Bank - Stress

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Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C.
Hull 2010
Chapter 1 – Introduction
This book addresses derivatives markets; how they work, can be used and what determines prices in
them.
Futures contracts
Futures contract = Agreement to buy or sell an asset at a certain time in the future for a certain price.
Long = agreement to buy
Short = agreement to sell
Spot price = immediate delivery
History of future markets
1848 the Chicago Board of Trade (CBOT) was established to bring farmers and merchants together to
agree on the price for production in the near future. Speculators at the time traded the contract.
The Chicago Mercantile Exchange organized futures trading in foreign currencies.
Open-outcry-system is the traditional trading on the exchange floor (trading pit).
Electronic trading has led to algorithmic trading (automated high-frequency trading).
The over-the-counter market (ten times larger than exchange)
OTC is an alternative to exchanges; a telephone and computer network of dealers.
Financial institutions act as market makers for the more commonly traded instruments:
They quote
Bid price = a price at which the financial institution is prepared to buy
Offer price = a price at which the financial institution is prepared to sell
Advantage of OTC:
- Terms of a contract do not have to be those specified by an exchange; market participants
are free to negotiate any mutually attractive deal.
Disadvantage of OTD:
- Credit risk (risk that the contract will not be honored)
Forward contracts
A forward is the same as a future, but is traded on the OTC market.
Spot traders are trading a foreign currency for immediate delivery.
Options (Both exchange and OTC)
Call option – gives the holder the right to buy an asset by a certain date for a certain price
Put option – gives the holder the right to sell an asset by a certain date for a certain price
The contract contains the following:
- Exercise- or strike-price
- Expiration- or maturity-date
Maturity dates is for European options only, American can be exercised at any time.
An option gives the right to do something, but the holder does not have to exercise its right
Option premium = the up-front price that has to be paid by the investor for the option contract.
Properties of options:
As strike-price increases, the price of a call option decreases, but a put option increases. Both
options become more valuable as time to maturity increases.
In the USA an option contract is a contract to buy or sell 100 shares. The price is always given for one
share.
History of options markets
Selling and option is also known as writing an option
The Put and Call Brokers and Dealers Association
- Brought buyers and seller together for options
Deficiencies:
1. There was no secondary market (no right to sell the option to another party)
2. No mechanism to guarantee that the writer of the option would honor the contract.
These days the NASDAQ is the premier exchange for trading foreign options. Most exchanges also
offer combinations of futures contracts and options on these contracts.
Types of trader:
Hedgers – use derivatives to reduce the risk from potential future movements in the market
Comparison:
Futures – neutralize risk by fixing the price to be paid or received.
Options – provide insurance; protecting for adverse movements while allowing to benefit from
favorable price movements
Speculators – Use derivatives to bet on future directions of the market
Futures possibilities:
Possibility 1 = An upfront investment
Possibility 2 = Take a long position on futures contracts (small margin account as initial investment)
Option possibilities:
Options can be ten times more profitable, but are greater in potential loss.
However, the loss is limited to the amount paid for the options.
Arbitrageurs – take offsetting positions in two or more instruments to lock in a profit
e.g. the stock price in NY is $162 and P100 in London when the exchange rate is $1.650 per pound.
At 100 shares the profit (in absence of transaction costs) is $300.
Forces of supply and demand will cause the price of dollar to rise so opportunities don’t last long.
In practice very small arbitrage opportunities are observed.
Hedge funds – have become big users of derivatives for all three purposes
Similar to mutual funds; invest funds on behalf of clients. However, they accept only from financially
sophisticated people and do not publicly offer their securities.
Mutual funds are subject to regulations requiring:
- Shares are redeemable at any time
- Investment policies are disclosed
- Use of leverage is limited
- No short positions are taken
- Etc.
Hedge funds are relatively free of these regulations
Fees are typically 1-2% of investment + 20% of the profits.
Funds of funds – Invest in a portfolio of hedge funds
The hedge fund manager must:
1. Evaluate the risks to which the fund is exposed
2. Decide which risks are acceptable and which will be hedged
3. Devise strategies to hedge unacceptable risks
Labels:
Long/short equities: Purchase undervalued securities and short overvalued in such a way that the
exposure to the overall direction of the market is small
Convertible arbitrage: Long position on a convertible bond combined with an actively managed short
position in the underlying equity.
Distressed securities: Buy securities issued by companies in or close to bankruptcy
Emerging markets: Invest in debt or equity of companies in developing countries
Global macro: Carry out trades that reflect anticipated global macroeconomic trends
Merger arbitrage: Trade after an M&A is announced so that a profit is made if the deal takes place.
Dangers
Traders who have a mandate to hedge or follow an arbitrage strategy become (consciously or
unconsciously) speculators.
Risk limits should be set and activities of traders monitored.
Chapter 2 – Mechanics of Futures and Forward Markets
This chapter covers how futures markets work.
- Specification of contracts
- Operation of margin accounts
- Organization of exchanges
- Regulation of markets
- The way in which quotes are made
- Treatment of transactions for accounting and tax purposes
Forward contracts are also discussed
Closing out futures positions
Most futures are never delivered. Most traders choose to close out their position which means
entering into the opposite trade to the original one.
Gain or loss is determined by the change in the futures price between closing out and going
short/long.
The possibility of final delivery however ties the futures price to the spot price.
Specification of a futures contract
When ready to deliver, the short party files a notice of intention to deliver with the exchange.
1. The asset (commodity’s range of grades and quality)
2. The contract size (the amount of the asset) size depends for hedgers
3. Delivery arrangements (location determines price if it’s far from the source of commodity)
4. Delivery months (period that delivery can be made)
5. Price quotes (e.g. in dollars and cents)
6. Price limits and Position limits (Price limit is to prevent large price movements because of
speculation. A limit move is a move in either direction equal to the daily price limit. The
contract is said to limit up or limit down).
(Position limit is the number of contracts a speculator may hold; purpose is to prevent the
exercise of undue influence on the market)
Convergence of futures price to spot price
When the delivery period approaches, the futures price converges to the spot price.
When futures price above the spot price a clear arbitrage opportunity exists:
1. Short the futures contract
2. Buy the asset
3. Make delivery
When futures price below the spot price:
Companies interested in requiring the asset will wait for delivery to be made.
The operation of margins
Margins are a tool to avoid contract default (e.g. opposite party has no money).
1. Daily settlement
`The investor deposits money in the margin account when the contract is entered (initial margin). At
the end of each trading day, the margin account is adjusted to reflect the investor’s gain or loss.
Referred to as daily settlement or marking to market.
The maintenance margin ensures the balance never goes negative; if the account falls below this, the
investor receives a margin call from which he needs to deposit a variation margin. May this margin
not be provided; the broker closes out the position.
- Margin levels are determined by the variability of the price of the underlying asset.
- Usually maintenance margin is about 75% of the initial margin.
Day trade = closing out the position on the same day
Spread transaction = take both a short and long position in a contract
2. The clearing house and Clearing margins
Clearing house is an intermediary in futures transactions.
The house keeps track of all transactions and calculates the net position of each of its members. Both
broker and clearing house member are required to maintain a margin account.
The latter is clearing margin and is adjusted just as margin accounts of investors.
However, the balance is maintained at an amount to the original margin times the number of
contracts outstanding.
- On gross base: sum of the long and short positions
- On net base: long and short offsets against each other. (see example book)
OTC markets
Reducing credit risk in over-the-counter markets:
1. Collateralization
Similar to margin; transaction is valued every day and the positive is paid by the other party, or you
own the negative amount to the other party.
2. Use of clearing houses in OTC markets
Since the 2007-2009 crises some OTC transactions are obligated to use clearing houses.
The clearing house takes on the credit risk of both party A and B and manages it by requiring an
initial margin and daily variation margins from them.
Systemic risk = risk a large financial institution fails and leads to failures by other large financial
institutions can a collapse of the financial system.
Market quotes
- Prices
Open- opening price
High – highest of the day
Low – lowest of the day
Settlement- closing price
Change – change in settlement price from the previous day
Volume- number of contracts traded
Open interest – number of contracts outstanding
- Patterns of futures prices
Normal market = a market where the futures price is an increasing function of the time to maturity
Inverted market = market where the futures price decreases with the maturity of the futures contract
Sometimes because of seasonality the market shows a mixture of normal and inverted markets.
Delivery
Period during which delivery can be made is determined by the exchange
The short position determines when to deliver -issues a notice of intention to deliver to the exchange
The exchange passes this notice to the party with the oldest outstanding long position.
If transferrable, long investors have half an hour time to find another party who accepts notice.
First notice day = day on which notice can be submitted to exchange
Last notice day = last day on which notice can be submitted
Last trading day = to avoid delivery; long investors close out the contract prior to 1st notice day
- Cash settlement
When settled in cash, the final settlement is equal to the spot price of the underlying asset at either
opening or close of trading on that day.
E.g. predetermined on the third Friday of the delivery month and settlement at the opening price.
Types of traders and types of orders
FCM’s (future commission merchants) - following instructions from clients and charge commissions –
e.g. hedgers, speculators or arbitrageurs.
Locals – trading on own account
Speculators can be classified as:
- Scalpers- watch for short term trends and profit from small changes
- Day traders- hold position for less than one day (no overnight risk of media)
- Position traders- hold for longer periods, hope for major movements
- Orders
Market order = a request to the broker to trade immediately at the best price available
- Limit order – specifies a particular price (limit)
- stop-loss order – specifies a particular price (to close out a position)
- stop-limit order – 2 prices, e.g. stop price of $40 and limit price of $41.
- MIT (market-if-touched order) or board order- After a price more favorable than the specified
price, this order sets a stop-loss and a market-if-touched order to ensure profits at a
favorable price.
- Discretionary order or market-not-held order – A market order but execution may be delayed
at the broker’s discretion to get a better price
Unless otherwise stated, an order expires at the end of the trading day.
Time-of-day order – specifies a particular period of time
Open order or good-till-cancelled order – is in effect until executed
Fill-or-kill order – must be executed immediately on receipt or not at all.
Regulation
CFTC (Commodity Futures Trading Commission) regulates the USA futures market.
- Licenses futures exchanges and approving contracts
Approves also all contracts and changes to existing contracts.
NFA (National Futures Association) prevents fraud by monitoring trades
-
Trading irregularities
An investor group may ‘corner the market’; taking huge long futures positions and try to exercise
control over the supply of the underlying commodity.
When maturity is approached, the group doesn’t close out the position so the number of outstanding
futures exceeds the amount of commodity available for delivery. Short position will find it difficult to
deliver and become desperate to close out their positions.
The result is a large rise in both futures and spot prices.
Dealt with by increasing margin requirement or imposing position limits.
Accounting and tax
- Accounting
You account for the gain or loss amount reached at the time of closing the books e.g. December
2011, and once again on the actual closing date of the contract that is calculated with the next year
e.g. 2012.
- Tax TOO BORING TO READ PP. 40/41 2.10 TAX
Forward vs. Futures contracts
Forward
Private contract between 2 parties (OTC)
Not standardized
Usually one specified delivery date
Settled at end of contract
Delivery or cash settlement usually takes place
Some credit risk
-
Futures
Traded on exchange
Standardized contract
Range of delivery dates
Settled daily
Contract is usually closed out prior to maturity
Almost no credit risk
The forward contract will make its profit at one time at the final maturity whereas the future
contract will lose and gain spread over the period and at the end make the same profit.
CH3 Hedging strategies using futures
Hedge to reduce risk
Risks related to fluctuations, foreign exchange rate, level of stock market and other variables.
Perfect hedge = one that completely eliminates the risk.
Basic principles
Take a position that neutralizes the risk as far as possible.
e.g. take a short position on a commodity that may either increase or decrease in price.
- Short hedges
Involves a short position in futures contract (example above)
Do this when you already (or will) own the asset and expect to sell in the future.
This leads to a loss if the value increases but a gain if the value decreases.
- Long hedges
When you know you will purchase in the future and want to lock a price down.
Arguments for and again hedging
Interest, exchange and commodity prices are difficult to predict so these risks can be hedged.
Why some risks are not hedged:
- Hedging and shareholders
Shareholders can hedge themselves and take a diversified portfolio. However, the transaction and
commission costs may be expensive.
- Hedging and competitors
When competitive pressures within the industry may be such that the price of goods/services
fluctuates to reflect raw material costs/interest/exchange rates, a company that hedges can expect
to have fluctuating profit margins.
- Other considerations
Although a hedging position is perfectly logical, it might be difficult to justify in the case of a loss.
Basis risk
In practice hedging is not so straight forward:
1. The asset whose price has to be hedged may not be exactly the same as the asset underlying
the futures contract.
2. The hedger may be uncertain as to the exact date to buy/sell the asset.
3. The hedge may require the futures contract to be closed out well before maturity.
- The basis
The basis in a hedging situation = Futures price – spot price
Strengthening of the basis = when the spot price increases by more than the futures price
Weakening of the basis = when the futures price increases by more than the spot price
To examine the basis risk, we use the following notation:
S1: Spot price at time t1 (hedge put in place)
S2: Spot price at time t2 (closed out)
F1: Futures price at time t1
F2: Futures price at time t2
b1: Basis at time t1
b2: Basis at time t2
In a future, the effective price obtained for the asset with hedging is S2 + F1 – F2 = F1 + b2
-
Investment assets (currencies/stock indices/gold) have a lower basic risk because of the
arbitrage argument.
- Consumption commodities have more basic risk because of imbalance between supply and
demand and difficulties of storing.
When the asset underlying the hedge increases risk, the price of basic risk can be calculated as
following:
The price of the underlying asset is S1* and S2*
= S2 + F1 – F2
Written as: F1 + (S2*- F2) + (S2 - S2*)
Basis risk can lead to both and improvement or worsening of a hedger’s position.
- Short hedge = basis strengthens; position improves – basis weakens; position worsens
- Long hedge = basis strengthens; positions worsens – basis weakens; position improves
- Choice of contract
Two components for choice:
1. The choice of the underlying asset
2. Choice of delivery month
The asset being hedged should match the exactly the asset underlying the futures contract.
In other circumstances, you should carefully analyze which prices of futures contracts are most
closely related with the price of the asset hedged.
When the expiration of the hedge corresponds with the delivery month, this contract is chosen
usually. Sometimes especially not because futures prices may be quite irregular during the delivery
month. A longer hedge also takes to risk to take the delivery.
In general basis risk increases as the time between hedge expiration and delivery month increases.
Rule of thumb is to choose these close to each other.
Cross hedging
When the asset being hedged and the underlying asset are different.
Hedge ratio = the ratio of the size of the position in future contracts to the size of exposure.
h* = the ratio of average change in S for a particular change in F (optimal ratio is h* = 1)
- Minimum variance hedge ratio (optimal hedge ratio)
Depends on volatility
∆S = change in spot price during the hedge’s lifetime
∆F = change in futures price during the hedge’s lifetime
The formula for the best fit slope:
p = the correlation coefficient between ∆S and ∆F
- Optimal number of contracts
QA = Size (units) of position being hedged
QF = Size (units) of one futures contract
N* = optimal number of contracts for hedging
From equation of optimal hedge ratio, the futures contracts should be on h* x QA of the assets
The number of futures contracts required is:
- Tailing the hedge
Calculate the impact of daily settlement.
h* VA
N* = -------------VF
V = value of asset being hedged or futures contract.
Stock index futures
Stock index – tracks changes in the value of a hypothetical portfolio of stocks
- Hedging an equity portfolio
VA = current value of the portfolio
VF = current value of one futures contract (futures price x number of contracts)
When there is no optimal hedge ratio (h*=1), we can use the beta parameter ᵝ
Beta = the slope when the return on the asset is regressed against the return of a stock index
B = 0 = portfolio mirrors the return on the index
B = 2 = changes in return from portfolio are twice as high as changes in return from the index
B = 0,5 = the changes are half as great
To hedge a portfolio with beta 2, you need twice as much contracts.
A portfolio with beta 0,5 only needs half the contracts to hedge it.
VA
N* = ᵝ -------------VF
- Reasons for hedging an equity portfolio
If the hedger feels the stock have been chosen well, the performance of the market may be
uncertain, but the stock in the portfolio will outperform the market (after appropriate adjustments
for the beta in the portfolio).
- Locking in the benefit of stock picking
To lock in returns of your portfolio, you can short ᵝ VA/ VF index futures contracts
Rolling the hedge forward
When the expiration data is later than the delivery dates of all futures contracts.
Close out the futures contract and short a new futures contract.
CH4 – Interest rates
Types of rates
- Treasury
Rates earned from treasury bills/bonds. Rate at which the government borrows in its own currency
- LIBOR (London InterBank Offered Rate)
The rate of interest at which a bank is willing to make a deposit to another bank.
A bank needs an AA-credit-rating.
LIBID = the London Interbank Bid Rate = the rate at which banks are prepared to accept deposits.
The rates are determined by supply and demand.
- Repo (repurchase agreement)
Securities are sold to another company and contracted to buy them back later at a slightly higher
price.
Measuring interest rates
e.g. A deposit of $100 receives 10% interest
Annual compounding = $100 x 1.1 = $110
Semi-annual compounding = $100 x 1,05 x 1,05 = $110,25
Quarterly compounding = $100 x 1,0254 = $110,38
For annual compounding over n year: A(1+R)n R= interest rate and n = number of years
If compounded m times per annum: A(1+R/m)mn
- Continuous compounding
When m is infinity
AeRn
e = in the calculator (2.71828, eX exponential function)
To discount (express as % of face value) = multiplying by e-Rn
Book p. 83 = equation to convert an annual compounding rate to continuously compounded rate and
visa versa.
Zero rates (for n years) or spot rate
No intermediate payments but can be compounded per annum continuously.
Bond pricing
Most bonds pay coupons to the holder periodically. The bond’s principal (face value) is paid at the
end of its life. The theoretical price is calculated as the present value of all cash flows that will be
received.
e.g. the discount value is calculated semiannually (in case of semiannual compounding).
AeRn:
- Bond yield
With only a single discount rate, expressed as y.
- Par yield
The rate that causes the bond price to equal its face value. Suppose the coupon is C per annum (or
C/2 per six months). The value of the bond is equal to its face value when (2-years semiannually
compunded:
Determining treasury zero rates
Bootstrap method=
CH6 – Interest rate futures
First part: Treasury bond and Eurodollar futures contracts (traded in USA)
Second part: Duration measure and how it can measure sensitivity of a portfolio to interest rates
Interest rate futures contracts can be used to hedge exposure to interest rate movements.
DAY COUNT AND QOUTATION CONVENTIONS
Day counts
The way in which interest accrues over time.
Reference period = the time between coupon payments on a bond.
Day count is expressed as X/Y x Interest earned in reference period.
X = Number of days between days
Y = Number of days in reference period
There are three day-count conventions mostly used:
1. Actual/actual (in period)- US treasury bonds
- Interest earned is based on the actual days elapsed to the actual number of days in the
period between payments.
2. 30/360 – US corporate and municipal bonds
- 30 days per month, and 360 days per year. (less days is higher interest)
3. Actual/360 – US treasury bills and other money market instruments
- Reference period is 360 days. So calculation is dividing actual number of elapsed days by
360 and multiplies by the rate.
Quotations
Prices are sometimes quoted using a discount rate, which is the interest earned as a percent of the
final face value.
e.g. quoted as 8, means an interest rate of 8% of the face value per 360 days. Suppose face value is
$100. Then interest earned over 91 days = 100 x 0.08 x 91/360 = 2.0222
This corresponds to a true rate of interest of 2.002/(100-2.0222) = 2.064% for the 91day period.
Relationship between the cash price and quoted price is:
P=360/n x (100-Y)
P=quoted price
Y= cash price
n= remaining life in calendar days.
US treasury bonds
Cash price = Quoted price + Accrued interest since last coupon date
TREASURY BOND FUTURES
Quoted the same way as treasury bonds
At multiples of 1/32 (a tick) e.g. 90-05 at face value 100.000= 90.156,25
Conversion factor
Defines the price received for the bond
Cash received = (Most recent settlement price x Conversion factor) + Accrued interest
Cheapest-to-deliver bond
Purchase price and cash received for bonds calculations are given above.
The Cheapest-to-deliver bond is where the outcome is least for:
Quoted price – (Most recent settlement price x Conversion factor)
Determining the futures price
• Factors that affect the futures price:
– Delivery can be made any time during the delivery month
– Any of a range of eligible bonds can be delivered
But if we assume both cheapest-to-deliver bond and delivery date are known, then F0 = (S0 – I )erT
– F0 is the futures price
– S0 is the cash price of the bond
– I is the present value of coupons during the life of futures
– T is time till futures matures
– r is interest rate applicable to a time period of T
Eurodollar futures
A dollar deposited at a foreign bank outside the USA
Forward vs future interest rate
Futures contract is settled daily
Forward contract is realized at the end.
In the short maturity (1 year) there is no difference, but for longer dated contracts, there are
differences in settlement at time.
Convexity adjustment
Differences between two rates
Forward rate = Future rate -0,5 SD2 T1T2
T1 = time to maturity of the futures contract
T2 = time to maturity of the rate underlying the futures contract
SD = standard deviation of change in short-term interest in one year
DURATION
A measure of how long on average the holder of the bond has to wait before receiving cash payment
A coupon-bearing bond maturing in n years has a duration of less than n years because the holder
receives some of the payments prior to year n.
- Duration of a bond that provides cash flow ci at time ti is
 ci e  yti 
D   ti 

B 
i 1 
n
Where B is its price and y is its yield (continuously compounded).
The bond price is the present value of all payments
The duration is therefore a weighted average of the times when payments are made.
The weight applied to time t is the PV of cash flow at time t as a % of the bond price.
Consider a 3-year 10% coupon bond with a face value of $100. Bond yield of 12%
Time (yrs) cash flows
PV
weight time *weight
0,5
5
4,709
0,050
0,025
1
5
4,435
0,047
0,047
1,5
5
4,176
0,044
0,066
2
5
3,933
0,042
0,083
2,5
5
3,704
0,039
0,098
3
105
73,256
0,778
2,333
10% Total
= 5% cash flow every six
months
94,213
1
2,653
Present value = 5e-0.12x0.5=4.709
Sum PV = bond price at 94,213
Weight = number in column 3 / 94,213 (sum weight is always 1)
Duration = sum column 5 (2,653)
Duration interpretation
• Measures how sensitive a bond value is with respect to the change of interest rate.
– the longer the duration, the more sensitive is the bond price to interest rate movements
– B = bond price, D = Duration and ∆y is change of the bond yield. Duration measures the
bond price sensitivity to a small parallel shift in the yield curve, because only a
parallel shift can be summarized by a same change in the bond yield.
– Given D=2.653, when the yield increases by 10 basis point (0.1%), bond price will decline
by 0.265% of the previous price, or decline by a dollar amount of
0.265%*94.213=$0.250
Duration based hedging
• This involves hedging against interest rate risk by matching the durations of the assets and
the hedge
• It provides protection against small parallel shifts in the zero curve
• Relationship holds for a bond portfolio
•
Relationship holds for a bond futures
CH7 – Swaps
A swap is an agreement between two companies to exchange cash flows in the future.
1. Interest rate swap
2. Currency swap
Mechanics of interest rate swaps
Plain vanilla interest rate swap = the most common type of swap.
- A company agrees to pay cash flows equal to interest at a predetermined fixed rate on a
notional principal for a number of years. In return it receives interest at a floating rate.
Floating rate = an interest rate that changes on a periodic basis
- The notional principle (e.g. 100mil) is not exchanged, only the interest.
- Interest rate is set at the beginning of the period to which it applies, and paid at the end.
LIBOR
The floating rate in most swap agreements is the LIBOR (for maturities till 12 months).
Typical uses of an interest rate swap
Converting a liability/borrowing or an investment from:
– fixed rate to floating rate
You have to pay 5.2% on a debt to a lender, then you can enter a swap that you receive a
fixed rate and pay a floating, like Intel does.
– floating rate to fixed rate
You will have to pay a LIBOR rate to a lender, then you can enter a swap that you receive a
LIBOR rate and pay a fixed, like MS does.
Swap can similarly be used to transform an asset
A financial institution is usually involved as intermediary
Market maker
Display bid and offer prices from their own accounts. They must carefully quantify and hedge the risk
they are taking. Quotes by a swap market maker:
Bfix = Value of fixed-rate bond underlying the swap
Bfl = Value of floating-rate bond underlying the swap
The comparative-advantage argument
Some companies have comparative advantages in fixed-rate markets while other have the advantage
in floating-rate markets. Suppose:
• AAA-Corp wants to borrow floating (AAA rating)
• BBB-Corp wants to borrow fixed (BBB rating)
Difference in borrowing fixed rates is higher than in floating rates. AAA’s advantage is in fixed rates.
•
•
AAA-Corp: pay LIBOR-0.35%
BBB-Corp: pay 4.95% fixed rate
When a financial institution is involved:
•
•
•
AAA-Corp: pay LIBOR-0.33%
BBB-Corp: pay 4.97%
F.I.: profit 0.04%
Valuation of interest rate swaps
• Interest rate swaps is worth 0 when initiated.
Two valuation approaches:
1. Difference between two bonds
2. Valued as a portfolio of forward rate agreements (FRAs)
In terms of bond prices:
With a long position in fixed-rates and a short position in floating-rate bond:
Vswap= Bfix- Bfl
With a long position in floating-rates and a short position in fixed-rate bond:
Vswap= Bfl- Bfix
In terms of FRA’s
As time passes, floating rates changes unexpectedly, swap has a value, and can be valued as a
portfolio of forward rate agreements (FRAs)
• Each exchange of payments in an interest rate swap is an FRA
• The FRAs can be valued on the assumption that today’s forward rates are realized, and
today’s forward rate can be derived from today’s LIBOR curve.
1. Use LIBOR rate to calculate forward rates for each of the LIBOR rates that will determine
swap cash flows
2. Calculate swap cash flows on the assumption that the LIBOR rates will equal the forward
rates
3. Discount these swap cash flows (using LIBOR) to obtain the swap value
Example
• Pay six-month LIBOR, receive 8% (s.a. compounding) on a principal of $100 million
• Remaining life 1.25 years
• LIBOR rates for 3-months, 9-months and 15-months are 10%, 10.5%, and 11% (cont comp)
• 6-month LIBOR on last payment date was 10.2% (s.a. compounding)
Currency swap
Exchanging principal and interest payments in one currency for principal and interest payments in
another currency.
E.g. An agreement to pay 5% on a sterling principal of £10,000,000 & receive 6% on a US$ principal of
$18,000,000 every year for 5 years
Similarly one can use this swap to transform liabilities and assets, motivated by comparative
advantage.
• In a currency swap the principal is exchanged!
- Principal amounts are usually chosen to be approximately equivalent using the exchange rate
at the swap’s initiation
- The exchange of the principal happens at the beginning in the opposite direction of interest
cash flow, and at the end of the swap in the direction of interest cash flow
Cash flow of currency swap:
Typical uses of currency swaps
Conversion from borrowing in one currency to a borrowing in another currency, or from an
investment in one currency to an investment in another currency
• Without swap: IBM can issue $18mln of US$-dominated bonds at 6% interest. (borrowing in
US$ with an interest of 6%)
• With swap: in addition to bond issue
– 1. exchange $18mln (using bond issue proceeds) for receiving £10mln,
– 2. pay 5% interest in £, receive 6% interest in $ (used to pay bond interest)
– 3. pay £10 mln, receive $18 mln (used to pay bond principal).
Valuation of Currency Swaps
Like interest rate swaps, currency swaps can be valued as a portfolio of forward contracts
Assuming that forward exchange rates will be realized:
Example
• All Japanese LIBOR/swap rates are 4%
• All USD LIBOR/swap rates are 9%
• 5% is received in yen; 8% is paid in dollars. Payments are made annually
• Principals are $10 million and 1,200 million yen
• Swap will last for 3 more years
• Current exchange rate is 110 yen per dollar
Credit risk
• A swap is worth zero to a company initially
• At a future time its value is liable to be either positive or negative
• The company has credit risk exposure only when its value is positive, because the value to
the counterparty is negative, and it might default.
CH8 – Securitization and the credit Crisis of 2007
•
•
•
Financial derivatives: Helps transferring risk in the economy to the party who is willing to
take it (for compensation).
Securitization: A process of taking an illiquid asset, or group of assets (residential
mortgages), and transforming them into a security.
In 1960s, Mortgage-backed security (MBS) was created so that proceeds from the sale of
MBS can be used to finance the loan to mortgage borrower. So banks can lend faster than
what their deposits allows.
Asset Backed Security (Simplified)
Getting Principal back depends on loss on the underlying asset
Rating agencies rated the securities
Finding investors for the Mezzanine tranches was more difficult, leading to the creation of ABS’s of
ABS’s: ABS CDOs (Collateralized Debt Obligation)
In the ABS CDO, the senior tranche accounts for 65% of the principle amount of the ABS mezzanine
tranche. The senior tranche of the ABS CDO is also rated AAA; meaning that the total of the AAArated instruments created are about 90% [80% + (65% of 15%)]
Before the credit crunch:
• Starting in 2000, mortgage originators in the US relaxed their lending standards and created
large numbers of subprime first mortgages.
• This, combined with very low interest rates, increased the demand for real estate and prices
rose.
• To continue to attract first time buyers and keep prices increasing they relaxed lending
standards further (e.g.100% mortgages, ARMs, teaser rates)
• Subprime mortgages were frequently securitized, and sold to investors with little information
about the mortgages, except loan-to-value ratio and FICO score which can be manipulated.
• Quality of the mortgage is low. E.g. NINJAs, liar loans, non-recourse borrowing
• Banks found it profitable to invest in the AAA rated tranches because the promised return
was significantly higher than the cost of funds and capital requirements were low
When the bubble busted:
• In 2007 the bubble bursts. Some borrowers could not afford their payments when the teaser
rates ended. Others had negative equity and recognized that it was optimal for them to
exercise their put options.
• Mortgage is non-recourse, meaning lender can take possession of the house when borrower
defaults, but other assets of the borrower are off-limits. This also means borrower can sell
the house for the outstanding principal on the mortgage. (essentially an American put
option, taken advantages of by many speculative defaulters)
• U.S. real estate prices fell and products (ABS, ABS CDO), created from the mortgages, that
were previously thought to be safe began to be viewed as risky
• Investors (pension funds, banks) in these products incurred big losses. Insurance company
(like AIG) who provide the guarantees also loses.
The loss banks led to credit crisis. Their capital eroded, more risk-averse, credit spreads increased
Key mistakes:
• Default correlation goes up in stressed market conditions
• Assumption that a BBB tranche is like a BBB bond
Need to Align Interests of Originators and Investors
• There is evidence that mortgage originators used lax lending standards because they knew
loans would be securitized
• For a rebirth of securitization it is necessary to align the interests of originators and investors
•
Regulators are insisting that when credit risk is transferred a certain percentage (5% to 10%)
of each tranche is retained by the originator
Role of Compensation Plans
• Short term compensation (the end-of-year bonus) is the most important part of the
compensation for many employees of financial institutions
• This creates short term horizons for decision making
• Financial institutions are now realizing that bonuses should be based on performance over 3
to 5 years.
Transparency
• ABSs and ABS CDOs were complex inter-related products
• Once the AAA rated tranches were perceived as risky they became very difficult to trade
because investors did not understand the risks
• To help them understand the risks, creators of the products should provide a way for
potential purchasers to assess the risks (e.g., by providing software)
Need for Models
• Most financial institutions did not have models to value the tranches they traded (It appears
that they did not follow their own procedures on this)
• Without a valuation model risk management is virtually impossible
More Emphasis on Stress Testing
• We need more emphasis on stress testing and managerial judgement; less on the
mechanistic application of VaR models (particularly when times are good)
• Senior management must be involved in the development of stress test scenarios
CH10 – properties of stock options
There are 6 factors that affect the price of a stock option:
1.
2.
3.
4.
5.
6.
Current stock price
Strike price
Time to expiration
Volatility
Risk free interest rate
Dividends expected to be paid
S0
K
T
ơ
r
•
•
•
•
c:
p:
C:
P:
European call option price
European put option price
American Call option price
American Put option price
•
•
ST :
D:
Stock price at option maturity
Present value of dividends during option’s life
Summary of the effect on the price of a stock option of increasing one variable while keeping all
others fixed
Higher stock price
Higher strike price
Longer time to expiration
Higher volatility
Higher risk-free rate
-
call increases, put decreases
call decreases, put increases
both increase, but call double as much/long as put
both increase, but call starts higher
call increases, put decreases
Upper and lower bounds
If an option price is above the upper bound or below the lower bound, there are profitable
opportunities for arbitrageurs.
An American option is worth at least as much as the corresponding European option:
C≥c
c  S0
P≥p
Upper bound to call option price:
CS
0
Upper bound to put option price:
PK
p  Ke  rT
The worst that can happen to a call option is that is expires worthless, its value cannot be negative:
Lower bound to call option:
c ≥ S0 –Ke -rT
Lower bound to put option:
p ≥ Ke -rT–S0
Put-call parity
Consider the following 2 portfolios:
– Portfolio A: European call on a stock + zero-coupon bond that pays K at time T
– Portfolio C: European put on the stock + the stock
Values of the portfolios:
Portfolio A
Portfolio C
ST > K
ST < K
Call option
ST − K
0
Zero-coupon bond
K
K
Total
ST
K
Put Option
0
K− ST
Share
ST
ST
Total
ST
K
Put-call parity result: Equation 10.6, page 236
Both are worth max(ST , K ) at the maturity of the options
They must therefore be worth the same today. This means that
c + Ke -rT = p + S0
Arbitrage opportunities
Suppose that
C=3
S0= 31
T = 0.25
r = 10%
K = 30
D=0
What are the arbitrage possibilities when
p = 2.25 ?
p=1?
• c + Ke -rT = p + S0
Early Exercise
• Usually there is some chance that an American option will be exercised early
• An exception is an American call on a non-dividend paying stock
• This should never be exercised early
•
An Extreme Situation
For an American call option:
S0 = 100; T = 0.25; K = 60; D = 0
Should you exercise immediately?
• What should you do if
You want to hold the stock for the next 3 months?
You do not feel that the stock is worth holding for the next 3 months?
Reasons For Not Exercising a Call Early (No Dividends)
• No income is sacrificed
•
You delay paying the strike price
•
Holding the call provides insurance against stock price falling below strike price
Should Puts Be Exercised Early?
Yes, always exercise when a put is deep in the money.
• Protect against price fall, but price can never be negative
• Earn interest on proceeds
The Impact of Dividends on Lower Bounds to Option Prices
(Equations 10.8 and 10.9, pages 243-244)
c  S 0  D  Ke  rT
p  D  Ke  rT  S 0
CH11 – Trading strategies involving options
With options you may create several payoff functions: the payoff as a function of the stock price.
These trading strategies ignore the time value of money. The profit is shown as the final payoff minus
the initial cost. In theory, you should calculate the present value of final payoffs minus initial costs.
Strategies involving a single option and stock
The following give the difference between payoff and profit of holding one option.
The following profit patterns are the same as in CH9 for single option. Put-call parity gives an
understanding why this is so: S0 – c = Ke(-rT) + D - p
a.
b.
c.
d.
Long stock + short call
Short stock + long call
Long put + long stock
Short put + short stock
Spreads
1. Bull spread (limits both upside and downside risk) (hope stock price increases)
Buy a c1 with lower strike; sell a c2 with a higher strike, Negative initial cash flow
Using calls:
3 types of bull-spreads exist:
1. Both calls initially out of the money (most aggressive, small probability of high payoff)
2. One call initially in the money, the other call initially out of the money
3. Both calls initially in the money (more conservative)
Using puts:
2. Bear spreads (hope stock price declines) (limit both upside and downside potential)
Strike price of option bought is higher than strike price of option sold
Initial cash outflow
Using puts:
Using calls:
3. Box spreads
Combination of bull call spread (2 different strike prices K1 & K2) and a bear put spread (with
the same two strike prices).
- Only work with European options
- Payoff is always K2 - K1 and the present value is (K2 - K1)e-rT
4. Butterfly spreads (if stock moves are assumed to be unlikely)
Options with 3 different strike prices
e.g. high call 1, low call 3, and halfway between sell two call options at price 2.
Price 2 is generally close to the current stock price.
Butterfly leads to profit if the stock price stays at price 2, but gives loss if the price moves in
either direction.
Has a negative initial cash flow
Using calls:
Using puts:
5. Calendar spreads (same strike price but different expiration dates)
Up to now the options all expired at the same time.
Short call, long (great maturity) call with same strike price.
Profit is made when stock price at expiration of the short-maturity option is close to the
strike price of the short-maturity. Loss occurs at a significant rise or decline or stock price.
Neutral – strike price close to current stock price
Bullish – Higher strike price
Bearish – Lower strike price
6. Diagonal spreads (both expiration date and strike price are different)
Bull, bear and calendar are created from a long position in one option, and a short in another.
Combinations
An option trading strategy that takes position in both calls and puts on the same stock
1. Straddle (Buy a call and a put with the same K (strike price), initial cash outflow)
When expecting a large move in stock price but don’t know which direction. However, carefully
consider if this jump is already reflected in the option price.
2. Strips and Straps (buy one call and 2 puts with same K, or buy two calls and 1 put with same K)
Strip – considers a big move > decrease in stock price
Strap – considers a big move > increase in stock price
3. Strangles (Buy a put at a lower strike, and a call at a higher strike
Compared with a straddle, stock price has to move further away
In order to make a profit)
- When expecting a large move in stock price but don’t know which direction. However,
carefully consider if this jump is already reflected in the option price.
CH12 – Introduction to Binomial Trees
Technique to price an option
Diagram of possible paths that can be followed by the stock price over the life of the option.
1. Explains nature of no-arbitrage arguments
2. Binomial tree numerical procedure to value American options
3. Principle of risk-neutral valuation
A one-step binomial model (and a no-arbitrage argument)
A stock price of $20 will be $22, or $18 in three months.
A three month call option has a strike price of $21
Portfolio is riskless when 22∆ - 1 = 18∆ or ∆ = 0.25 [1-0/22-18]
(22 x .25 – 1 = 4.5, and 18 x .25 = 4.5)
∆ is the number of shares necessary to hedge a short position in one option
Because it has no risk, the return must equal the risk-free interest rate (e.g. 12%):
4.5e-0.12x3/12=4.367
So the portfolio that is long 0.25 shares and short 1 position is worth 4.367
To calculate the option price (denoted as f):
Current stock price is $20 > 20 x 0.25 – f = 5 – f
5 – f = 4.367
f = 0.633
A generalization
A derivative lasts for time T and is dependent on a stock:
S0=stock price
f = option price
u > 1; d < 1
The portfolio is riskless when
=
or:
Value of portfolio at time T =
Value of portfolio today =
The cost of setting up portfolio = S0∆-f
Hence S0∆-f =
P = the risk neutral probability of up and downward movements =
Two-step binomial trees
The stock price starts at $20 and in each of the two time steps may go up/down by 10%.
Each time step is three months and the risk-free interest rate is 12%
Option strike price is $21
Objective is to calculate the value of the option
Value at note B =
Value at note A =
Note C is zero because it leads to both E and F
= 1.2823
The value of the option is 1.2823
A generalization
S0 during each step moves up u or goes down d; so after two up movements the value is fuu
The length of a step is now ∆t (otherwise calculate p again for the next node)
A put example
These procedures can be used to price both puts and calls
American options
The procedure is to work back from end of the tree to the beginning, testing at each node to see
whether early exercise is optimal.
Steps in the binominal tree valuation:
1. Work out possible stock prices at all nodes
2. Calculate the payoff of the option at maturity
3. Work backwards, calculate the value of the option at earlier nodes
(Using risk-neutral probabilities), use other value if early exercise is possible and gives a
higher value.
4. Do this until the initial node
Choosing u and d
In practice they are determined by the volatility:
Ơ = stock price volatility
∆t = length of one time step on the tree
CH13 – Valuing stock options: The black-Scholes-Merton model
Model that influenced the way to price options
Black & Scholes – used capital pricing model to relate required return on the option to the required
return on the stock (depends on stock price and time)
Merton – sets up a riskless portfolio of an option and underlying stock and argues that the return
must be the risk-free return.
Assumptions about how stock prices evolve
What is the probability distribution in a day, week, month or year?
The BSM-model considers a non-dividend-paying stock and assumes normal distribution on returns.
The returns in two different, non-overlapping periods are assumed to be independent.
µ = Expected return on the stock
ơ = Volatility of the stock price
S = Stock price
Mean of return
= µ ∆t
Standard deviation
= ơ√∆t
Assumption of the model =
∆S/S ~ N(µ∆t, ơ2∆t)
Normal distribution
= N(m,v)
Variance
= ơ2∆t
The lognormal distribution
The assumption is that stock price at any future time has a lognormal distribution (always positive
and skewed)
These assumptions imply ln ST is normally distributed with mean: ln S 0  (    2 / 2)T
And standard deviation:  T

ln ST  N ln S 0  (    2 2)T ,  2T
Alternatively, ST is log normally distributed or
(Because the logarithm of ST is normal)
ln

ST
 N (    2 2)T ,  2T
S0
Expected return
R is the continuously compounded return per annum, E(R)=µ – ơ2/2


Expected value of the stock price is E(ST)=S0eµT
ln
Expected return =
or

ST
 RT ~ N (    2 2)T ,  2T
S0

R ~ N (    2 2),  2


Volatility
The standard deviation of the return in time ∆t (expressed in years) is:  t
If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price
change in one day? 25x√1/252 = 1.57% x 50 = $78.74
Because volatility is much higher when the market is open, time is measured in trading days (a 252)
when options are valued.
Estimating volatility from historical data
n+1 = number of observations
Si = stock price at end of ith interval, where i=0,1,2,3….n
t = length of time interval in years
1.
2.
3.
4.
Take observations S0, S1,….Sn on the variable of each trading day
Define the continuously compounding rate as ---------------------
Calculate SD, s, of the µi’s
The historical volatility per year estimate is: s x √252
 S 
ui  ln  i 
 Si 1 
We can estimate how much this estimate will vary from the standard deviation of this sampling
distribution, which we call the standard error (SE) of the estimate: ̂ / 2n
=
1
√
−1
∑=1

2
1
−
(∑
=1

(−1)
2
)
14
15
16
17
18
19
20
20.90
21.25
21.40
21.40
21.25
21.75
22.00
1.00000
1.01675
1.00706
1.00000
0.99299
1.02353
1.01149
daily std. Dev
vol
0.00000
0.01661
0.00703
0.00000
-0.00703
0.02326
0.01143
0.01216
0.19302
The concept underlying Black-Scholes
• The option price and the stock price depend on the same underlying source of uncertainty
• We can form a portfolio consisting of the stock and the option which eliminates this source
of uncertainty
• The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate
Assumptions for BM model
1. Stock price is log-normally distributed
2. No transaction costs and taxes
3. No dividend on the stock during the life of option
4. No riskless arbitrage opportunities
5. Security trading is continuous
6. Investor can borrow and lend at the same risk free rate
7. Short-term risk free rate is constant.
The Black-Scholes Formulas:


Implied volatility: implied by the option prices, market opinion about stocks’ future volatility,
forward looking
Historical volatility: direct measure of the movement of the stock price over the history,
realized volatility, backward looking
CH15 – Options on Stock Indices and Currencies (15.1 & 15.2)
Refresh memory; payoffs by several options:
Options on stocks indices
- Some track the movement of the market as a whole; others are based on the performance of
a particular sector (IT, oil gas, telecoms, etc.).
An index option contract is 100 x the index (settled in cash).
Portfolio insurance
Limiting the downside risk, but maintain the upside potential
Beta of 1.0 implies that the portfolio return mirrors that from the index.
Example 1
Portfolio has Beta 1.0
It’s worth is $500.000
The index stands at 1000
What trade is necessary to insure against the portfolio falling below $450,000?
The number of options to buy is calculated by:
500.000
1 ∗ 100∗1000 = 5 contracts
To calculate the strike price of the option you need to buy:
500.000−450.000
= 10% (maximum decline you can stand)
500.000
When Beta is not 1.0
In this case, Beta put options have to be bought for each $100S0 in the portfolio
E.g. Beta is 2.0:
500.000
2 ∗ 100∗1000 = 10 Contracts rather than 5 as before.
Calculation of the expected value including interest rates etc. p.325
CH17 – The Greek letters
Each Greek letter measures a different dimension to the risk in an option position.
The aim is to manage the Greeks so that all risks are acceptable.
Greek letters
Delta
Gamma
Theta
Rho
Vega
Measure different dimensions of the risk in option position such as S, r, t, σ.
Example:
A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend- paying
stock S0 = 49, K = 50, r = 5%, σ = 20%, T = 20 weeks
The Black-Scholes-Merton value of the option is $240,000
How does the bank hedge its risk?
Naked and Covered positions:
Naked = do nothing
Covered = buy 100.000 shares today
Both strategies leave exposure to significant risk
Stop-loss strategy:
Buying as soon as the price rises above K, and selling as soon as the price falls below K.
Doesn’t work well for hedging purposes.
Delta hedging:
∆ = rate of change of the option price with respect to the change in the underlying asset.
Delta of a call =
∆
∆
Delta neutral is ∆ = 0
Rebalancing is adjusting the hedge periodically. Dynamic hedging is contrasted by Static hedging
where the hedge is set-up and never adjusted.
Delta hedging involves maintaining a delta neutral portfolio
For every Call option, you buy Δ shares
For every Put option you short Δ shares.
The delta of a European call on a non-dividend-paying stock is N (d 1)
The delta of a European put on the stock is [N (d 1) – 1]
Example of Delta Hedging:
S=$100, c=$10, now a trade sells 20 call option contract (right to buy 2000 shares). Delta now is 0.6.
How to hedge his position in calls?
Buy 0.6*2000=1200 shares.
To verify: stock goes up by $1, stock position: +1200, call position: -0.6*2000
Delta of call position: 0.6*(-2000)=-1200, meaning when S increases by , trader loses 1200
Delta of stock position: 1*1200=1200, meaning when S increases by , trader gains 1200
Total position is delta neutral.
Theta (time)
Theta (Θ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to
the passage of time, often quoted to say the change when one day passes.
Usually negative for an option
Θ = ∆ value of portfolio / ∆ amount of time passed
Gamma (change of delta to the underlying asset)
Gamma (Γ) is the rate of change of delta (Δ) with respect to the price of the underlying asset
**Delta hedging assumes the price to move to X1 but actually moves to X2, this is a hedging error.
Gamma measures the curvature of the relationship between option price and stock price.**

If absolutely value of gamma is small, delta changes slowly, adjustment in delta hedging need
to be made only relatively infrequently.
 If absolute value of gamma is large, adjustment should be made more frequently, and it is
very risky to leave a delta-neutral portfolio for any length of time.
A position in the underlying asset has a 0 gamma.
Vega (volatility)
Vega (ν) is the rate of change of the value of a derivatives portfolio with respect to volatility
 Vega is positive for long option positions, negative for short positions.
 A position in the underlying asset has a 0 Vega.
Rho (interest rate)
Rho is the rate of change of the value of a derivative with respect to the interest rate.
 Call option has a positive rho, and put option has a negative rho.
Managing Delta, Gamma, & Vega
Delta can be changed by taking a position in the underlying asset
To adjust gamma and Vega, it is necessary to take a position in an option or other derivative
Hedging in Practice
 Traders usually ensure that their portfolios are delta-neutral at least once a day
 Gamma and Vega neutral are less easy to achieve, as other options traded in large volume at
competitive prices are needed.
 Whenever the opportunity arises, they improve gamma and Vega
 As portfolio becomes larger hedging becomes less expensive, due to lower bid-offer spread
CH20 – Value at Risk
VaR is an attempt to prove a single number summarizing the total risk in a portfolio.
“I am X% certain that there will not be a loss of more than V dollars in the next N days”
VaR is the loss level that will not be exceeded with a specified probability
Expected shortfall is the expected loss given that the loss is greater than the VaR level
The time horizon
VaR has two parameters; time horizon N, and confidence level X.
N-day VaR = 1-day VaR x √N
Advantages of VaR
It captures an important aspect of risk in a single number
It is easy to understand
It asks the simple question: “How bad can things get?”
Historic Simulation
- Create a database of the daily movements in all market variables.
- The first simulation trial assumes that the percentage changes in all market variables are as
on the first day
- The second simulation trial assumes that the percentage changes in all market variables are
as on the second day
And so on
Suppose we use 501 days of historical data
Vi = the value of a market variable on day i
There are 500 simulation trials
The i’th trial assumes that the value of the market variable tomorrow is
The portfolio’s value tomorrow is calculated for each simulation trial
The loss between today and tomorrow is then calculated for each trial (gains are negative losses)
The losses are ranked and the one-day 99% VaR is set equal to the 5th worst loss
Alternative to Historical Simulation: The model-building approach
Make assumptions about the probability distributions of return on the market variables
**
• In option pricing we express volatility as volatility per year
• In VaR calculations we express volatility as volatility per day
• In practice we assume that ơday is the SD of the percentage change in one day
Microsoft Example (single asset case)
• We have a position worth $10 million in Microsoft shares
• The volatility of Microsoft is 2% per day (about 32% per year)
• We use N = 10 and X = 99
• The standard deviation of the change in the portfolio in 1 day is $200,000
• The standard deviation of the change in 10 days is
• We assume that the expected change in the value of the portfolio is zero (This is OK for short
time periods)
• We assume that the change in the value of the portfolio is normally distributed
• Since N(–2.33)=0.01, the VaR is 2.33  632,456  $1,473,621
AT&T Example (single asset case)
• Consider a position of $5 million in AT&T
• The daily volatility of AT&T is 1% (approx. 16% per year)
• The S.D per 10 days is 50,000 10  $158,144
• The VaR is 158,114  2.33  $368,405
Portfolio (two-asset case)
• Now consider a portfolio consisting of both Microsoft (10 mln) and AT&T (5 mln)
• Suppose that the correlation between the returns is 0.3
• A standard result in statistics states that  X Y   X2   Y2  2 X  Y
• In this case sX = 200,000 and sY = 50,000 and r = 0.3. The standard deviation of the change in
the portfolio value in one day is therefore 220,227
(√200.0002 + 50,0002 + 2 x 0.3 x 200,000 x 50,000)
• The 10-day 99% VaR for the portfolio is 220,227  10  2.33  $1,622,657
• The benefits of diversification are
(1,473,621+368,405)–1,622,657=$219,369
Model Building vs Historical Simulation Approaches
• Model building approach has the disadvantage that it assumes that market variables have a
multivariate normal distribution
• Historical simulation is computationally slower and cannot easily incorporate volatility
updating schemes
Stress Testing
• This involves testing how well a portfolio would perform under some of the most extreme
market moves seen in the last 10 to 20 years
Back-Testing
• Tests how well VaR estimates would have performed in the past
• We could ask the question: How often was the loss greater than the VaR level
Assignment 1
Members
Group number: 4
Open question: Find a real firm and identify the risks it faces. Choose one major risk it
faces and discuss how the firm manages the risk.
Heineken Holding N.V.
Activities of this beer (and other beverages) producing company are exposed to a variety of
financial risks, these include:
-
currency risk,
interest rate risk,
credit risk
liquidity risk,
market risk,
“Some of the risk management strategies include the use of derivatives, primarily in the form of
spot and forward exchange contracts and interest rate swaps, but options can be used as well. It
is the Group policy that no speculative transactions are entered into.” (Heineken Annual report
2012, Notes to the financial statements, note 31-37).
For this open question the foreign currency risk is selected to be discussed.
This is a form of risk that arises from fluctuations in price of a currency relative to another.
Heineken has assets and operations globally and faces currency risk if its positions are not
hedged in the activities of sales, purchases and borrowings in another currency.
Heineken hedges 90% of its cash-flows by forecasting sales and purchases. To hedge the foreign
currency risk, mainly forward exchange contracts are used (with a maturity of no more than one
year after the fiscal year).
Example
If Heineken wants to acquire barley from a company in China, which Heineken must pay
for in 180 days in the amount of $500,000. The risk exists that there might be a decrease
in the exchange rate of American Dollar versus Chinese Yuan during the intervening 180
days. To hedge against this risk, Heineken may long a forward contract with its bank to
buy the $500,000 in 180 days, at the current exchange rate.
Problem 1.12.
The use of derivates (in this case futures) can reduce the risk that comes from potential future
movements in the market. Hedging by shorting a future will neutralize risk by fixing the price to
be received. Therefore, the mining company could take a short position in future contracts on
gold. This means the company has the option to sell their gold every 2 months against a certain
price with a lower risk. By doing so, the company does not have the risk of selling at a lower
price in the future. However, if the price of gold increases the company will not have the
opportunity to benefit from this because the price is fixed.
Problem 1.20.
(a) 100.000.000 * (0.008-0.0074)= 60.000
The spot rate is 0.0006 lower than the forward rate, the trader gains 60,000 yen.
(b) 100.000.000 * (0.008-0.0091)=-110.000
The spot rate is 0.0011 higher than the forward rate, the trader loses 110,000 yen.
Problem 1.30.
It is advised to buy the three-month call options (with a strike price of $95 that currently sells
for $4.70) only if there is a high probability that the stock price will rise above $100 in three
month. The calculations below explain this in more detail.
Cost per share for call option strategy = 95+ 4.70
= $99.70 (2000 shares)
Cost for share in the strategy for buying stock =
= $94 (100 shares)
Call options strategy becomes more profitable with a stock price of;
99,70-94 = 5.70 *100 = $570
570 / 1900 = 0.3
99.7 + 0,3= $100
When the stock price rises above $100 per share, the profit of the call option strategy becomes
higher than when the investor is buying the stock right away. Note; the call options strategy
involves taking a greater potential loss when the stock price decreases.
Problem 2.27.
A margin is cash or marketable securities deposited by an investor with his or her broker. The
margin is adjusted to reflect the daily settlement. When the margin account becomes lower than
the maintenance margin ($2000) a margin call will follow. In this case, because the company
holds a short position, a margin call will be demanded if the future price is higher than 270
cents. The calculations below explain this in more detail.
100 (3000 – 2000) / 5000 = 20 cents
20 + 250 = 270 cents
When the margin account increases to $4500, the company is able to withdraw $1500. The
margin account increases to $4500 when the future prices is 220 cents. The calculations below
explain this in more detail.
100 * (1500 / 5000) = 30 cents
250 - 30 = 220 cents
Problem 3.17.
The influence the weather can have on corn production is not a valid reason to hedge the price
risk. This because the weather has an influence on the whole market or at least a large part of
the market. Therefore, if the harvest of the farmer is lower as expected this will also be the case
for the other farmers. Consequently, the total corn offered in the market will be very low
resulting in high price. When the farmer has hedged the price risk he will receive a relatively low
price because of short futures position.
Problem 3.27.
a) The number of contracts that should be shorted are calculated according to the
following formula that takes into consideration the Beta:
VA
N* = ᵝ ------------VF
50.000.000
Nr. Of contracts = 0.87 -------------------- = 138.20
1259 x 250
We round to the nearest whole number of contracts; 138 contracts should be shorted.
Value of portfolio
Current 3 month futures price
One month futures price
Risk-free rate per annum
Dividend rate per annum
Beta of portfolio
Number of contracts
= $50.000.000,00
=
$1.259,00
=
100,2500%
=
6,0000%
=
3,0000%
=
0,87
=
138
Value of index in two months
Futures price of index today
Futures price of index in two months
Gain in futures position $
Return on market
Expected return on portfolio
Expected portfolio value in two months including dividends $
Total value of position in two months $
1000
1100
1200
1300
1400
1259
1259
1259
1259
1259
1002,5
1102,75
1203
1303,25
1403,5
$431.750,00
$453.125,00
$474.500,00 $495.875,00 $517.250,00
-0,195
-0,115
-0,035
0,045
0,125
-0,16835
-0,09875
-0,02915
0,04045
0,11005
-$8.417.500,00 -$4.937.500,00 -$1.457.500,00 $2.022.500,00 $5.502.500,00
$8.849.250,00
5390625
1932000
-1526625
-4985250
Problem 4.11.
The semiannual coupon from this bond is:
100* (0.5* 0.04) = $2
The zero rates are continuously compounded. The cash price of the coupon is therefore:
2e-0.04*0.5 + 2e-0.042*1.0 + 2e-0.044*1.5 + 2e-0.046*2.0 + 102e-0.048*2.5 = $98.04
Problem 4.28.
a)
To convert the rate with a compounding frequency the following formula can be used:
Rc = m ln (1 + ) =

In case of a 6 month period: 2* ln (1+0.04/2) = 0.039605= 3,9605%
In case of a 12 month period: 2* ln (1+ 0.045/2) = 0.0445012 = 4.45012%
In case of a 18 month period: 2* ln (1+0.0475/2) = 0.046944712 = 4.6944712%
In case of a 24 month period: 2* ln (1+ 0.05/2) = 0.049385225= 4.9385225%
b)
To calculate the forward interest rate we use the following formula (For R1 and R2 we use the
rates calculated for 18 and 24 months in question A);
R2T2 – R1T1
4.9385225 x 2 – 4.6944712 x 1,5
RF = --------------------- = RF ------------------------------------------ = 5,6709404
T2 – T1
2 - 1,5
This is the rate for annually continuous compounding. To calculate the equivalent rate with
compounding semi-annually we use:
Rm = m(eRc/m -1) = Rm = 2(e5,6709404/2-1) = 0.0575210462 x 100 = 5.75210462
The value of the FRA is:
VFRA = L(RK-RF)(T2-T1)e-R2T2 =
VFRA = 1.000.000 x (,06 - ,0575210462) (2 - 1,5) e -,049385225 x 2
VFRA = 1122,9048971753
Assignment 2
Members
:
Group number: 4
1. A stock is expected to pay a dividend of $1 per share in two months and in five months. The
stock price is $50, and the risk-free rate of interest is 8% per annum with continuous compounding
for all maturities. An investor has just taken a short position in a six-month forward contract on the
stock.
a) What are the forward price and the initial value of the forward contract?
Here we are trying to define the forward price of an asset that provides a known income.
Dividend per share at two and five months = $1 and r = .08.
The present value of the dividends, I, is given by:
I = 1e-0.08x2/12 + 1e-0.08x5/12 = 1.953971262
The variable T is 6 months, so the forward price, F0, is given by:
F0 = (50 - 1.953971262)e0.08x6/12 = 50.00682437
F0 = 50.01
Luenberger: “The forward price is the price that applies at delivery. This price is negotiated so that
the initial payment is zero; that is, the value of the contract is zero when it is initiated”.
Luenberger, D. Investment Science. Oxford University Press, 1998.
John C. Hull: “The value of a forward contract at the time it is first entered into is zero. At a later
stage it may prove to have a positive or negative value.”
Hull, J.C. Fundamentals of futures and options markets. Pearson 7th edition, 2011.
Tuckman: “By definition, the forward price is such that the buyer and seller are willing to enter into
the forward agreement without any initial exchange of cash. This implies that the initial value of the
forward contract is zero. Over time, however, the value of the forward position may rise or fall.”
Tuckman, B. Serrat, A. Fixed income securities. John Wiley & Sons 3d edition, 2011.
b) Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8%
per annum. What are the forward price and the value of the short position in the forward
contract?
For the value of our forward contract that provides a known income with present value I:
f = S0 – I – Ke –rT
After three months have passed, there are two months left until the next dividend payment and
three months more until maturity.
The present value of dividends I, is given by:
I = 1xe-0.08x2/12 =.986755161
We can then solve the formula by:
f = 48 - .986755161 – 50,01e-0.08x3/12= -2,006735652
2. A bank offers a corporate client a choice between borrowing cash at 12% per annum and
borrowing gold at 3% per annum. (If gold is borrowed, interest must be repaid in gold. Thus, 100
ounces borrowed today would require 103 ounces to be repaid in one year.) The risk-free interest
rate is 9.25% per annum, and storage costs are 0.5% per annum. Discuss whether the rate of
interest on the gold loan is too high or too low in relation to the rate of interest on the cash loan.
The interest rates on the two loans are expressed with annual compounding. The risk-free interest
rate and storage costs are expressed with continuous compounding.
To equalize both calculations, we will use the current stock-price of gold on the exchange market.
Gold World Spot price in USD per ounce = $1.564,76
ÌEX.nl commodities exchange market: http://www.iex.nl/Koersen/Commodities.aspx
Thus, in case of 100 ounce, the amounts in dollars are $156.476,00
If cash is borrowed, the repayment at the 12% loan is $175.253,12
If gold is borrowed, the repayment should include the storage costs, risk-free interest rate and the
three per cent borrowing rate.
At repayment, the gold-loan requires 103 ounces to be repaid.
103 ounces x $1.564,76 = 161.170,28
The value at these costs is
161.170,28e(.0975+.005)1 = $178.566,57
$178.566,57 - $175.253,12 = $3.313,64
This shows that the cash loan is cheaper than the borrowing of gold. It would therefore indicate that
the interest rate on gold is too high.
3. A trader owns gold as part of a long-term investment portfolio. The trader can buy gold for $950
per ounce and sell gold for $949 per ounce. The trader can borrow funds at 6% per year and invest
funds at 5.5% per year. (Both interest rates are expressed with annual compounding.) For what
range of one-year forward prices of gold does the trader have no arbitrage opportunities? Assume
there is no bid–offer spread for forward prices.
An arbitrage opportunity is at hand when the trader can buy the commodity when the futures value
is higher than the spot price and costs of borrowing.
In this situation, the trader may borrow funds at 6% per year to buy gold at $950.
The spot price plus costs of borrowing are 1.06*950=1007.
If the price of a one-year forward contract is relatively high, the trader could make use of this
opportunity.
Similarly the trader may sell its gold at $949 and invest at 5.5% per year. The turnover on sales and
dividend at the end of year would be $1001,195
If the price of a one-year forward contract is relatively low, the trader may make use of this
opportunity.
In between these cases there exists no opportunity for arbitrage, that is; if the price of a one-year
forward contract is between $1001,195 and $1007.
4. Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and a 10-year
zero-coupon bond with a face value of $6,000. Portfolio B consists of a 5.95-year zero-coupon bond
with a face value of $5,000. The current yield on all bonds is 10% per annum.
(a) Show that both portfolios have the same duration.
To calculate the duration of the bond portfolio we may use the following formula;
Where;
Ti = 1 and 10
Ci = 2000 and 6000
Y = .10
The formula can be filled in as follows:
1 x 2000e-.10x1 + 10 x 6000e-.10x10
2000e-.10x1 + 6000e-.10x10
The outcome; 23.882,42/4.016,93 = 5,94544
Portfolio B consists of only one bond with a 5,95 year maturity. The duration is therefore the same
for portfolio A and B.
(b) Show that the percentage changes in the values of the two portfolios for a 0.1% per annum
increase in yields are the same.
The value of a portfolio can be calculated by the sum of dividends until maturity and the return on
the sales of the portfolio.
In case of portfolio A:
2000e-0,1x1 + 6000e-0,1x10 = 4016.95
In case the yield increases at 0.1% per annum, the value of portfolio A is:
2000e-0,101x1 + 6000e-0,101x10 = 3993.18
The difference in value is 23.77.
23,77 / 40,1695 = 0,59%
In case of portfolio B:
5000e-0,1x5,95 = 2757.81
In case the yield increases at 0.1% per annum, the value of portfolio B is:
5000e-0,101x5,95 = 2741.45
The difference in value is 16.36
16.36 / 27.58 = 0,59%
The percentage change in values for both portfolios are the same.
(c) What are the percentage changes in the values of the two portfolios for a 5% per annum
increase in yields?
In case of portfolio A:
Original value is = 4016.95
In case the yield increases at 5% per annum, the value of portfolio A is:
2000e-0,15x1 + 6000e-0,15x10 = 3060.20
In case of portfolio B:
Original value = 2757.81
In case the yield increases at 5% per annum, the value of portfolio B is:
5000e-0,15x5,95 = 2048.15
Difference in value for A = 956.75 / 40.1695 = 23.82%
Difference in value for B = 709.66 / 27.58 = 25.73%
5. A three-month Eurodollar futures quote changes from 97 to 95, what is the gain or loss to an
investor who is short two contracts.
One contract is $1000.000 So you get the interest of 1000.000. The interest rate is that of LIBOR.
When an investor takes a short position, he or she profits from a decline in the future (maturity
date).
When a quote decreases by 1 basis point, the investor who has a short position gains $25 per
contract (by design).
In this case, the quote decreases by 20 basis points. Per contract then, the investor gains: 20x $25=
$500
With 2 contracts, this amount is: 2x $500= $1000
6. It is June 25, 2010. The futures price for the June 2010 CBOT bond futures contract is 118-23.
There are two bonds. Bond A maturing on January 1, 2026 and paying a coupon of 10%, has a
conversion factor of 1.40. Bond B maturing on October 1, 2031, paying coupon of 7%, has a
conversion factor of 1.12.
a. Suppose that the quoted prices of the bond A and B are 169.00 and 136.00, respectively. Which
bond is cheaper to deliver?
Bond A:
Bond B:
Maturity: jan 1 2026
Maturity: oct 1 2031
Coupon payment: 10%
Coupon payment: 7%
Conversion factor: 1.40
Conversion factor 1.12
Quoted price: 169.00
Quoted price: 136.00
To calculate this, the following must be done: quotes futures price x conversion factor
Because we were not able to able to calculate the quoted futures price. Therefore, a value of 118.5 is
used.
Bond A:
118.5x1.40= 165.9
169.00 – 165.9 = 3.1 (less than quoted bond price)
Bond B:
118.5x1.12= 132.72
136 – 132.72 = 3.28 (less than quoted bond price)
This means that the first bond is cheaper to deliver.
Assuming that the cheapest to deliver bond is actually delivered, what is the cash price received
for the bond?
The first bond is cheaper to deliver, so for this bond the cash price is calculated.
The price for this bond is: 165.9
The accrued interest for this bond is:
Semiannual payment: 5%
Count conventions for this a, in this case: 176/181= 0.97237569
5x 0.97237569= 4.86187845
165.9 + 4.86187845= 170.7619 (rounded)
7. Company A wishes to borrow U.S. dollars at a fixed rate of interest. Company B wishes to
borrow sterling at a fixed rate of interest. They have been quoted the following rates per annum
(adjusted for differential tax effects):
Sterling
US Dollars
Company A
10.9%
7.0%
Company B
10.6%
6.2%
Design a swap that will net a bank, acting as intermediary, 20 basis points per annum
and that will produce a gain of 15 basis points per annum for each of the two companies.
Both parties borrow in the market where they have a comparative advantage;


Company A borrows Sterling at 10.9% per annum
Company B borrows US Dollar at 6.2% per annum
The total gain to all parties is; (7.0 - 6.2) – (10.9 -10.6) = 0.5% per annum (50 basis points).
The swap should be designed in such manner that the total gain is distributed as described
below;
 Bank
– 20 basis points
 Company A – 15 basis points
 Company B – 15 basis points
A possible design which satisfies this distribution is given below; `
£10.45%
£10.9%
£10.9%
Company A
Bank
$6.85%
Gains
Company A;
Company B;
Bank;
7 – 6.85
= 0.15
10.6 – 10.45 = 0.15
10.45 – 10.9 = -0.45
6.85 – 6.2 = 0. 65
0.20
Company B
$6.2%
$6.2%
8. In an interest rate swap, a financial institution pays 10% per annum and receives three-month
LIBOR in return on a notional principal of $100 million with payments being exchanged every three
months. The swap has a remaining life of 14 months. The average of the bid and offer fixed rates
currently being swapped for three-month LIBOR is 12% per annum for all maturities. The threemonth LIBOR rate one month ago was 11.8% per annum. All rates are compounded quarterly.
What is the value of the swap?
For the valuation of an interest swap, two approaches can be used. For this assignment the
method ‘valuation in terms of bond prices’ will be used.
Valuation in terms of bond prices
In a swap with a long position in a floating-rate bond and a short position in a fixed-rate bond,
the value of a swap can be calculated as follows;
Vswap = Bfl –Bfix
(Bfix = value fixed bond, Bfl = value floating bond rate)
For the calculation of Bfl and Bfix the correct discount rates must be calculated. For this, the
quarterly compounded rate must be converted to a continuous compounded rate. The
following formula can be used for this;
Rc = m ln (1 + 
) = 4* ln (1+0.12/4) = 0.118235209= 11.8235209%

Now Bfix and Bfl = (L + k*)e -r*t* can be calculated;
L= 100
k*= 100 * 0.118* (3/12)=2.95
r*=11.8235209%
t*= 2/12
Bfl = (100 + 2.95) e -0.118235209 *2/12 =100.9411391
Bfix = 2.5e-0.118235209 *2/12 + 2.5e-0.118235209 *5/12 + 2.5e-0.118235209 *8/12 +2.5e-0.118235209 *11/12 +
102.5e-0.118235209 *14/12 = 98.67763652
Now Vswap = Bfl –Bfix can be calculated;
Vswap = 100.9411391 - 98.67763652 =$2.26350258 mln
Assignment 3
Members
:
Group number: 4
Open question; describe the 8 components of ERM and explain them with examples.
COSO’s Enterprise Risk Management helps firms to assess and improve internal control systems by
providing key principles and concepts, a common language and clear direction and guidance.
The ERM framework is represented in a three-dimensional matrix, showing a relationship between
firms’ objectives, represented by eight components and the firms’ units.
In an ongoing process of identifying events, and strategy formulation, management may effectively
deal with uncertainty and the associated risks and opportunities, enhancing the capacity to build
value.
The following provides a sum-up of the eight interrelated components plus examples in the context
of University Twente and how it may apply ERM.
1. Internal Environment – Determines a firms’ philosophy towards risk management and
establishing risk culture; including people, integrity and ethical values.
University Twente’s philosophy to Risk Management is facilitating a flow of information and
stressing communication along the organization, and providing staff and people with the
capabilities and/or tools to overcome barriers. This realizes that risk and control is everyone’s
job, as well is the fundamental respect and integrity for employees and students. Incorporating
ERM into daily practices will ensure high ethical standards by living the Universities’ core values.
Integrity of culture is enhanced by demonstrating ethics and values, correct measurement and
rewarding of performance, and system access and security.
2. Objective Setting – Considers risk tolerance; management risk strategy and its view of how
much risk the board is willing to accept.
When considering several options for strategies, Twente University will consider these against
its vision which encompasses contributing to the student’s education level. In such case, two out
of three options could result into cost-savings, but negatively affect the time of teachers to
answer questions and opening times of the library; influencing the facilities and level of
education. The university will then focus on the other option which doesn’t decrease costs, but
remains a steady level of education and may be beneficial to marketing tools and public reports.
The University is willing to accept this risk of not lowering costs, but improving education
standards
3. Event Identification – identifies risks and opportunities both internally and externally that
could affect strategy or the achievement of objectives.
The University of Twente strategic objectives is to become the world’s leading hub for nanotechnology and its related objective is to hire 100 qualified staff to boost innovation in this field.
A potential event is an unexpected heat in the job market which could cause fewer job-offers
being accepted, resulting in too few staff. The identification of this event will benefit procedure
and the University´s organization may in time identify the event, and outsource the hiring of
staff to a qualified country in which the job market is more stable.
4. Risk Assessment – Analyses risks, and allows the firm to understand to which extend potential
events may affect objectives. Assesses the likelihood and impact of the event on both an
inherent (in the absence of actions taken by management) and residual basis.
Twente University may perform researches in foreign countries and have identified the event of
a change in foreign exchange rate in a particular banana-republic. For the purpose of risk
assessment, the event is analysed and found that the likelihood of occurring is quite high. In
such case, the University might choose to hedge against this risk, in which case the residual
impact is only $5.000,- whereas the inherent risks’ impact was $15.000,-. A report is delivered to
the board of the University, explaining the impact the risk could have on the strategic plan.
5. Risk Response – The policy that a firm’s management had earlier outlined on risk tolerance
and appetite is now used to select the correct response to the identified risks. Management
may choose to avoid, accept, reduce or share risk.
In the case of the abovementioned research in the banana-republic, the University of Twente
has to make a decision regarding this risky event. When chosen to avoid, the University may
decide to no longer do research here, and move to a country with a stable currency, beneficial
to the University. In case of accepting the risk the University tolerates the loss and covers the
loss by itself. When reducing the risk, the University may choose to diversify its research
facilities globally and not be fully represented in the banana-republic, resulting in a balanced
portfolio and reduced exposure to risk of changing exchange rate. When shared, the University
will enter in a partnership with the local banana-university and/or taking an insurance that will
cover a significant part of the expected loss.
6. Control Activities – Policies and procedures throughout the organization that support the
effectively implementing of risk responses.
When Twente University has selected a response to the risk, for example the risk of foreign
exchange rate; the policies and procedures will support the implementation of the risk
response. In case of risk acceptance, the treasury department of the University will continually
report to the board, to reassess exposure and whether or not to hedge the situation. In case of
avoidance, a contract that has to be signed by the involved parties will state that the country
where research is implemented belong to the Times top-50 wealthy countries.
7. Information and Communication – The identification, capturing and communicating of
information in a form and timeframe that enables people to carry out their responsibilities on
risk management. The information flow occurs down, across and up the entire organization.
In the case of the foreign exchange rate risk for the University of Twente, professors that are
leading researches taking place in foreign countries have to review newspapers and journals on
a semi-annual basis for the particular country, and report to the board about the economic and
political stability of the country. An integrated data system may automatically collect and
compress information that is delivered as a report. For the process of researching in foreign
countries, a flowchart of all processes with key controls is noted.
8. Monitoring – The ongoing monitoring of ERM by management activities and/or separate
evaluations.
For the University of Twente the monitoring may consist of mostly day-to-day review of
information from daily activities. For example reviewing reports on the Universities key research
indicators, education standards and use of student facilities; giving feedback on the
effectiveness of the ERM components. The importance of monitoring is that the University
board asks itself the question: “Are the controls sufficiently qualifying risks so that the University
can attain its objectives?”
Problem 9.23.
The price of a stock is $40. The price of a one-year European put option on the stock with a strike
price of $30 is quoted as $7 and the price of a one-year European call option on the stock with a strike
price of $50 is quoted as $5. Suppose that an investor buys 100 shares, shorts 100 call options, and
buys 100 put options. Draw a diagram illustrating how the investor’s profit or loss varies with the
stock price over the next year. How does your answer change if the investor buys 100 shares, shorts
200 call options, and buys 200 put options?
Table 1 and Figure 1 illustrate the profit and loss in the first case; when buying 100 shares and put
options, and shorting 100 call options. The change of the answer is illustrated in Table 2 and Figure 2
which represents the buy of 100 shares, 200 puts and shorting 200 calls.
Table 1. Change of profit and loss in the first case along the variety of stock price over the next year
Stock price
10
20
30
40
50
60
70
Put option
(100)
1300
300
-700
-700
-700
-700
-700
Call option
(100)
500
500
500
500
500
-500
-1500
Buy stock
(100)
-3000
-2000
-1000
0
1000
2000
3000
Profit
-1200
-1200
-1200
-200
800
800
800
Figure 1. Graph of change in profit and loss in the first case along the variety of stock price over the
next year (stock price x10)
Table 2. Change of profit and loss in the second case along the variety of stock price over the next
year
Stock price
Put option
Call option
Buy stock
Profit
(200)
(200)
(100)
10
2600
1000
-3000
600
20
600
1000
-2000
-400
30
-1400
1000
-1000
-1400
40
-1400
1000
0
-400
50
-1400
1000
1000
600
60
-1400
-1000
2000
-400
70
-1400
-3000
3000
-1400
Figure 2. Graph of change in profit and loss in the second case along the variety of stock price over
the next year (stock price x10)
Problem 10.10
What is a lower bound for the price of a two-month European put option on a non-dividend-paying
stock when the stock price is $58, the strike price is $65, and the risk-free interest rate is 5% per
annum?
Strike price (K): $65
Risk free rate (r): 5% (per annum)
Stock price (S0): 58
Time to expiration (T): 2 months
Ke-rT – S0
Lower bound is:
65e-0.05x2/12 - 58 = $6.46
Problem 10.14.
The price of a European call that expires in six months and has a strike price of $30 is $2.
The underlying stock price is $29, and a dividend of $0.50 is expected in two months and
again in five months. The term structure is flat, with all risk-free interest rates being 10%.
What is the price of a European put option that expires in six months and has a strike price of
$30?
Call:
Time to expiration (T): 6 months
Strike price (K): $30
Put:
6 months
$30
Price of option ( c): 2
(p)….
Stock price (S0): $29
Dividend (D): $0.50
Dividend time: 2 months, 5 months
Risk free rate (per annum) (r): 10%
10%
c + Ke-rT+ D = p+S0
2 + 30e-0.1x6/12+ (0.50e-0.1x2/12 + 0.50-0.1x5/12) = ... + 29
2 + 28.5369 + (0.4917 + 0.4796) = ... + 29
31.5082= … + 29
31.5082 – 29= 2.51 (rounded)
The put price then, is $2.51
Problem 10.22.
A European call option and put option on a stock both have a strike price of $20 and an
expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per
annum, the current stock price is $19, and a $1 dividend is expected in one month. Identify the
arbitrage opportunity open to a trader.
Facts;








Current stock price (S0): $19
Strike price call options (K): $20
Strike price put options (K): $20
Time to expiration of options (T); 3 months (European; can be exercised only on the maturity
date)
Risk-free interest rate (r): 10% per annum
Value of European call option to buy one share(c): $3
Value of European call option to buy one share(p): $3
Dividend : $1 expected in 1 month
The put- call parity implies that when the equation given below does not hold, there are arbitrage
opportunities.
c + D + Ke-rT = p + S0
c + D + Ke-rT = 3 + 1 + 20e-0.1*3/12 = 23.50619824
p + S0=3 +19 = 22
As can be seen in the calculation p + S0 < c + D + Ke-rT . This means that the call is overpriced relative
to the put. In order to exploit this arbitrage opportunity the following arbitrage strategy should be
applied; short the call and buy both the put and the stock. This strategy requires an initial investment
of; 19 + 3 – 3 = 19 . When the investment is financed at the risk-free rate, a repayment of 19e0.1
*3/12
=19.48098729 is required at the end of the three months. After 3 months either the call (stock
price > $20) or the put (stock price < $20) is exercised. In both cases the stock sells for $20 leading to
a gain of;
20 –19.48098729 = 0.51901271 + D =0.51901271 + 1 = $1.51901271
Problem 11.19.
Three put options on a stock have the same expiration date and strike prices of $55, $60, and
$65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can
be created. Construct a table showing the profit from the strategy. For what range of stock
prices would the butterfly spread lead to a loss?
“A spread trading strategy involves taking a position in two or more options of the same type. A
butterfly spread involves positions in options with three different strike prices. It can be created by
buying a call option with a relatively low strike price, K1, buying a call option with a relatively high
strike price, K3, and selling two call options with a strike price, K2, halfway between K1 and K3. A
butterfly spread leads to a profit if the stock price stays close to K2, but gives rise to a small loss if
there is a significant stock price move in either direction. It is therefore an appropriate strategy for an
investor who feels that large stock prices moves are unlikely. ”
Hull, J.C. Fundamentals of futures and options markets. Pearson 7th edition, 2011.
For this assignment, the above means that a butterfly spread can be created by selling two $60 puts
and buying a $55 put and a $65 put. The costs of creating the spread are; (5*2) – 3 – 8 = -1 = $1.
Based on this, the profit per range of stock prices can be calculated. The red marked stock price
ranges lead to losses;
Stock price
Payoff
Payoff – Cost of spread=
Profit
ST < 55
0
-1
55 ≤ ST < 60
ST – 55
ST – 55–1
60 ≤ ST < 65
65 – ST
65–1 – ST
ST ≥ 65
0
-1
Assignment 4
Members
Group number: 4
1. A stock price is currently €50. It is known that at the end of two months it will be either €53 or
€48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value
of a two-month European call option with a strike price of €49?
To calculate todays value of the call option, we have to take into account the risk neutral probability
of the upward movement to a stock price of €53. The value of the portfolio in two months is 4 (4953).
The risk neutral probability is:
= e-10x2/12 – (48/50) / (53/50)-(48/50) = .5681
The value of the portfolio is then: e-10x2/12 x .5681 x 4 = 2.235
2. A stock price is currently €40. Over each of the next two three-month periods it is expected to go
up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous
compounding.
a. What is the value of a six-month European put option with a strike price of €42?
b. What is the value of a six-month American put option with a strike price of €42?
A: The nodes of the binominal tree at the upwards movement would look as following:
40---(t=.25)---- 44 ----(t=.25) -----48.4
P = e-12x3/12 – (36/40) / (44/40)-(36/40) = .97045 – 0.90 / 1.1 – 0.90 = 0.6523
The value of the second node (44) is then: e-.12x.25(0.6523x6.4+.3477x0) = 4.759
The value at the starting node (40) is then: e-12x.25(0.6523x4.0514+.3477x0) = 2.118
B: What is the value of a six-month American put option with a strike price of €42?
The value of an American option is the greater of an European option. An earlier optimal exercise is
possible at the second node. Following the above formula, the value of the 6-month American put
option is 2.537
3. Suppose that observations on a stock price (in euros) at the end of each of 15 consecutive weeks
are as follows:
30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0,
32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2
Estimate the stock price volatility.
Week Closing stock price (€), Price relative, Weekly return, Weekly return²,
Si
Si / Si - 1
Ui = ln(Si / Si - 1) Ui² = ln(Si / Si - 1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
S=√
30,20
32,00
31,10
30,10
30,20
30,30
30,60
33,00
32,90
33,00
33,50
33,50
33,70
33,50
33,20
1
∗
14−1
0,011450 −
1
∗
14(14−1)
1,059603
0,971875
0,967846
1,003322
1,003311
1,009901
1,078431
0,996970
1,003040
1,015152
1,000000
1,005970
0,994065
0,991045
0,057894
0,003352
-0,028528
0,000814
-0,032683
0,001068
0,003317
0,000011
0,003306
0,000011
0,009852
0,000097
0,075508
0,005701
-0,003035
0,000009
0,003035
0,000009
0,015038
0,000226
0,000000
0,000000
0,005952
0,000035
-0,005952
0,000035
-0,008996
0,000081
∑ Ui = 0,094708 ∑ Ui² = 0,011450
0,094708²= √ 0,0008807692 − 0,0000492835 =
=√ 0,0008314856 = 0.0288354929
σ = 0.0288354929 * √ 52 = 0,2079356964 = 20,79356964%
4. Consider an option on a non-dividend-paying stock when the stock price is €30, the exercise price
is €29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to
maturity is four months.
a. What is the price of the option if it is a European call?
b. What is the price of the option if it is an American call?
What is the price of the option if it is a European call?
S0 = 30
K = 29
r = 0,05
σ = 0,25
T = 4/12
d1 =
d2 =
30
29
ln( )+ (0,05+
0,252
)
2
∗ 4/12
0,25 ∗√ 4/12
30
29
ln( )+ (0,05 −
d1
d2
N(d1)
N(d2)
0,252
)
2
0,25 ∗√ 4/12
= 0,4225156773
∗ 4/12
= 0,27817811
0,422516
0,278178
0,663676 (obtained by NORMSDIST function in excel)
0,609562 (obtained by NORMSDIST function in excel)
c = 30 * 0,663676 – 29e-0,05*4/12 * 0,609562 = 2,525162 = $ 2,525162
What is the price of the option if it is an American call?
On the basis of the citation shown below, it can be concluded that the price of the American
call option equals the price of the European call option(calculated in the above calculations).
‘’Because the American call price, C, equals the European call price, c, for a non dividendpaying stock, equation (13.5) also gives the price of an American call.’’
Hull, J.C. Fundamentals of futures and options markets. Pearson 7th edition, 2011.
5. Consider a position consisting of a $300,000 investment in gold and a $500,000 investment in
silver. Suppose that the daily volatilities of these two assets are 1.8% and 1.2%, respectively, and
that the coefficient of correlation between their returns is 0.6. What is the 10-day 97.5% value at
risk for the portfolio? By how much does diversification reduce the VaR?
The variance in the portfolio is as follows:
0.0182 x 3002 + 0.0122 x 5002 + 2 x 0.6 x 0.018 x 300 x 0.012 x 500= 104.4
Standard deviation of the portfolio is: √104.4= 10.2
The 1-day 99.75% VaR is: 10.2 x 1.96= 19.99
The 10-day 99.75% VaR is: √10 x 19.99= 63.2139
(These numbers are in $000, and are rounded).  The 10-day 99.75% VaR, then, is: $63214
By how much does diversification reduce the VaR?
The 10-day 99.75% VaR of the gold position:
300.000 x 0.018= 5400
The 10-day 97.5% VaR of this position is: √10 x 1.96 x 5400= $33470 (rounded)
The 10-day 99.75% VaR of the silver position:
500.000 x 0.012= 6000
The 10-day 97.5% VaR of this position is: √10 x 1.96 x 6000= $37188 (rounded)
Therefore, the diversification benefit is: $33470 + $37188= $70658
$70658 - $63214= $7444
Clarification of BSM
1. Black-Scholes-Merton formula for an option on a non-dividend paying stock:
c is the price of a European call option, p is the price of a European put option, S0 is stock
price, K is strike price, r is continuously compounded risk free rate. T is time to expiration of
the option.
c  S0 N ( d1 )  K e  rT N ( d 2 )
p  K e  rT N ( d 2 )  S0 N ( d1 )
ln( S0 / K )  ( r   2 / 2)T
 T
ln( S0 / K )  ( r   2 / 2)T
d2 
 d1   T
 T
where d1 
2. Black-Scholes-Merton formula for an option on a stock that distributes known
dividends during the life of the option:
c  ( S 0 - D) N ( d 1 )  K e  rT N ( d 2 )
p  K e  rT N ( d 2 )  ( S 0 - D) N ( d 1 )
ln(( S 0 - D) / K )  ( r   2 / 2)T
 T
ln(( S 0 - D) / K )  ( r   2 / 2)T
d2 
 d1   T
 T
D is the present value of the dividends during the life of the option.
where d 1 
3. Black-Scholes-Merton formula for an option on a stock paying a dividend yield at rate
q: (page 330 of the book, not required)
c  S 0 e -qT N ( d 1 )  K e  rT N ( d 2 )
p  K e  rT N ( d 2 )  S 0 e -qT N ( d 1 )
ln( S 0 e -qT / K )  ( r   2 / 2)T
where d 1 
 T
ln( S 0 e -qT / K )  ( r   2 / 2)T
d2 
 d1   T
 T
equivalently
ln( S 0 / K )  ( r  q   2 / 2)T
d1 
 T
ln( S 0 / K )  ( r  q   2 / 2)T
d2 
 d1   T
 T
Course recap
•
Two papers, lecture notes, and textbook (Chp 1-13, 15, 17, 20, 24)
•
Not mandatory readings of the textbook:
–
Chapter 2: 2.8, 2.9
–
Chapter 3: Section “Tailing the hedge” in 3.4
–
Chapter 5: section “convenience yield” in 5.11, 5.13, 5.14
–
Chapter 6: 6.3, section “hedging floating-rate loan” in 6.5
–
Chapter 7: 7.2, 7.3, 7.5, 7.6, 7.11
–
Chapter 9: 9.6, 9.7, 9.8, 9.9, 9.10
–
Chapter 11: section “calendar spreads” and “diagonal spreads” in 11.2
–
Chapter 12: 12.8, 12.9
–
Chapter 13: 13.11
–
Chapter 15: 15.3, 15.4, 15.5, 15.6
–
Chapter 17: First 2 paragraphs (incl. formulas) in section 17.12
–
Chapter 20: 20.4, 20.5, 20.6, 20.7
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