasury zero rates  bootstrapping
97.5
)
100
𝑟0.25 =
−0.25
𝑏𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒
ln(
)
𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙
𝑟𝑇 =
−𝑇
ln(
Forward rates
Forward interest rates are the rates of interest implied by current zero rates for periods of time in
the future.
Company wants to fund a project that pays back in 9 months
A. Take a 6-month loan and roll over
B. Take a 9-month loan
Implied forward rate:
For short-term investments (maturity shorter than one year), we use
simple compounding:
$1 invested today yields $(1+rTT), where T = maturity and rT = zero rate
associated with maturity.
(1 + 𝑟0,9 ∗ 0.75) = (1 + 𝑟0,6 ∗ 0.5) ∗ (1 + 𝐹6,9 ∗ 0.25)
Longer maturity: Company wants to fund a project that pays
back in 2 years
A. Take a 1-year loan and roll over
B. Take a 2-year loan
FORWARD RATE AGREEMENTS
An FRA is an over the counter agreement that a certain interest rate will apply to either borrowing
or lending a certain principal during a specified future period of time.
Valuation
The value of the FRA where RK is EARNED is: 𝑉𝐹𝑅𝐴 = 𝐿(𝑅𝐾 − 𝑅𝐹 )(𝑇2 − 𝑇1 )𝑒 −𝑅2𝑇2
For a company receiving interest at the floating rate and PAYING at Rk:
𝑉𝐹𝑅𝐴 = 𝐿(𝑅𝐹 − 𝑅𝐾 )(𝑇2 − 𝑇1 )𝑒 −𝑅2𝑇2
4.3 The 6-month and 1-year zero rates are both 10% per annum. For a bond that has a life of 18
months and pays a coupon of 8% per annum (with semiannual payments and one having just been
made), the yield is 10.4% per annum. What is the bond’s price? What is the 18- month zero rate?
All rates are quoted with semiannual compounding.
8%
Suppose the bond has a face value of $100. 2 100 = 4. There would be a cash flow of $4 after 6
month, 4$ after a year, and $104 (coupon + principal) after 18 month. The price of the bond is
obtained by discounting these cash flows at 10.4% per annum. The bond’s price is:
4
4
104
+
+
= $96.74
2
0.104
0.104
0.104 3
1+
(1 + 2 )
(1 + 2 )
2
If the 18 months zero rate is R:
4
4
104
+
+
= $96.74 → 𝑅 = 0.1042 = 10.42%
2
0.1
0.1
𝑅 3
1+ 2
(1 + 2 )
(1 + 2 )
Suppose that zero interest rates with continuous compounding are as follows:
Maturity
Rate (% per
Forward rate
annum)
1
2
𝐹1,2 = 3𝑥2 − 2𝑥1 = 4%
2
3
3
3.7
𝐹2,3 = 3.7𝑥3 − 3𝑥2 = 5.1%
4
4.2
𝐹3,4 = 4.2𝑥4 − 3.7𝑥3 = 5.7%
5
4.5
𝐹4,5 = 4.5𝑥5 − 4.2𝑥4 = 5.7%
4.5 Use the rates to value an FRA where you will PAY 5% (compounded annually) for the third year
on $1 million.
RF = 5.1%
L = 1 million RK = rate agreed on FRA = 5%
RF = 5.1% with continuous compounding, with annual: 𝑒 0.051∗1 − 1 = 5.232%
The 3-year int rate is 3.7% with continuous compounding  RM = 3.7%
𝑉𝐹𝑅𝐴 = 𝐿(𝑅𝐹 − 𝑅𝐾 )(𝑇2 − 𝑇1 )𝑒 −𝑅2𝑇2 = 1,000,000(5.232% − 5%)(1)𝑒 −0.037∗3 = 2,078.85
4.21 Explain why an FRA is equivalent to the exchange of a floating rate of interest for a fixed rate
of interest.
A FRA is an agreement that a certain specified interest rate, Rk, will aply to a certain principal L for
a certain specified future time period. Suppose that the rate observed in the market for the future
time period at the beginning of the time period proves to be R M. If the FRA is an agreement that RK
will apply when the principal is invested, the holder of the FRA can borrow the principal at R M and
then invest it at Rk. The net cash flow at the end of the period is then an inflow of RKL and an outflow
of RML. If the FRA is an agreement that Rk will apply when the principal is borrowed, the holder of
the FRA can invest the borrowed principal at RM. The net cash flow at the end of the period is then
an inflow of RML and an outflow of RKL. In either case we see that the FRA involves the exchange of
a fixed rate of interest on the principal of L for a floating rate of interest on the principal.
CHAPTER 7: SWAPS
A swap is an agreement between two companies to exchange cash flows in the future.
Consider a hypothetical 3-year swap initiated on March 5, 2007, between Microsoft and Intel. M
agrees to pay I an interest rate of 5% per annum on a principal of 100, and I pays M the 6-month
LIBOR rate on the same principal. Microsoft is the fixed-rate-payer and Intel the floating-rate payer.
Payments are exchanged every 6 months and the 5% rate is quoted with semiannual compounding.
In total, there are 6 exchanges of payment. The fixed payments are always 0.5x5%x100=2.5. The
floating-rate payments on a payment date are calculated using the 6-month LIBOR rate prevailing
6 months before the payment date. The principal is not exchanged, it’s only used for calculations.
Using the swap to transform a liability
Microsoft could use the swap to transform a floating-rate loan into a fixed-rate loan. Suppose that
M has arranged to borrow 100$ at LIBOR + 0.1% (10 basis point):
1. It pays LIBOR + 0.1% to its outside lenders
2. It receives LIBOR under the terms of the swap
3. It pays 5% under the terms of the swap.
These 3 sets of cash flows net out to an int rate payment of 5.1%. Thus, for M, the swap could have
the effect of transforming borrowings at a floating rate of LIBOR + 0.1% into borrowings at a fixed
rate of 5.1%.
For Intel, the swap could have the effect of transforming a fixed-rate loan into a floating-rate loan.
Suppose that Intel has a 3-year $100 million loan outstanding on which pays 5.2%. After it has
entered into the swap, it has the following three sets of cash flows:
1. It pays 5.2% to its' outside lenders.
2. It pays LIBOR under the terms of the swap.
3. It receives 5% under the terms of the swap.
These three sets of cash flows net out to an interest rate payment of LIBOR + 0.2%. Thus, for Intel,
the swap could have the effect of transforming borrowings at a fixed rate of 5.2% into borrowings
at a floating rate of LIBOR + 0.2%.
Using the swap to transform an asset
Consider Microsoft. The swap could have the effect of transforming an asset earning a fixed rate of
interest into an asset earning a floating. Suppose M owns a 100 million in bonds that will provide
interest at 4.7% per annum over the next 3 years.
1. It receives 4.7% on the bonds
2. It receives LIBOR under the terms of the swap
3. It pays 5% under the terms of the swap
Microsoft can transform an asset earning 4.7% into an asset earning LIBOR – 0.3%
Determining LIBOR/swap zero rates
The value of a newly issued floating-rate bond that pays 6-month LIBOR is always equal to its
principal value when the LIBOR/swap zero curve is used for discounting.
Interest rate swaps
Suppose that both Google and Microsoft neet $10 million to finance an investment with an economic
life of 5 years. Because of factors related to the nature of the
investment and the firm’s operations, Google prefers to have a
floating–rate liability. Google can borrow at a fixed rate of 10% or a
floating rate equal to LIBOR; Microsoft can borrow at a fixed rate of
11.25% or a floating rate of LIBOR + 0.5%.
Google can borrow at 1.25% less at fixed rate, but only 0.50% less at floating rate. It has a
comparative advantage borrowing at fixed rate. The difference 1.25% − 0.50% = 0.75% is the
surplus that the three parties in the swap (Google, Microsoft, and the swap bank) can split. If surplus
= 0, we can’t arrange a swap.
We arrange a swap such that each party earns 0.25% surplus. Since google wanted a floating-rate
liability, the swap should lead to Google borrowing at LIBOR -0.25% and Microsoft at 11%.
Google: we need to solve 10% + LIBOR – X =
LIBOR -0.25%  X = 10.25%.
Microsoft: we need to solve LIBOR + 0.50% − LIBOR +X =11%  X = 10.50%
Google is to borrow from outside investors at a fixed rate of 10%. It will then enter a swap with the
bank, in which it will pay LIBOR, receiving 10.25%. As a result of the swap, its net borrowing cost
becomes 10% + LIBOR − 10.25% = LIBOR − 0.25%, i.e. 25 basis points cheaper than Google could
obtain without the swap. Google has used the swap to turn a fixed–rate liability into a floating–rate
one, as desired.
Microsoft is to borrow from outside investors at a floating rate of LIBOR + 0.50%. It will then enter
into a swap with the bank, in which it will pay 10.5%, receiving LIBOR. This is convenient for
Microsoft, because as a result of the swap its net borrowing cost becomes LIBOR + 0.50% +10.5%−
LIBOR =11%
Note, finally, that this is convenient for the bank too. Because it receives LIBOR + 10.50% from the
two firms, and pays 10.25% + LIBOR, its net profit from the swap is 25 basis points too.
There are many possibilities, changing the exchange between Microsoft and the bank. We can
distribute the 25 basis point in many ways.
Valuation of interest rate swaps
To value the swap, we just need to price the two bonds. Pricing the fixed rate bond is
straightforward: compute the present value of all future cash flows.
Suppose we are valuing a fixed–for–floating interest rate swap for 8% against
6–month LIBOR, with notional principal $100 million. The swap makes semi–
annual payments, and has a residual life of 1.25 years. On the last reset date, the
LIBOR rate was 10.2%. The term structure of LIBOR rates is as reported in the
table.
Fixed rate: at each reset date the fixed payments are 1 × 8% × $100 million = $4 million. The
present value of the fixed rate leg is therefore:
Bfix = 4e−0.25×10.0% + 4e−0.75×10.5% + 104e−1.25×11.0% = $98.238 million
Floating rate leg: LIBOR was 10.2% on the last reset date  the floating rate leg pays a coupon of:
K*= ½ x 10.2% x $100 million = $5.1 million
In addition, the claim on the residual floating rate payments is a par bond, i.e. its value is precisely
$100 million. The present value of the total claims on the floating rate leg is thus:
Bfloat = (100 + 5.1)e −0.25 × 10% = $102.505 million
L = principal; the next exchange of payments is at time t*; k*= floating payment at time t* that was
determined at the last payment date. Bfloat = (L + k*)e −r*×t* where r* is the LIBOR/swap zero rate
for a maturity of t*. In the example, t* = 0.25 because we have 3 months till next payment.
The value of the swap, from the point of view of the floating rate payer, is: V = Bfix - Bfloat = -$4.267
m
For a fixed rate payer: V = Bfloat – Bfix = $4.267
Approach 2: Hedging portfolio
A portfolio of FRAs hedges the risk borne by the floating–rate payer in the interest rate swap. By
our familiar no arbitrage logic, the value of the interest rate swap is then equal to the value of the
hedging portfolio of FRAs.
Let us go back to our earlier example; and assume semiannual compounding. The “recipe” to value
an FRA is: assume that future spot rates will be equal to today’s forward rates, and discount. So the
first step is obtaining forward rates. The forward rate for the period from 3 to 9 months from now
is:
F3,9 =
0.75 × 10.5% − 0.25 × 10.0%
0.5
= 10.75% with continuous compounding
Passing to semiannual compounding:
Rc = 10.75%
𝑅𝑐
Rm= rate with compounding m times per year  𝑅𝑚 = 𝑚 (𝑒 𝑚 − 1) = 2(𝑒 10.75%∗0.5 − 1)
The implied FRA rate is 2(𝑒 0.5∗10.75% − 1) = 11.04%. This implies a floating rate payment on the
reset date 9 months from now of $100 million × 11.04%x0.5 = $5.522 million.
1.25×11%−0.75×10.5%
F9,15=
1.25−0.75
= 11.75% → 𝐹𝑅𝐴 = 2(𝑒 0.5×11.75% − 1) = 12.102%
Floating rate payment 15 months from now: 100x12.102%x0.5= $6.051 million
Taking present values and adding up the payments, the value of the floating rate leg is:
5.1𝑒 −0.1×0.25 + 5.522𝑒 −0.105×0.75 + 6.051𝑒 −0.11×1.25 = $15.351 million
Subtracting this from the value of the fixed rate leg: 4e−0.25×10.0% + 4e−0.75×10.5% + 4e−1.25×11.0%=
$11.084
The value of the swap is −$4.267 million, just as before.
CURRENCY SWAPS
Unlike the case of interest rate swaps, in a currency swap the principals are exchanged, first at the
inception of the swap, then at the end of the swap. In between, fixed interest payments are
exchanged. Typically, currency swaps are used to convert a liability or an investment from currency
into another.
Company X wishes to borrow U.S. dollars at a fixed rate of interest. Company Y wishes to borrow
Japanese yen at a fixed rate of interest. The amounts required by the two companies are roughly
the same at the current exchange rate. The companies
have been quoted the following interest rates, which
have been adjusted for the impact of taxes:
Design a swap that will net a bank, acting as intermediary, 50 basis points per annum. Make the
swap equally attractive to the two companies and ensure that all foreign exchange risk is assumed
by the bank.
Valuation of currency swaps
We can view the swap as a portfolio of two bonds, denominated in different currencies. Denoting
by D the “domestic” currency and F the “foreign” one, the value of the swap, from the point of view
of the domestic currency receiver, is then:
VSwap = BD − S0(D/F)BF
where S0(D/F) is the spot exchange rate at the time we make the valuation. Clearly, the terms in the
above expression are reversed if we take the perspective of the “domestic” currency payer. B D and
BF are the value of the bonds defined by the domestic/foreign cash flows on the swap.
For instance, suppose that the term structure is flat in the U.S. and in Japan, at r¥ = 4% and r$ =
9% with continuous compounding. You would like to value a swap in which you are to receive ¥5%
and pay $8%, once a year. The notional principals are ¥1,200 million and $10 million; the remaining
life of the swap is 3 years, and the spot exchange rate is S0(¥/$) = ¥110/$. (110 yen =1$)
0.08x10=$0.8
0.05x1200=60
The payments on the Yen bond are ¥60 each year. Combined with the principal payment of ¥1,200
at maturity, their present value is B¥ = ¥60 × (e−5%×1 + e−5%×2 + e−5%×3 ) + ¥1, 200e−5%×3 = ¥1,
230.55. Similarly, the present value of the U.S. dollar bond is B$ = $0.80 × (e −9%×1 + e−9%×2 +
e−9%×3) + $10e−9%×3 = $9.64. Therefore, the value of the swap to you is S0(¥/$) × B¥ − B$ =
1230.55
($0.0091/¥) × ¥1, 230.55 − $9.64 = $1.54 million or 110 − 9.6439 = 1.5430 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
Hedging portfolio: valuation as portfolio of forward contracts
Going back to our example, we need to compute forward ex- change rates using the familiar
expression F = S0e(r−rf )T.
0.009091𝑒 (0.09−0.04)1 = 0.009557
Simple calculations yield F1 = $0.0096/¥, F2 = $0.0100/¥, F3 = $0.0106/¥. The implied net swap
cash flows (Yen inflows minus U.S. dollar out- flows) are –$0.23, –$0.20, and $2.51 million; in
present value terms, again a $1.54 million value for the swap.
Dollar value of yen cash flows: 60x0.009557=0.5734 million
Net cash flow at the end of year 1: 0.8 - 0.5734=-0.2266 million dollars. This has a PV of:
−0.2266𝑒 −0.09×1 = −0.2071
This is the value of forward contract corresponding to the exchange of cash flows at the end of year
1.
7.8 Explain why a bank is subject to credit risk when it enters into two offsetting swap contracts.
7.10 A financial institution has entered into an interest rate swap with company X. Under the terms
of the swap, it receives 10% per annum and pays six-month LIBOR on a principal of $10 million for
five years. Payments are made every six months. Suppose that company X defaults on the sixth
payment date (end of year 3) when the LIBOR/swap interest rate (with semiannual compounding)
is 8% per annum for all maturities. What is the loss to the financial institution? Assume that sixmonth LIBOR was 9% per annum halfway through year 3.
Every 6 months, the bank receives: 0.5x10%x10,000,000 = $500,000
At the end of year 3, the bank was due to receive the $500,000 and pay 0.5x9%x$10m = $450,000
Immediate loss = $50,000.
To value the remaining swap, we assume that forward rats are realized. All forward rates are 8%
per annum. The remaining cash flows are therefore valued on the assumption that the floating
payment is 0.5 x 0.08 x 10m= $400,000
Net payment received: $500,000 - $400,000 = $100,000
The total cost of default is therefore the cost of foregoing the following cash flows:
3-year: $50,000 + years 3.5;4;4.5;5 = 4 x $100,000 = $450,000
Discounting these cash flows to year 3 at 0.5x8% = 4% per six months, we obtain the cost of the
default:
1
1
1
1
500,000 + 100,000 (
+
+
+
) = 412989
2
3
1 + 0.04 1.04
1.04
1.044
7.12 A financial institution has entered into a 10-year currency swap with company Y. Under the
terms of the swap, the financial institution receives interest at 3% per annum in Swiss francs and
pays interest at 8% per annum in U.S. dollars. Interest payments are exchanged once a year. The
principal amounts are 7 million dollars and 10 million francs. Suppose that company Y declares
bankruptcy at the end of year 6, when the exchange rate is $0.80 per franc. What is the cost to the
financial institution? Assume that, at the end of year 6, the interest rate is 3% per annum in Swiss
francs and 8% per annum in U.S. dollars for all maturities. All interest rates are quoted with annual
compounding.
Forward rate when interest rates are compounded annually:
𝐹0 = 𝑆0 (
1+𝑟
1+𝑟𝑓
𝑇
)  Third column
Year
6
7(1 year forw)
8(2-year forw)
9
10
Dollar
paid
7m x 0.08
560,000
560,000
560,000
560,000
7,560,000
𝑆0 = 0.8; r = 0.08; rf = 0.03
CHF
received
10m x 0.03
300,000
300,000
300,000
300,000
10, 300,000
Forward rate
F0
0.8
0.8388
0.8796
0.9223
0.9670
Dollar equiv of CHF
received
F0 x CHF received
240,000
261,600
263,900
276,700
9,960,100
Cash flow
lost
5th – 2nd
-320,000
-308,400
-296,100
-283,300
2,400,000
Discounting the numbers in the final column to the end of year 6 at 8% per annum, cost of the
default:
−320,000 −
308,400 296,100 283,300 2,400,100
−
−
+
= $679,800
1.08
1.082
1.083
1.084