Chapter 1
Chapter 1
Chapter 1
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<strong>Chapter</strong> 1
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5. Define a forward contract. Explain at what time are cash flows generated for this<br />
contract. How is settlement determined?<br />
Answer: A forward contract is an agreement to buy or sell an asset at a future date<br />
(denoted T ), at a specified price called the delivery price (denoted F ). Denote the initial<br />
date (the inception date or the date of the agreement) by t = 0. At inception there are<br />
no cash flows on a forward contract. At maturity, if the then-prevailing spot price ST of<br />
the underlying asset is greater than F , then the buyer (the “long position”) has gained<br />
ST − F via the forward while the seller (the “short position”) has correspondingly lost<br />
ST − F . Depending on contract specifications, the settlement may either be in cash<br />
(the seller pays the buyer ST − F ) or physical (the seller delivers the asset and receives<br />
F ). If ST < F , the buyer loses F − ST and the seller gains this quantity.<br />
6. Explain who bears default risk in a forward contract.<br />
Answer: Default arises if, at maturity, one of the parties fails to fulfill their obligations<br />
under the contract. Default risk only matters for the party that is ”in the money” at<br />
maturity, that is, that stands to profit at the locked-in price in the contract. (If the<br />
spot price at maturity is such that a party would lose from performing on the obligation<br />
in the contract, counterparty default is not a problem.) Prior to maturity, since either<br />
party may finish in-the-money, both parties are exposed to default risk.<br />
7. What risks are being managed by trading derivatives on exchanges?<br />
Answer: An important one is counterparty default risk. In a typical futures exchange,<br />
the exchange interposes itself between buyer and seller and guarantees performance<br />
on the contract. This reduces significantly the default risk exposure of both parties.<br />
Further, daily settling of marked-to-market gains and losses ensures that the loss to the<br />
exchange from an investor’s default is limited to at most one day’s settlement amount<br />
(and because of maintenance margins is usually less than even this; see <strong>Chapter</strong> 2 for a<br />
description of the margining process).<br />
8. Explain the difference between a forward contract and an option.<br />
Answer: A forward contract is an agreement to buy or sell an asset at a future date<br />
(denoted T ), at a specified delivery price (denoted F ). The agreement is made at time<br />
t = 0 for settlement at maturity T . An option is the right but not the obligation to buy<br />
(a “call” option) or sell (a “put” option) an asset at a specified strike price on or before<br />
a specified maturity date T . In comparing a long forward contract to a call option, the<br />
main difference lies in the fact that the forward buyer has to buy the stock at the forward<br />
price at maturity, whereas in a call option, the buyer is not required to carry out the
Sundaram & Das: Derivatives - Problems and Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5<br />
purchase if it is not in his interest to do so. The forward contract confers the obligation<br />
to buy, whereas the option contract provides this right with no attendant obligation.<br />
9. What is the difference between value and payoff in the context of derivative securities.<br />
Answer: The value of a derivative is its current fair price or its worth. The payoff (or<br />
payoffs) refers to the cash flows generated by the derivative at various times during its<br />
life. For example, the value of a forward contract at inception is zero: neither party pays<br />
anything to enter into the contract. But the payoffs from the contract at maturity to<br />
either party could be positive, negative, or zero depending on where the spot price of<br />
the asset is at that point relative to the locked-in delivery price.<br />
10. What is a short position in a forward contract? Draw the payoff diagram for a short<br />
position at a forward price of $103, if the possible range of the underlying stock price is<br />
$50-150.<br />
Answer: A short position in a forward is where you are the seller of the forward contract.<br />
In this case, you gain when the price of the underlying asset at maturity is below the<br />
locked-in delivery price. The payoff diagram for this contract is as shown in the following<br />
picture. When the price of the stock at maturity is the delivery price of $103, there are<br />
neither gains nor losses.<br />
Payoff<br />
60<br />
40<br />
20<br />
0<br />
-20<br />
-40<br />
-60<br />
A Short Forward Contract's Payoff<br />
Stock Price<br />
50 60 70 80 90 100 110 120 130 140 150<br />
11. Forward prices may be derived using the notion of absence of arbitrage, and market<br />
efficiency is not necessary. What is the difference between these two concepts?<br />
Answer: Absence of arbitrage means that a trading strategy cannot be found that creates<br />
cash inflows without any cash outflows, i.e., creates something out of nothing. Efficiency,
Sundaram & Das: Derivatives - Problems and Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6<br />
as that term is used by financial economists, implies more: not only the absence of<br />
arbitrage but that asset prices reflect all relevant information.<br />
12. Suppose you are holding a stock position, and wish to hedge it. What forward contract<br />
would you use, a long or a short? What option contract might you use? Compare<br />
the forward versus the option on the following three criteria: (a) uncertainty of hedged<br />
position cash-flow, (b) Up-front cash-flow and (c) maturity-time regret.<br />
Answer: If a forward contract is to be used, then a short forward is required. Alternatively,<br />
a put option may also be used. The following describes the three criteria for the<br />
choice of the forward versus the option.<br />
• Cash-flow uncertainty is lower for the futures contract.<br />
• The futures contract has no up-front cash-flow, whereas the option contract has<br />
an initial premium to be paid.<br />
• There is no maturity-time regret with the option, because if the outcome is undesirable,<br />
the option need not be exercised. With the futures contract there is a<br />
possible downside.<br />
13. What derivatives strategy might you implement if you expected a bullish trend in stock<br />
prices? Would your strategy be different if you also forecast that the volatility of stock<br />
prices will drop?<br />
Answer: If you expect prices to rise, there are several different strategies you could<br />
follow: you could go long a forward and lock in a price today for the future purchase;<br />
you could buy a call which gives you the right to buy the stock at a fixed strike price;<br />
or you could sell a put today, receive a premium, and keep the premium as your profit if<br />
prices trend upward as you expect.<br />
The volatility issue is a bit trickier. As we explain in <strong>Chapter</strong> 7, both call and put options<br />
increase in value with volatility, so if you expect volatility to decrease, you do not want<br />
to buy a call: when volatility drops, what you have bought automatically becomes less<br />
valuable.<br />
14. What are the underlyings in the following derivative contracts?<br />
(a) A life insurance contract.<br />
(b) A home mortgage.<br />
(c) Employee stock options.<br />
(d) A rate lock in a home loan.
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Answer: The underlyings are as follows:<br />
(a) A life insurance contract: the event of one’s demise.<br />
(b) A home mortgage: mortgage interest rate.<br />
(c) Employee stock options: equity price of the firm.<br />
(d) A rate lock in a home loan: mortgage interest rate.<br />
15. Assume you have a portfolio that contains stocks that track the market index. You now<br />
want to change this portfolio to be 20% in commodities and only 80% in the market<br />
index. How would you use derivatives to implement your strategy?<br />
Answer: One would use futures to do so. We would short market index futures for<br />
20% of the portfolio’s value, and go long 20% in commodity futures. A collection of<br />
commodity futures adding up to the 20% would be required.<br />
16. In the previous question, how do you implement the same trading idea without using<br />
futures contracts?<br />
Answer: Futures contracts are traded on exchanges and are known as “exchange-traded”<br />
securities. An alternative approach to achieving the goal would be to use an over-thecounter<br />
or OTC product, for example, an index swap that exchanges the return on the<br />
market index for the return on a broadly defined commodity index.<br />
17. You buy a futures contract on the S&P 500. Is the correlation with the S&P 500 index<br />
positive or negative? If the nominal value of the contract is $100,000 and you are<br />
required to post $10,000 as margin, how much leverage do you have?<br />
Answer: The futures contract is positively correlated with the stock index. The leverage<br />
is 10 times. That is, for every $1 invested in margin, you get access to $10 in exposure.
<strong>Chapter</strong> 2
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16. What is the “closing out” of a position in futures markets? Why is closing out of<br />
contracts permitted in futures markets? Why is unilateral transfer or sale of the contract<br />
typically not allowed in forward markets?<br />
Answer: To close out a position in a futures market, an investor must take an offsetting<br />
opposite position in the same contract. (For example, to close out a long position in 10<br />
S&P 500 index futures contracts with expiry in March, an investor must take a short<br />
position in 10 S&P 500 index futures contracts with expiry in March.) Once a position<br />
is closed out, the investor no longer has any obligations remaining.<br />
Credit risk is key to allowing investors to close out contracts. In a futures exchange, the<br />
exchange interposes itself between buyer and seller as the guarantor of all trades; thus,<br />
there is little credit risk involved. In forward markets, allowing investors to unilaterally<br />
transfer their obligations could exacerbate credit risk, so it is typically disallowed.<br />
An obligation under a forward contract may be eliminated in one of two ways: (a) the<br />
contract may be unwound with the same counterparty or (b) the contract may be offset<br />
by entering into an equal and opposite contract with a third party. The latter is the<br />
analog of the unilateral close-out of futures contracts. However, while close-out of the<br />
futures contract leaves the investor with no net obligations, offset of a forward contract<br />
leaves the investor with obligations on both contracts.<br />
17. An investor enters into a long position in 10 silver futures contracts at a futures price of<br />
$4.52/oz and closes out the position at a price of $4.46/oz. If one silver futures contract<br />
is for 5,000 ounces, what are the investor’s gains or losses?<br />
Answer: Effectively, the investor buys at $4.52 per oz and sells at $4.46 per oz, so takes<br />
a loss of $0.06 per oz. Per contract, this amounts to a loss of (5000 × 0.06) = $300.<br />
Over 10 contracts, this results in a total loss of $3,000.00.<br />
18. What is the settlement price? The opening and closing price?<br />
Answer: The opening price for a futures contract is the price at which the contract is<br />
traded at the begining of a trading session. The closing price is the last price at which<br />
the contract is traded at the close of a trading session. The settlement price is a price<br />
chosen by the exchange as a representative price from the prices at the end of a session.<br />
The settlement price is the official closing price of the exchange; it is the price used to<br />
settle gains and losses from futures trading and to invoice deliveries.<br />
19. An investor enters into a short futures position in 10 contracts in gold at a futures price<br />
of $276.50 per oz. The size of one futures contract is 100 oz. The initial margin per<br />
contract is $1,500, and the maintenance margin is $1,100.
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(a) What is the initial size of the margin account?<br />
(b) Suppose the futures settlement price on the first day is $278.00 per oz. What is<br />
the new balance in the margin account? Does a margin call occur? If so, assume<br />
that the account is topped back to its original level.<br />
(c) The futures settlement price on the second day is $281.00 per oz. What is the new<br />
balance in the margin account? Does a margin call occur? If so, assume that the<br />
account is topped back to its original level.<br />
(d) On the third day, the investor closes out the short position at a futures price of<br />
$276.00. What is the final balance in his margin account?<br />
(e) Ignoring interest costs, what are his total gains or losses?<br />
Answer: Futures position: short 10 contracts<br />
Size of one contract: 100 oz<br />
Initial margin per contract: $1,500<br />
Maintenance margin per contract: $1,100<br />
Initial futures price: $276.50 per oz<br />
(a) Initial size of margin account = 1, 500 × 10 = 15, 000.<br />
(b) If the settlement price is $278 per oz, the short position has effectively lost $1.50<br />
per oz. This is a loss of 1.50 × 100 = 150 per contract. Since the position has<br />
10 contracts, the overall loss is 150 × 10 = 1, 500. Thus, the new balance in the<br />
margin account is 15, 000 − 1, 500 = 13, 500. A margin call does not occur since<br />
this new balance is larger than the maintenance margin of $11,000.<br />
(c) When the settlement price moves to $281 per oz, the short position effectively<br />
loses another $3 per oz. The loss per contract is 3 × 100 = 300, so the overall<br />
loss is 300 × 10 = 3, 000. Thus, the balance in the margin account is reduced<br />
to 13, 500 − 3, 000 = 10, 500. Since this is less than the maintenance margin, a<br />
margin call occurs. Assume the account is topped back to $15,000.<br />
(d) When the position is closed out at $276 per oz, the short position makes a gain of<br />
281−276 = 5 per oz. This translates to a gain of 500 per contract, and, therefore,<br />
to an overall gain of 5,000. Thus, the closing balance in the margin account is<br />
15, 000 + 5, 000 = 20, 000.<br />
(e) The investor began with a margin account of $15,000, and deposited another<br />
$4,500 to meet the margin call, for a total outlay of $19,500. Since the margin<br />
account balance at time of close out is $20,000, his overall gain (ignoring interest<br />
costs) is $500.<br />
20. The current price of gold is $642 per troy ounce. Assume that you initiate a long position<br />
in 10 COMEX gold futures contracts at this price on 7-July-2006. The initial margin is
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5% of the initial price of the futures, and the maintenance margin is 3% of the initial<br />
price. Assume the following evolution of gold prices over the next five days, and compute<br />
the margin account assuming that you meet all margin calls.<br />
Date Price per Ounce<br />
7-Jul-06 642<br />
8-Jul-06 640<br />
9-Jul-06 635<br />
10-Jul-06 632<br />
11-Jul-06 620<br />
12-Jul-06 625<br />
Answer: The initial margin is $321, and the maintenance margin is $193. The following<br />
is the evolution of the margin account. Note that there is one margin call that takes<br />
place on 11-July-2006.<br />
Initiation Price = 642<br />
Initial Margin (5%) = 321<br />
Maintenance Margin (3%) = 192.6<br />
Number of contracts = 10<br />
Margin Account<br />
Opening Daily Profit Adjusted Margin Call Closing<br />
Date Gold Price Balance and Loss Balance Deposit Balance<br />
7-Jul-06 642<br />
8-Jul-06 640 321 -20 301 0 301<br />
9-Jul-06 635 301 -50 251 0 251<br />
10-Jul-06 632 251 -30 221 0 221<br />
11-Jul-06 620 221 -120 101 220 321<br />
12-Jul-06 625 321 50 371 0 371<br />
21. When is a futures market in “backwardation”? When is it in “contango”?<br />
Answer: A futures market is said to be in backwardation if the futures price is less than<br />
the spot price. It is in contango if futures price is above spot.<br />
22. Suppose there are three deliverable bonds in a Treasury Bond futures contract whose<br />
current cash prices (for a face value of $100,000) and conversion factors are as follows:<br />
(a) Bond 1: Price $98,750. Conversion factor 0.9814.
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(b) Bond 2: Price $102,575. Conversion factor 1.018.<br />
(c) Bond 3: Price $101,150. Conversion factor 1.004.<br />
The futures price is $100,625. Which bond is currently the cheapest-to-deliver?<br />
Answer: Since the long position will pay the futures price of 100,625 times the conversion<br />
factor in settlement, the short position prefers to deliver the bond on which the ratio of<br />
the sale price to the purchase price is highest. Essentially, this means the bond delivered<br />
is cheapest relative to the sale price. We compute this ratio for all three bonds as follows:<br />
100, 625 × 0.9814<br />
98, 750<br />
100, 625 × 1.018<br />
102, 575<br />
100, 625 × 1.004<br />
101, 150<br />
= 1.0000<br />
= 0.99865<br />
= 0.99879<br />
Hence, the first bond is the cheapest to deliver.<br />
23. You enter into a short crude oil futures contract at $43 per barrel. The initial margin is<br />
$3,375 and the maintenence margin is $2,500. One contract is for 1,000 barrels of oil.<br />
By how much do oil prices have to change before you receive a margin call?<br />
Answer: If the margin account falls to a value of $2500 then a call will occur. Therefore,<br />
the loss on the position must be equal to $3375-$2500=$875 for a margin call. Solving<br />
the following equation<br />
1000 (P − 43) = 875<br />
gives P = 43.875, which is the price at which a margin call will take place.<br />
24. You take a long futures contract on the S&P 500 when the futures price is 1,107.40,<br />
and close it out three days later at a futures price of 1,131.75. One futures contract is<br />
for 250× the index. Ignoring interest, what are your losses/gains?<br />
Answer: The gain is<br />
250(1131.75 − 1107.40) = $6087.50
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25. An investor enters into 10 short futures contract on the Dow Jones Index at a futures<br />
price of 10,106. Each contract is for 10× the index. The investor closes out five<br />
contracts when the futures price is 10,201, and the remaining five when it is 10,074.<br />
Ignoring interest on the margin account, what are the investor’s net profits or losses?<br />
Answer: Exercise for the reader.<br />
26. A bakery enters into 50 long wheat futures contracts on the CBoT at a futures price<br />
of $3.52/bushel. It closes out the contracts at maturity. The spot price at this time is<br />
$3.59/ bushel. Ignoring interest, what are the bakery’s gains or losses from its futures<br />
position?<br />
Answer: Each CBoT Wheat contract is for 50,000 bushels and so the settlement gain<br />
is<br />
50 × 50, 000 × (3.59 − 3.52) = $175, 000<br />
27. An oil refining company enters into 1,000 long one-month crude oil futures contracts on<br />
NYMEX at a futures price of $43 per barrel. At maturity of the contract, the company<br />
rolls half of its position forward into new one-month futures and closes the remaining<br />
half. At this point, the spot price of oil is $44 per barrel, and the new one-month futures<br />
price is $43.50 per barrel. At maturity of this second contract, the company closes out<br />
its remaining position. Assume the spot price at this point is $46 per barrel. Ignoring<br />
interest, what are the company’s gains or losses from its futures positions?<br />
Answer: Exercise for the reader.<br />
28. Define the following terms in the context of futures markets: market orders, limit orders,<br />
spread orders, one-cancels-the-other orders.<br />
Answer: See section 2.3 of the book.<br />
29. Distinguish between market-if-touched orders and stop orders.<br />
Answer: See section 2.3 of the book.<br />
30. You have a commitment to supply 10,000 oz of gold to a customer in three months’<br />
time at some specified price and are considering hedging the price risk that you face. In<br />
each of the following scenarios, describe the kind of order (market, limit, etc.) that you<br />
would use.
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will be sustained. Thus, unless there is high volatility and a reversal of direction, this<br />
approach may not be profitable and might turn out to be loss-making.<br />
32. The spread between May and September wheat futures is currently $0.06 per bushel.<br />
You expect this spread to widen to at least $0.10 per bushel. How would you use a<br />
spread order to bet on your view?<br />
Answer: If the price differential between September and May futures is currently $0.06<br />
and is expected to widen to $0.10, then we should enter into a long position in the<br />
September contract and a short position in the May contract. When the spreads widens<br />
we close out both contracts.<br />
33. The spread between one-month and three-month crude oil futures is $3 per barrel. You<br />
expect this spread to narrow sharply. Explain how you would use a spread order given<br />
this outlook.<br />
Answer: Assuming the three-month minus one-month spread will narrow, we should go<br />
long the one-month contract and short the three-month contract. When the spread<br />
narrows, we buy back the short three-month contract and sell back the long one-month<br />
contract. We capture (ignoring interest) the difference between $3 and the new spread.<br />
34. Suppose you anticipate a need for corn in three months’ time and are using corn futures<br />
to hedge the price risk that you face. How is the value of your position affected by a<br />
strengthening of the basis at maturity?<br />
Answer: The basis is the futures price minus the spot price. A strengthening of the<br />
basis occurs if the basis increases. If this occurs, the position in the question is positively<br />
affected since you are long futures. In notational terms, you go long futures today (at<br />
price F0, say) and close it out at T (at price FT , say) for a net cash flow on the futures<br />
position of FT −F0. In addition, you buy the corn you need at the time-T spot price ST ,<br />
leading to a total net cash flow of (FT − F0) − ST = (FT − ST ) − F0. A strengthening<br />
of the basis FT − ST at maturity improves this cash flow.<br />
35. A short hedger is one who is short futures in order to hedge a spot cash flow risk. A<br />
long hedger is similarly one who goes long futures to hedge an existing risk. How does<br />
a weakening of the basis affect the positions of short and long hedgers?<br />
Answer: The short hedger is short futures and long spot, so gains if the basis weakens.<br />
The long hedger is long futures and short spot, so loses in this case.
<strong>Chapter</strong> 3
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Answer: The spot price of wheat is $3.60. Since there are no storage costs, we compute<br />
the theoretical forward price of wheat as 3.60 exp(0.08 × 3/12) = 3.6727 which is less<br />
than the forward price. Hence, there is an arbitrage opportunity.<br />
In order to arbitrage this situation, we would undertake the following strategy:<br />
• Sell wheat forward at 3.90.<br />
• Buy wheat spot at 3.60.<br />
• Borrow 3.60 for three months .<br />
At inception, the net cash-flow is zero. At maturity, we deliver the wheat we own and<br />
receive the forward price of $3.90. We return the borrowed amount with interest for<br />
a cash outflow of 3.60 exp(0.08 × 0.25) = 3.6727.This results in a net cash inflow of<br />
0.2273. The following table summarizes:<br />
Cash Flows<br />
Source Initial Final<br />
Short Forward - +3.9000<br />
Long spot −3.6000 -<br />
Borrowing +3.6000 −3.6727<br />
Net - +0.2273<br />
Note that it makes no difference if the contract is cash-settled instead of settled by<br />
physical delivery. If it is cash-settled, letting WT denote the spot price of wheat at date<br />
T , we receive 3.90 − WT on the forward contract, sell the spot wheat we own for WT ,<br />
and repay the borrowing, for exactly the same final cash flow.<br />
5. A security is currently trading at $97. It will pay a coupon of $5 in two months. No<br />
other payouts are expected in the next six months.<br />
(a) If the term structure is flat at 12%, what should the be forward price on the security<br />
for delivery in six months?<br />
(b) If the actual forward price is $92, explain how an arbitrage may be created.<br />
Answer: We have that S = 97, and the PV of holding benefits is 5 exp(−0.12 ×<br />
(2/12)) = 4.9010. Thus, the forward price should be<br />
(97 − 4.9010) exp(0.12 × (6/12)) = 97.794.<br />
Since the forward price is 92, it is mispriced (under-priced). The arbitrage is as follows.<br />
At inception:<br />
• Buy forward at 92.
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• Sell short spot at 97<br />
• Invest P V (5) = 4.901 for three months at 12%.<br />
• Invest 97 − P V (5) = 92.099 for six months at 12%.<br />
• In three months, use the cash inflow of 5 from the investment to pay the coupon<br />
due on the shorted security.<br />
• In six months, receive the cash from the six-month investment. Pay the delivery<br />
price of 97 on the forward and receive unit of the security, Use this to close the<br />
short spot position.<br />
The initial and interim cash flows are zero, and the final cash flow is positive as the<br />
following table shows:<br />
Cash Flows<br />
Source Initial Interim Final<br />
Long forward 0 - −92.00<br />
Short spot +97.00 - -<br />
3-month investment −4.901 +5 -<br />
6-month investment −92.099 - +97.794<br />
Net - - +5.794<br />
6. Suppose that the current price of gold is $365 per oz and that gold may be stored<br />
costlessly. Suppose also that the term structure is flat with a continuously compounded<br />
rate of interest of 6% for all maturities.<br />
(a) Calculate the forward price of gold for delivery in three months.<br />
(b) Now suppose it costs $1 per oz per month to store gold (payable monthly in<br />
advance). What is the new forward price?<br />
(c) Assume storage costs are as in part (b). If the forward price is given to be $385<br />
per oz, explain whether there is an arbitrage opportunity and how to exploit it.<br />
Answer: The answers to the three parts are given below:<br />
(a) The forward price of gold is<br />
365 exp(0.06 × 0.25) = 370.52.<br />
(b) With storage costs we need to first find the present value of the holding costs (M).<br />
These are:<br />
1 + 1 exp(−0.06 × 1/12) + 1 exp(−0.06 × 2/12) = 2.9851.
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The forward price then is<br />
(S + M) exp(rT ) = (365 + 2.9851) exp(0.06 × 0.25) = 373.55.<br />
This is higher because the storage costs have been factored in.<br />
(c) If the forward price is 385, there is an arbitrage which is exploited as follows. At<br />
inception:<br />
• Sell forward at 385.<br />
• Buy spot at 365.<br />
• Pay storage costs = 1.<br />
• Invest 1 exp(−0.06 × 1/12) = 0.9950 for one month.<br />
• Invest 1 exp(−0.06 × 2/12) = 0.9900 for two months.<br />
• Borrow 365 + 1 + 0.9950 + 0.9900 = 367.985 for three months.<br />
The net cash flow at inception is zero.<br />
At the end of one month, realize $1 from the investment of 0.995 made at time<br />
zero, and use this to pay off the storage costs. There are no net cash flows.<br />
At the end of two months, realize $1 from the investment of 0.99 made at time<br />
zero, and use this to pay off the storage costs. Again, there are no net cash flows.<br />
On maturity, deliver the spot holding to close out the forward contract by physical<br />
delivery. Cash flows at maturity are:<br />
385 − 367.985 exp(0.06 × 3/12) = 11.454.<br />
This is positive irrespective of the time-T spot price of gold.<br />
The following table summarizes all the cash-flows.<br />
cash-flows<br />
Source Initial Month 1 Month 2 Month 3<br />
Sell forward 0 - - 385<br />
Buy spot −365 - - -<br />
Month 1 storage cost −1 - - -<br />
Month 2 storage cost - −1 - -<br />
Month 3 storage cost – - −1 -<br />
Borrow +367.985 - - −373.5464<br />
Invest 0.995 for one month −0.995 +1 - -<br />
Invest 0.99 for two months −0.99 - +1 -<br />
Net 0 0 0 +11.454<br />
7. A stock will pay a dividend of $1 in one month and $2 in four months. The risk-free<br />
rate of interest for all maturities is 12%. The current price of the stock is $90.
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(a) Calculate the arbitrage-free price of (i) a three-month forward contract on the stock<br />
and (ii) a six-month forward contract on the stock.<br />
(b) Suppose the six-month forward contract is quoted at 100. Identify the arbitrage<br />
opportunities, if any, that exist, and explain how to exploit them.<br />
Answer: We are given: S = 90; r = 0.12 for all maturities; and that dividends of $1<br />
and $2 will be paid in one and four months, respectively.<br />
(a) First, consider the case of a three-month horizon. There is only one dividend to<br />
be considered, viz. the payment of $1 in one month. The present value of this<br />
dividend is<br />
exp{−(0.12)( 1<br />
)} × 1 = 0.99.<br />
12<br />
Since the dividend represents a cash inflow, we have M = −0.99. Therefore, the<br />
arbitrage-free forward price is<br />
F = (S + M)e rT = (90 − 0.99)e (0.12)(0.25) = 91.72.<br />
Now, consider the six-month horizon. There are two dividend payments that occur.<br />
The present value of the first dividend is 0.99, as we have seen above. The present<br />
value of the second dividend is<br />
exp{−(0.12)( 1<br />
)} × 2 = 1.92.<br />
3<br />
Therefore, the present value of the dividends combined is 0.99+1.92 = 2.91. Since<br />
the dividends represent a cash inflow, we must have M = −2.91.<br />
It follows that the arbitrage-free forward price for a six-month horizon is<br />
F = (S + M)e rT = (90 − 2.91)e (0.12)(0.50) = 92.475.<br />
(b) The six-month forward is quoted at 100, so it is overvalued relative to spot. To<br />
make an arbitrage profit, one should sell forward, buy spot, and borrow. Specifically:<br />
i. At time 0: Buy one unit of spot; borrow 87.09 for repayment in six months;<br />
borrow 1.92 for repayment in four months; and borrow 0.99 for repayment in<br />
one month.<br />
Net cash flow: 0.<br />
ii. In one month: receive dividend of $1; use this to repay the one-month loan.<br />
Net cash flow: 0.<br />
iii. In four months: receive dividend of $2; use this to repay the four-month loan.<br />
Net cash flow: 0.
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iv. In six months: Use the unit of spot to settle the short forward position; receive<br />
100 from the forward position; use 92.475 of this to repay the six-month loan.<br />
Net cash flow: 100 − 92.475 = 7.525.<br />
8. A bond will pay a coupon of $4 in two months’ time. The bond’s current price is<br />
$99.75. The two-month interest rate is 5% and the three-month interest rate is 6%,<br />
both in continuously compounded terms.<br />
(a) What is the arbitrage-free three-month forward price for the bond?<br />
(b) Suppose the forward price is given to be $97. Identify if there is an arbitrage<br />
opportunity and, if so, how to exploit it.<br />
Answer: The coupons represent a cash inflow, so the forward price is<br />
[99.75 − 4 exp(−0.05 × 2/12)] exp(0.06 × 3/12) = 97.231.<br />
If the forward price is 97, then the forward is underpriced relative to spot. An arbitrage<br />
exists and is exploited with the following strategy:<br />
• Buy forward at 97.<br />
• Sell spot at 99.75.<br />
• Invest the present value of the coupon for two months. The PV of the coupon is<br />
4 exp(−0.05 × 2/12) = 3.9668.<br />
• Invest the remaining proceeds for three months , i.e., invest 99.75 − 3.9668 =<br />
95.7832.<br />
The cash-flow at inception is zero.<br />
After two months, realize $4 from the investment of 3.9688 and use it to pay the coupon<br />
on the shorted bond. Net cash flow: zero.<br />
At maturity we have the following cash-flows:<br />
• Pay 97 on the forward contract and receive the bond. Use it to close out the short<br />
spot position.<br />
• Receive principal plus interest on the investment: 95.7832 exp(0.06 × 3/12).<br />
The net cash-flow is 0.2308, which is positive.<br />
9. Suppose that the three-month interest rates in Norway and the US are, respectively, 8%<br />
and 4%. Suppose that the spot price of the Norwegian Kroner is $0.155.
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(a) Calculate the forward price for delivery in three months.<br />
(b) If the actual forward price is given to be $0.156, examine if there is an arbitrage<br />
opportunity.<br />
Answer: The forward price of the Kroner is<br />
0.155 exp((0.04 − 0.08) × 3/12) = 0.15346.<br />
Since the forward price is actually 0.156, it is overpriced. We may exploit this by the<br />
following strategy:<br />
• Sell 1 Kroner forward at $0.156.<br />
• Buy PV(1 Kroner)= e −0.08×3/12 = 0.9802 spot at 0.155 × e −0.08×3/12 = $0.1519.<br />
• Invest PV(1 Kroner) for three months. Amount received at maturity = 1 Kroner.<br />
• Borrow $0.1519 for three months at 4%.<br />
• After three months, deliver Kroner and receive $0.156 from the forward. Repay<br />
dollar borrowing with interest for a total of $0.1534.<br />
The resulting cash flows are summarized in the following table:<br />
Cash flow in Kroner Cash flow in $<br />
At inception +0.9802 (purchase) −0.1519 (sale)<br />
−0.9802 (investment) +0.1519 (borrowing)<br />
Net: 0 Net: 0<br />
At T +1.00 (from investment) −0.1534 (repay borrowing)<br />
−1.00 (deliver to forward) +0.156 (receive from forward)<br />
Net: 0 Net: +0.0026<br />
Since all cash flows are zero or positive, we have the required arbitrage.
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10. Consider a three-month forward contract on pound sterling. Suppose the spot exchange<br />
rate is $1.40/£, the three-month interest rate on the dollar is 5%, and the three-month<br />
interest rate on the pound is 5.5%. If the forward price is given to be $1.41/£, identify<br />
whether there are any arbitrage opportunities and how you would take advantage of<br />
them.<br />
Answer: We are given the information that S = 1.40, r = 0.05 and d = 0.055. From<br />
this data, the arbitrage-free forward price of a three-month forward contract should be<br />
F = e (r−d)T S = e (0.05−0.055)(1/4) (1.40) = 1.3983.<br />
Thus, at the given forward price of $1.41/£, the forward contract is overvalued relative<br />
to spot. To take advantage of the opportunity, we should sell forward, buy spot, and<br />
borrow to finance the spot purchase. Specifically:<br />
• Enter into a short forward contract to deliver pounds in three months at $1.41/£.<br />
• Buy e −dT = 0.9863 pounds spot at the spot price of $1.40/£.<br />
Cost: $(1.40)(0.9863) = $1.3809.<br />
• Invest the £0.9863 for three months at 5.5%.<br />
Amount received after three months: £1.<br />
• Borrow $1.3809 for three months at 5%.<br />
Amount due in three months: $(e (0.05)(1/4) (1.3809) = $1.3983.<br />
The resulting cash flows are summarized in the following table:<br />
Cash flow in £ Cash flow in $<br />
At inception +0.9863 (from purchase) −1.3809 (to purchase £)<br />
−0.9863 (investment) +1.3809 (borrowing)<br />
Net: 0 Net: 0<br />
At T +1.00 (from investment) −1.3983 (repay borrowing)<br />
−1.00 (deliver to forward) +1.41 (receive from forward)<br />
Net: 0 Net: +0.0117<br />
Since all cash flows are zero or positive, we have the required arbitrage.
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11. Three months ago, an investor entered into a six-month forward contract to sell a stock.<br />
The delivery price agreed to was $55. Today, the stock is trading at $45. Suppose the<br />
three-month interest rate is 4.80% in continuously compounded terms.<br />
(a) Assuming the stock is not expected to pay any dividends over the next three<br />
months, what is the current forward price of the stock?<br />
(b) What is the value of the contract held by the investor?<br />
(c) Suppose the stock is expected to pay a dividend of $2 in one month, and the<br />
one-month rate of interest is 4.70%. What are the current forward price and the<br />
value of the contract held by the investor?<br />
Answer: The answers to the three parts are:<br />
(a) The current forward price is<br />
45 exp(0.048 × 3/12) = 45.543.<br />
(b) The value of the contract is P V (K −F ). Since K −F = 55.000−45.543 = 9.457,<br />
the contract value is 9.457 exp(−0.048 × 3/12) = 9.3442.<br />
(c) Now suppose the stock is expected to pay a dividend of $2 in one month. The<br />
present value of this dividend payment is<br />
e −(0.047)(1/12) 2 = 1.992.<br />
Since the dividend payment represents a cash inflow, we have M = −1.992. Thus,<br />
the arbitrage-free forward price is now<br />
F = e rT (S + M) = e (0.048)(1/4) (45 − 1.992) = 43.527.<br />
The value of holding a short position in this forward contract with a delivery price<br />
of K = 55 is now<br />
PV(K − F ) = e −(0.048)(1/4) (55 − 43.527) = 11.336.<br />
12. An investor enters into a forward contract to sell a bond in three months’ time at $100.<br />
After one month, the bond price is $101.50. Suppose the term-structure of interest<br />
rates is flat at 3% for all maturities.<br />
(a) Assuming no coupons are due on the bond over the next two months, what is the<br />
forward price on the bond?<br />
(b) What is the marked-to-market value of the investor’s short position?<br />
(c) How would your answers change if the bond will pay a coupon of $3 in one month’s<br />
time?
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Answer: The answers to the three parts are:<br />
(a) The forward price at the end of one month is<br />
101.50 exp(0.03 × 2/12) = 102.01.<br />
(b) The marked-to-market value of the original contract is<br />
P V (F − K) = (100 − 102.01) exp(−0.03 × 2/12) = −2.00.<br />
(c) If there is a coupon one month from now, then the re-estimated forward price is:<br />
(101.50 − 3 exp(−0.03 × 1/12)) exp(0.03 × 2/12) = 99.001.<br />
In this case, the value of the contract to sell forward at 100 is<br />
(100 − 99.001) exp(−0.03 × 2/12) = 0.994.<br />
13. A stock is trading at $24.50. The market consensus expectation is that it will pay a<br />
dividend of $0.50 in two months’ time. No other payouts are expected on the stock over<br />
the next three months. Assume interest rates are constant at 6% for all maturities. You<br />
enter into a long position to buy 10,000 shares of stock in three months’ time.<br />
(a) What is the arbitrage-free price of the three-month forward contract?<br />
(b) After one month, the stock is trading at $23.50. What is the marked-to-market<br />
value of your contract?<br />
(c) Now suppose that at this point, the company unexpectedly announces that dividends<br />
will be $1.00 per share due to larger-than-expected earnings. Buoyed by the<br />
good news, the share price jumps up to $24.50. What is now the marked-to-market<br />
value of your position?<br />
Answer: The answers to the three parts are as follows:<br />
(a) The dividends represent a cash inflow, so the arbitrage free forward price of the<br />
three-month forward is obtained by using the formula P V (F ) = S + M. Substituting<br />
for the various input values, this gives us the original forward price as<br />
[24.50 − 0.50 e −0.06×2/12 ] × e 0.06×3/12 = 24.368.<br />
Denote this locked-in delivery price by K.
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(b) After one month, the contract has two remaining months of life. Given that the<br />
new spot price is S = 23.50 (and that interest rates are unchanged), the forward<br />
price of the same contract is<br />
F = [23.50 − 0.50 e −0.06×1/12 ] × e 0.06×2/12 = 23.234.<br />
Hence, the marked-to-market value of the original contract is<br />
P V (F − K) = (23.234 − 24.368) exp(−0.06 × 2/12) = −1.1227<br />
i.e., a loss of $1.1227 per share. On 10,000 shares, the value of the position is<br />
$10, 000 × −1.1227 = −$11, 227.<br />
(c) If the dividends change to 1.00, we need to rework the forward price and re-assess<br />
the position. The new forward price will be:<br />
[24.50 − 1.00 e −0.06×1/12 ] × e 0.06×2/12 = 23.741.<br />
At this forward price, the value of the original contract is<br />
(23.741 − 24.368) × e −0.06×2/12 = −0.6208<br />
or a loss of $0.6208 per share, for a total loss of $6,208 on the position.<br />
14. Suppose you are given the following information:<br />
• The current price of copper is $83.55 per 100 lbs.<br />
• The term-structure of interest rates is flat at 5%, i.e., that the risk-free interest<br />
rate for borrowing/investment is 5% for all maturities in continuously-compounded<br />
and annualized terms.<br />
• You can take long and short positions in copper costlessly.<br />
• There are no costs of storing or holding copper.<br />
Consider a forward contract in which the short position has to make two deliveries:<br />
10,000 lbs of copper in one month, and 10,000 lbs in two months. The common delivery<br />
price in the contract for both deliveries is P , that is, the short position receives P upon<br />
making the one-month delivery and P upon making the two-month delivery. What is<br />
the arbitrage-free value of P ?<br />
Answer: Let Q denote the quantity delivered each month (i.e., Q = 10, 000 lbs). To<br />
replicate this contract, we need to buy 2Q units of copper today and store it. After one<br />
month, we deliver the first Q units, and after one more month, the second Q units. The<br />
cost of this replication strategy is the current spot price of 2Q units, which is<br />
2 × 100 × 83.55
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This must equal the present value of the cash outflows on the forward strategy, which is<br />
P e −0.05×1/12 + P e 0.05×2/12 = P × (e −0.05×1/12 + e −0.05×2/12 )<br />
Equating these, we can solve for P :<br />
P =<br />
2 × 8, 355<br />
exp(−0.05 × 1/12) + exp(−0.05 × 2/12)<br />
This is the arbitrage free value of P .<br />
= 8, 407.40<br />
15. This question generalizes the previous one from two deliveries to many. Consider a<br />
contract that requires the short position to make deliveries of one unit of an underlying<br />
at time points t1, t2, . . . , tN. The common delivery price for all deliveries is F . Assume<br />
the interest rates for these horizons are, respectively, r1, r2, . . . , rN in continuouslycompounded<br />
annualized terms. What is the arbitrage-free value of F given a spot price<br />
of S?<br />
Answer: The answer follows the same logic as we had in the previous question, i.e.,<br />
F =<br />
N × S<br />
N exp(−rti i=1 ti)<br />
Such a contract (one which calls for multiple deliveries at a fixed price F ) is a “commodity<br />
swap.” Commodity swaps are usually settled in cash, rather than by physical delivery<br />
as we’ve assumed here, though this does not change the arguments. Commodity swaps<br />
are discussed in <strong>Chapter</strong> 25.<br />
16. In the absence of interest-rate uncertainty and delivery options, futures and forward<br />
prices must be the same. Does this mean the two contracts have identical cash-flow<br />
implications? (Hint: Suppose you expected a steady increase in prices. Would you prefer<br />
a futures contract with its daily mark-to-market or a forward with its single mark-tomarket<br />
at maturity of the contract? What if you expected a steady decrease in prices?)<br />
Answer: Evidently not. For example, if prices ares steadily trending upward, a futures<br />
contract with its daily mark-to-market will result in earlier cash inflows to the long and<br />
cash outflows for the short. So if you were a long investor, you would prefer the futures<br />
to the forward (vice versa if you were a short investor).
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17. Consider a forward contract on a non-dividend-paying stock. If the term-structure of<br />
interest rates is flat (that is, interest rates for all maturities are the same), then the<br />
arbitrage-free forward price is obviously increasing in the maturity of the forward contract<br />
(i.e., a longer-dated forward contract will have a higher forward price than a shorterdated<br />
one). Is this statement true even if the term-structure is not flat?<br />
Answer: Consider two dates t1 and t2 where t1 < t2. Assume that the spot rates for<br />
these two dates are respectively, r1 and r2. Given a spot price of S, the two corresponding<br />
forward prices are F1 = Se r1t1 and F2 = Se r2t2 . When the term structure is flat, r1 = r2,<br />
and hence, it is easy to see that F1 < F2.<br />
For general curves of spot rates, i.e., when the term structure is not flat, suppose we<br />
want that F2 < F1. Then it must be that<br />
Se r2t2 < Se r1t1<br />
r2t2 < r1t1<br />
r2 < r1(t1/t2)<br />
Is this feasible? Mathematically, yes, we may find parameter values for which this<br />
condition holds, implying that when term structures are not flat, we may have longer<br />
term forward prices lower than shorter term ones. For example, r2 = 0.02, r1 = 0.06,<br />
t1 = 0.25 and t2 = 0.50, satisfies the condition.<br />
But economically, does this make sense? The answer is no. The condition that r1t1 ><br />
r2t2 means that an investor can make more investing for t1 years than for t2 years.<br />
This gives rise to an arbitrage opportunity where you borrow long term for t2 and invest<br />
short-term for t1.<br />
18. The spot price of copper is $1.47 per lb, and the forward price for delivery in three<br />
months is $1.51 per lb. Suppose you can borrow and lend for three months at an<br />
interest rate of 6% (in annualized and continuously-compounded terms).<br />
(a) First, suppose there are no holding costs (i.e., no storage costs, no holding benefits).<br />
Is there an arbitrage opportunity for you given these prices? If so, provide details<br />
of the cash flows. If not, explain why not.<br />
(b) Suppose now that the cost of storing copper for three months is $0.03 per lb,<br />
payable in advance. How would your answer to Part (a) change? (Note that<br />
storage costs are asymmetric: you have to pay storage costs if you are long copper,<br />
but you do not receive the storage costs if you short copper.)<br />
Answer:
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(a) When there are no holding costs, the forward price is<br />
1.47e 0.06×3/12 = 1.4922<br />
which implies that the quote of 1.51 per lb is an overstatement of the price.<br />
To take advantage of the opportunity, we go short the forward at 1.51, buy copper<br />
spot at 1.47, and borrow 1.47 to finance the copper spot purchase. This leaves a<br />
zero net cash flow at inception and has a cash inflow of 1.51 − 1.47e 0.06×3/12 =<br />
0.0178 at maturity.<br />
(b) When there are storage costs and these are asymmetric, the problem is trickier. If<br />
we treat the storage costs as a cost of carry, we arrive at the forward price<br />
(1.47 + 0.03)e 0.06×3/12 = 1.5227<br />
This makes it appear that the the given price of 1.51 is too low, i.e., that the forward<br />
is now underpriced, but this is illusory. If we try implementing the arbitrage strategy<br />
(long forward, short spot, invest), then we will have a cash outflow at maturity<br />
because we do not receive the storage costs when we are short copper.<br />
On the other hand, we also cannot create an arbitrage by the opposite strategy<br />
(short forward, long spot, borrow): this too leads to a cash outflow at maturity, in<br />
this case because we now have to pay storage costs.<br />
Thus, the asymmetric nature of storage costs wipes out any perceived arbitrage<br />
opportunity. Put differently, it is as if there are two “correct” theoretical arbitragefree<br />
forward prices: the price is 1.4922 if you plan to be long copper (and short<br />
spot) and 1.5227 if you plan to be short copper (and long spot).<br />
19. The SPX index is currently trading at a value of $1,265, and the FESX index (the<br />
Dow Jones EuroSTOXX Index of 50 stocks, referred to from here on as “STOXX”) is<br />
trading at e3,671. The dollar interest rate is 3% , and the euro interest rate is 5%.<br />
The exchange rate is $1.28/euro. The six-month futures on the STOXX is quoted at<br />
e3,782. All interest rates are continuously compounded. There are no borrowing costs<br />
for securities.<br />
(a) Compute the correct six-month forward futures prices of the SPX, STOXX, and<br />
the currency exchange rate between the dollar and the euro.<br />
(b) Is the futures on the STOXX correctly priced? If not, show how to undertake an<br />
arbitrage strategy assuming you are not allowed to undertake borrowing or lending<br />
transactions in either currency.<br />
Answer:
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(a) The required forward rates are:<br />
SPX forward price: 1265e 0.03×1/2 = 1284.1<br />
STOXX forward price: 3671e 0.05×1/2 = 3763.9<br />
Currency ($/e) forward: 1.28e (0.03−0.05)×1/2 = 1.2673<br />
(b) The STOXX forward contract is quoted at e3782, whereas its correct quote as<br />
computed above should be e3763.9. To exploit this error we undertake the following<br />
arbitrage strategy (sequence of trades), without using borrowing or lending<br />
in either currency:<br />
At time t = 0:<br />
i. Sell the STOXX forward at e3,782.<br />
ii. Buy the component stocks of the STOXX spot at e3,671.<br />
iii. Short the components of the SPX spot for $1,265. Do this for 3.7145 contracts<br />
(we will see why soon).<br />
iv. Convert the $1,265 (for 3.7145 contracts) into euros at the spot exchange rate<br />
of $1.28/euro. This results in 1, 265 × 3.7145/1.28 =e3, 671 which is exactly<br />
what is needed to buy the components of the STOXX above.<br />
v. Buy SPX forward at $1284.1.<br />
vi. Book a currency forward to sell 3,782 euros forward at an exchange rate of<br />
$1.2673/euro (the fair forward currency rate computed above).<br />
Notice the the total cash-flow in both currencies as a result of these six transactions<br />
is zero. We now move forward to maturity at the end of six months and examine<br />
the net cash-flow that is generated. We denote the spot value of the U.S. stock<br />
index as SPX and that of the euro stock index as STOXX. Below we describe the<br />
cash flows from each of the six components of the trading strategy we presented<br />
above:<br />
i. Close out the STOXX forward contract by delivery of the spot position: the<br />
cash-flow is 3,782 euros.<br />
ii. Buy back the components of the SPX index using the forward: cash-flow is<br />
−$3.7145 × 1, 284.10 and close out the short SPX position.<br />
iii. Sell the 3,782 euros forward, cash-flow is $3, 782 × 1.2673 = 4, 792.9.<br />
Thus, after all transactions netted off, we are left with a guaranteed gain of $23.1,<br />
representing the arbitrage profits.<br />
20. The current level of a stock index is 450. The dividend yield on the index is 4% (in<br />
continuously compounded terms), and the risk-free rate of interest is 8% for six-month
<strong>Chapter</strong> 4
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<strong>Chapter</strong> 4. Pricing Forwards & Futures II<br />
1. What is meant by the term “convenience yield”? How does it affect futures prices?<br />
Answer: Commodities are used in production and gets consumed in the process. Inventories<br />
of commodities are held by producers because this provides them with the<br />
flexibility to alter production schedules or as insurance against a stock-out that could<br />
cause business disruptions. The value of these options to consume the commodity out<br />
of storage ias needed is referred to as the commodity convenience yield. As a holding<br />
benefit, the convenience yield reduces the price of forwards and futures on the underlying.<br />
2. True or false: An arbitrage-free forward market can be in backwardation only if the<br />
benefits of carrying spot (dividends, convenience yields, etc.) exceed the costs (storage,<br />
insurance, etc.).<br />
Answer: True. Backwardation occurs when the present value of the benefits of carrying<br />
the physical commodity (including the convenience yield) outweigh the carrying costs.<br />
3. Suppose an active lease market exists for a commodity with a lease rate ℓ expressed<br />
in annualized continuously-compounded terms. Short-sellers can borrow the asset at<br />
this rate and investors who are long the asset can lend it out at this rate. Assume the<br />
commodity has no other cost of carry. Modify the arguments in the appendix to the<br />
chapter to show that the theoretical futures price is F = e (r−ℓ)T S.<br />
Answer: Suppose the forward price tat prevails (denoted, say, F ′ ) is not equal to F .<br />
Assume first that F ′ < F . Consider the following strategy:<br />
• Take a long forward position at F ′ . This involves no current cash-flow.<br />
• Borrow e −ℓT units of gold for T years; sell the borrowed gold at the spot price of<br />
S. Cash inflow today: e −ℓT S. Note that given the lease rate ℓ, the amount of gold<br />
due at maturity T is one unit.<br />
• Invest the cash of e ℓT S for maturity at T at the interest rate r.<br />
The net cash-flow at inception is zero. At date T , pay F ′ on the forward, receive one<br />
unit of gold, and use this to close the gold lease. Receive e rT × e −ℓT S = e (r−ℓ)T S from<br />
the investment. Net cash flow at T :<br />
e (r−ℓ)T S − F ′ .<br />
By hypothesis, this amount is positive, so we have an arbitrage profit. Similarly, if<br />
F ′ > e (r−ℓ)T S, reversing the above strategy would result in an arbitrage profit.
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Re-arranging we have that<br />
r = ℓ + (1/t) ln(F/S)<br />
So we can see that the new repo rates will be the old repo rates plus 0.005, which gives<br />
the two-month and five-month rates as:<br />
r2 = 0.034402, r5 = 0.036789.<br />
10. Copper is currently trading at $1.28/lb. Suppose three-month interest rates are 4% and<br />
the convenience yield on copper is c = 3%.<br />
(a) What is the range of arbitrage-free forward prices possible using<br />
S0e (r−c)T ≤ F ≤ S0e rT ? (1)<br />
(b) What is the lowest value of c that will create the possibility of the market being in<br />
backwardation?<br />
Answer:<br />
(a) Plug values into the equation above, i.e.,<br />
or<br />
1.28e (0.04−0.03)×3/12 ≤ F ≤ 1.28e 0.04×3/12<br />
1.2832 ≤ F ≤ 1.2929<br />
(b) The lowest value of c to create backwardation is r.<br />
11. You are given the following information on forward prices (gold and silver prices are per<br />
oz, copper prices are per lb):<br />
Commodity Spot One-month Two-month Three-month Six-month<br />
Gold 436.4 437.3 438.8 440.0 444.5<br />
Silver 7.096 7.125 7.077 7.160 7.220<br />
Copper 1.610 1.600 1.587 1.565 1.492
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iv. Lease the gold out for 3 months at 1% lease rate. Amount received at end of<br />
the lease: 1 oz.<br />
v. Borrow 359.10 for 3 months at 4%. Amount owed at maturity: 362.71<br />
At maturity, deliver the 1 oz. of gold received from the lessee to the forward contract<br />
and receive F = 366. Repay 362.71 on the borrowing. Net cash flow: +3.29.<br />
14. A three-month forward contract on a non-dividend-paying asset is trading at 90, while<br />
the spot price is 84.<br />
(a) Calculate the implied repo rate.<br />
(b) Suppose it is possible for you to borrow at 8% for three months. Does this give<br />
rise to any arbitrage opportunities? Why or why not?<br />
Answer: The implied repo rate is<br />
r = 1<br />
1<br />
[ln F − ln S] = [ln 90 − ln 84] = 0.27957,<br />
T 0.25<br />
or 27.98%. Since we can borrow at 8% for three months, there is a clear arbitrage<br />
opportunity:<br />
• Sell the forward at 90.<br />
• Borrow 84 at 8%.<br />
• Buy spot at 84.<br />
The net cash-flow at inception is zero. The net cash-flow at maturity is<br />
(90 − ST ) − 84 exp(0.08 × 3/12) + ST = 4.3031<br />
which is the difference between the repo rate and market borrowing rate on a base price<br />
of 84. To see this, note that<br />
84[exp(r × 3/12) − exp(0.08 × 3/12)] = 4.3031.
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• Reinvest all dividends into buying more of the index. Amount of index held in<br />
3 months: 1 unit.<br />
• Borrow 580.613 for three months at 6%.<br />
• Take a short position in the forward contract at F = 600.<br />
At maturity, deliver the unit of the index on the forward contact, and receive 600.<br />
Repay the borrowing: the cash outflow is 580.613 × e 0.06×1/4 = 589.39. The net<br />
cash flow of +10.61 represents arbitrage profits.<br />
17. A three month-forward contract on an index is trading at 756 while the index itself is at<br />
750. The three-month interest rate is 6%.<br />
(a) What is the implied dividend yield on the index?<br />
(b) You estimate the dividend yield to be 1% over the next three months. Is there an<br />
arbitrage opportunity from your perspective?<br />
Answer: The forward to spot relationship for this contract is as follows:<br />
750 exp[(0.06 − d)(0.25)] = 756<br />
Deriving the dividend yield from this results in d = 0.028127.<br />
Now if the dividend is only expected to be 1%, then the current forward price is too<br />
low. The arbitrage strategy is to buy forward, and sell spot. The details are left as an<br />
exercise. Be careful to account for the dividends in the strategy that you create for the<br />
risk-less arbitrage.<br />
18. The spot US dollar-euro exchange rate is $1.10/euro. The one-year forward exchange<br />
rate is $1.0782/euro. If the one-year dollar interest rate is 3%, then what must be the<br />
one-year rate on the euro?<br />
Answer: We exploit the following relationship:<br />
F = S exp[rUSD − rEuro]<br />
noting that time is one year. The equation to be solved is:<br />
1.0782 = 1.1000 exp[0.03 − rEuro]<br />
which means that rEuro = 0.05.
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19. You are given information that the spot price of an asset is trading at a bid-ask quote<br />
of 80 − 80.5, and the six-month interest rate is 6%. What is the bid-ask quote for the<br />
six-month forward on the asset if there are no dividends?<br />
Answer: There are two possible forward-spot arbitrage strategies: one where we buy<br />
forward (at F a ) and sell spot (at S b ), and the other where we sell forward (at F b ) and<br />
buy spot (at S a ), where the superscripts a and b refer to “ask” and “bid,” respectively.<br />
For the first strategy to not admit arbitrage profits, we must have<br />
F a ≥ e 0.06×1/2 × S b = 82.436<br />
For the second strategy not to be an arbitrage, we must have<br />
F b ≤ e 0.06×1/2 × S a = 82.952<br />
Any pair (F a , F b ) consistent with these inequalities (and, of course, F a ≥ F b ) can be<br />
an equilibrium bid-ask pair of forward prices.<br />
20. Redo the previous question if the interest rate for borrowing and lending are not equal,<br />
i.e., there is a bid-ask spread for the interest rate, which is 6.00–6.25%.<br />
Answer: Here, the first arbitrage strategy involves investing the short sale proceeds of<br />
S b at the lending rate of 6.00%, while the second strategy involves borrowing the spot<br />
purchase price of S a at the borrowing rate of 6.25%. These interest rates should be<br />
used in computing the inequalities. Carrying out the computations is left as an exercise.<br />
21. In the previous question, what is the maximum bid-ask spread in the interest rate market<br />
that is permissible to give acceptable forward prices?<br />
Answer: There is none.<br />
22. Stock ABC is trading spot at a price of 40. The one-year forward quote for the stock is<br />
also 40. If the one-year interest rate is 4% and the borrowing cost for the stock is 2%,<br />
show how to construct a risk-less arbitrage in this stock.<br />
Answer: First, we note that the correct forward price should be<br />
F (correct) = 40e 0.04−0.02 = 40.808<br />
Since the actual forward price is less than this, it is cheap. Hence we should buy it, and<br />
short the stock. To short the stock we will need to borrow it at a cost of 2%, but we
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will invest the proceeds (40) at 4%. Therein lies the source of the arbitrage profits. All<br />
these transactions at inception are net zero in cash-flow.<br />
We now examine the cash-flow at the end of the year. The forward position results in<br />
a cash-flow of −40 on purchase. The short spot position is closed out by delivering the<br />
stock to the lender. The net flow from the cost of borrowing the stock and the gains<br />
from lending the sales proceeds of the stock is 2% of $40. Hence, we gain a net amount<br />
of 0.808 (= −40 + 40e 0.04−0.02 ).<br />
23. You are given two stocks, A and B. Stock A has a beta of 1.5, and stock B has a beta of<br />
−0.25. The one-year risk-free rate is 2%. Both stocks currently trade at $10. Assume<br />
a CAPM model where the expected return on the stock market portfolio is 10%. Stock<br />
A has an annual dividend yield of 1% and stock B does not pay a dividend.<br />
(a) What is the expected return on both stocks?<br />
(b) What is the one-year forward price for the two stocks?<br />
(c) Is there an arbitrage? Explain.<br />
Answer: (a) We may use the CAPM to determine the expected return on both stocks,<br />
which are as follows.<br />
Stock A:<br />
Stock B:<br />
E(rA) = 0.02 + 1.5[0.10 − 0.02] = 0.14<br />
E(rB) = 0.02 − 0.25[0.10 − 0.02] = 0.0<br />
(b) The forward price for stock A is<br />
FA = 10e 0.02−0.01 = 10.101<br />
The forward price for stock B is<br />
FB = 10e 0.02−0.0 = 10.202<br />
(c) There is no arbitrage even though stock B has a forward price greater than that<br />
of stock A even though its expected return and dividend is zero. The forward price is<br />
based on a mathematical relationship between spot prices and interest rates, and does<br />
not have any relation to the expected growth rate of the stock.
<strong>Chapter</strong> 5
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5. In the presence of basis risk, is a one-for-one hedge, i.e., a hedge ratio of 1, always better<br />
than not hedging at all?<br />
Answer: Basis risk arises from the fact that the hedge and the underlying spot values are<br />
not perfectly correlated. Depending on this correlation, it may be better to not hedge<br />
than to hedge one-for-one. For example, this is certainly the case if price changes in the<br />
underlying spot asset and the hedge have no correlation with each other at all: in such<br />
a situation, hedging only results in additional uncertainty. For a more precise statement<br />
on when not hedging may be superior to hedging one-for-one, see Section 5.5.<br />
6. If the correlation between spot and futures price changes is ρ = 0.8, what fraction of<br />
cash-flow uncertainty is removed by minimum-variance hedging?<br />
Answer: As shown in Section 5.5, the fraction of unhedged cash flow variance removed<br />
by the minimum-variance hedge is ρ 2 , which in this case is 0.64 or 64%.<br />
7. The correlation between changes in the price of the underlying and a futures contract is<br />
+80%. The same underlying is correlated with another futures contract with a (negative)<br />
correlation of −85%. Which of the two contracts would you prefer for the minimumvariance<br />
hedge?<br />
Answer:<br />
The second one. As shown in Section 5.5, the fraction of unhedged cash flow variance<br />
removed by the minimum-variance hedge is ρ 2 , where ρ is the correlation of the spot<br />
price changes and price changes in the futures contract used for hedging. Since ρ 2<br />
increases as |ρ| increases, we should use a futures contract with the highest value of |ρ|.<br />
The only impact of the negative correlation is that the sign of the futures position gets<br />
reversed, i.e., we hedge a long spot exposure (a commitment to buy spot at maturity<br />
T ) with a short futures position and a short spot exposure (a commitment to sell spot<br />
at date T ) with a short futures position.<br />
8. Given the following information on the statistical properties of the spot and futures,<br />
compute the minimum-variance hedge ratio: σS = 0.2, σF = 0.25, ρ = 0.96.<br />
Answer: The minimum-variance hedge ratio is<br />
h ∗ = ρ σ(∆S)<br />
σ(∆F )<br />
= 0.96 × 0.2<br />
0.25<br />
= 0.768.<br />
This means to hedge a long spot exposure of size Q (i.e., to hedge a commitment to<br />
buy Q units spot at date T ), we use long futures contracts of size 0.768 Q units.
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9. Assume that the spot position comprises 1,000,000 units in the stock index. If the hedge<br />
ratio is 1.09, how many units of the futures contract are required to hedge this position?<br />
Answer: Note that the optimal hedge ratio is h ∗ = H/Q, where H is the number of<br />
units of the hedge, and Q is the number of units of the spot position. Hence, the<br />
required number of units in futures is<br />
H = h ∗ × Q = 1.09 × 1, 000, 000 = 1, 090, 000.<br />
In words, we enter into a futures contract that calls for the delivery of 1,090,000 units<br />
of the asset underlying the futures contract.<br />
10. You have a position in 200 shares of a technology stock with an annualized standard<br />
deviation of changes in the price of the stock being 30. Say that you want to hedge this<br />
position over a one-year horizon with a technology stock index. Suppose that the index<br />
value has an annual standard deviation of 20. The correlation between the two annual<br />
changes is 0.8. How many units of the index should you hold to have the best hedge?<br />
Answer: In the notation of the chapter, we are given that σ(∆S) = 30, σ(∆F ) = 20,<br />
and ρ = 0.8. So the minimum-variance hedge ratio is<br />
h ∗ = ρ σ(∆S)<br />
σ(∆F )<br />
= 0.8(30/20) = 1.20<br />
Hence, you need to short 1.2 × 200 = 240 units of the index to set up the hedge.<br />
11. You are a portfolio manager looking to hedge a portfolio daily over a 30-day horizon.<br />
Here are the values of the spot portfolio and a hedging futures for 30 days.
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Day Spot Futures<br />
0 80.000 81.000<br />
1 79.635 80.869<br />
2 77.880 79.092<br />
3 76.400 77.716<br />
4 75.567 77.074<br />
5 77.287 78.841<br />
6 77.599 79.315<br />
7 78.147 80.067<br />
8 77.041 79.216<br />
9 76.853 79.204<br />
10 77.034 79.638<br />
11 75.960 78.659<br />
12 75.599 78.549<br />
13 77.225 80.512<br />
14 77.119 80.405<br />
15 77.762 81.224<br />
16 77.082 80.654<br />
17 76.497 80.233<br />
18 75.691 79.605<br />
19 75.264 79.278<br />
20 76.504 80.767<br />
21 76.835 81.280<br />
22 78.031 82.580<br />
23 79.185 84.030<br />
24 77.524 82.337<br />
25 76.982 82.045<br />
26 76.216 81.252<br />
27 76.764 81.882<br />
28 79.293 84.623<br />
29 78.861 84.205<br />
30 76.192 81.429<br />
Carry out the following analyses using Excel:<br />
(a) Compute σ(∆S), σ(∆F ), and ρ.<br />
(b) Using the results from (a), compute the hedge ratio you would use.<br />
(c) Using this hedge ratio, calculate the daily change in value of the hedged portfolio.<br />
(d) What is the standard deviation of changes in value of the hedged portfolio? How<br />
does this compare to the standard deviation of changes in the unhedged spot<br />
position?
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Answer: The results are presented in the following tables. The first step is to compute<br />
the covariance matrix of the changes in spot and futures, as follows. Using Excel, we<br />
obtain:<br />
Covariance Matrix<br />
∆S ∆F<br />
∆S 1.276<br />
∆F 1.308 1.415<br />
Using these numbers to compute the correlation ρ and the hedge ratio h ∗ , we obtain:<br />
ρ = 0.9732<br />
h ∗ = 0.9732 × 1.276<br />
1.308<br />
= 0.9244<br />
Using a hedge ratio of h ∗ , we can calculate the daily changes (“P&L”) in the value of<br />
the hedged portfolio. For example, on day 1, this P&L is<br />
(79.635 − 80) − [0.9244 × (80.869 − 81)] = −0.243<br />
The following table summarizes these numbers:
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Day Spot Futures ∆S ∆F P&L<br />
0 80.000 81.000<br />
1 79.635 80.869 -0.365 -0.131 -0.243<br />
2 77.880 79.092 -1.756 -1.777 -0.113<br />
3 76.400 77.716 -1.479 -1.376 -0.207<br />
4 75.567 77.074 -0.834 -0.642 -0.241<br />
5 77.287 78.841 1.721 1.768 0.087<br />
6 77.599 79.315 0.312 0.474 -0.126<br />
7 78.147 80.067 0.547 0.752 -0.148<br />
8 77.041 79.216 -1.106 -0.851 -0.319<br />
9 76.853 79.204 -0.188 -0.012 -0.176<br />
10 77.034 79.638 0.180 0.434 -0.221<br />
11 75.960 78.659 -1.074 -0.979 -0.169<br />
12 75.599 78.549 -0.361 -0.111 -0.258<br />
13 77.225 80.512 1.626 1.964 -0.189<br />
14 77.119 80.405 -0.106 -0.107 -0.007<br />
15 77.762 81.224 0.643 0.820 -0.114<br />
16 77.082 80.654 -0.681 -0.571 -0.153<br />
17 76.497 80.233 -0.585 -0.420 -0.196<br />
18 75.691 79.605 -0.805 -0.629 -0.224<br />
19 75.264 79.278 -0.427 -0.327 -0.125<br />
20 76.504 80.767 1.240 1.488 -0.136<br />
21 76.835 81.280 0.330 0.513 -0.144<br />
22 78.031 82.580 1.196 1.300 -0.005<br />
23 79.185 84.030 1.153 1.450 -0.187<br />
24 77.524 82.337 -1.661 -1.693 -0.096<br />
25 76.982 82.045 -0.541 -0.292 -0.271<br />
26 76.216 81.252 -0.766 -0.793 -0.033<br />
27 76.764 81.882 0.548 0.629 -0.034<br />
28 79.293 84.623 2.529 2.742 -0.006<br />
29 78.861 84.205 -0.432 -0.419 -0.045<br />
30 76.192 81.429 -2.669 -2.776 -0.103<br />
The P&L has a variance of 0.009 which is less than 1% of the variance of the unhedged<br />
position of 1.276.<br />
12. Use the same data as presented above to compute the hedge ratio using regression<br />
analysis, again using Excel. Explain why the values are different from what you obtained<br />
above.<br />
Answer: The regression in Excel of daily changes δS on δF produces the following results.<br />
δS = −0.14 + 0.947 δF
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Finally, the current USD/SEK spot rate is 0.104, the current three-month USD/EUR<br />
forward rate is 1.071, and the current three-month USD/CHF forward rate is 0.602.<br />
(a) Which currency should the company use for hedging purposes?<br />
(b) What is the minimum-variance hedge position? Indicate if this is to be a long or<br />
short position.<br />
Answer:<br />
(a) Since the correlation of changes in the spot USD/SEK is higher with changes in<br />
forward USD/EUR than with changes in forward USD/CHF, the hedge will be<br />
better if the USD/EUR forward is used for hedging.<br />
(b) The optimal hedge ratio is<br />
h ∗ = ρ × σS<br />
σF<br />
= 0.90 × 0.007<br />
0.018<br />
= 0.35<br />
Since the correlation of USD/SEK and USD/EUR is positive, appreciation in the<br />
SEK should mostly be offset by appreciation in the EUR. Hence, the hedge position<br />
should be a long USD/EUR forward contract calling for the delivery of EUR (0.35×<br />
100) million = EUR 35 million.<br />
14. You use silver wire in manufacturing. You are looking to buy 100,000 oz of silver in<br />
three months’ time and need to hedge silver price changes over these three months. One<br />
COMEX silver futures contract is for 5,000 oz. You run a regression of daily silver spot<br />
price changes on silver futures price changes and find that<br />
δs = 0.03 + 0.89δF + ɛ<br />
What should be the size (number of contracts) of your optimal futures position. Should<br />
this be long or short?<br />
Answer: From the regression, the optimal hedge ratio is 0.89, so the size of the required<br />
futures position is 89,000 oz or 89, 000/5, 000 = 17.8 contracts. This should be a long<br />
position.<br />
15. Suppose you have the following information: ρ = 0.95, σS = 24, σF = 26, K = 90,<br />
R = 1.00018. What is the minimum-variance tailed hedge?
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(a) Ambiguous. The question does not indicate if profits on the hedge position are<br />
more likely if the term-structure is upward sloping.<br />
(b) Again, ambiguous. The forward price is convex in the interest rate, but the spot<br />
price too may be correlated with interest rates, so without more information on<br />
the nature of the underlying, it is not clear how the forward/futures price behaves<br />
when interest rates become more volatile.<br />
(c) Ambiguous. The performance of the hedge depends only on the correlation between<br />
price changes in the instrument used for hedging and price changes in the exposure<br />
being hedged, and not on the variances of price changes of the instruments used<br />
for hedging.<br />
However, if we add some more conditions, we can provide a qualified answer. For<br />
example, the size of the optimal hedge depends on the standard deviation of price<br />
changes of the contract used for hedging, and decreases as this standard deviation<br />
increases. So if the correlations are the same for both hedging instruments, the<br />
contract with greater volatility may be preferable since fewer positions are needed<br />
in an optimal hedge and this may reduce transactions costs.<br />
(d) Here, the answer is unambiguous. You want the correlation of spot to futures to<br />
be higher,as the hedged position will have a lower variance.<br />
23. You are trying to hedge the sale of a forward contract on a security A. Suggest a framework<br />
you might use for making a choice between the following two hedging schemes:<br />
(a) Buy a futures contract B that is highly correlated with security A but trades very<br />
infrequently. Hence, the hedge may not be immediately available.<br />
(b) Buy a futures contract C that is poorly correlated with A but trades more frequently.<br />
Answer: The question requires you to use your imagination to develop a model to<br />
trade off liquidity risk against basis risk. Here is one possible way to approach the<br />
problem (obviously not the only one). Suppose we denote the probability of being able<br />
to implement the hedge B by p. The probability of A remaining unhedged is then 1 − p.<br />
The variance of the unhedged position is σ 2 (∆A). The variance of the hedged position<br />
is σ 2 (∆A)(1 − ρ 2 AB ), where ρAB is the correlation between changes in A and B. Hence,<br />
the expected variance of the hedged position when B is used is<br />
p σ 2 (∆A)(1 − ρ 2 AB) + (1 − p)σ 2 (∆A).<br />
If C is used to hedge, the variance of the hedged position is<br />
σ 2 (∆A)(1 − ρ 2 AC)
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If we care about only the expected variance of the hedged portfolio, then, depending<br />
on which of these values is computed to be lower, the hedge instrument may be chosen<br />
accordingly. If the latter is lower, choose C, and if the former is lower, choose B. In<br />
particular, the first alternative is preferred when<br />
p σ 2 (∆A)(1 − ρ 2 AB) + (1 − p)σ 2 (∆A)σ 2 (∆A)(1 − ρ 2 AC)<br />
24. Download data from the Web as instructed below and answer the questions below:<br />
(a) Extract one year’s data on the S&P 500 index from finance.yahoo.com. Also<br />
download corresponding period data for the S&P 100 index.<br />
(b) Download, for the same period, data on the three-month Treasury Bill rate (constant<br />
maturity) from the Federal Reserve’s Web page on historical data:<br />
www.federalreserve.gov/releases/h15/data.htm.<br />
(c) Create a data series of three-month forwards on the S&P 500 index using the index<br />
data and the interest rates you have already extracted. Call this synthetic forward<br />
data series F .<br />
(d) How would you use this synthetic forwards data to determine the tracking error of<br />
a hedge of three-month maturity positions in the S&P 100 index? You need to<br />
think (a) about how to set up the time lags of the data and (b) how to represent<br />
tracking error.<br />
Answer: This exercise is left to the reader. For the definition of “tracking error,” see<br />
the answer to the next question.<br />
25. Explain the relationship between regression R 2 and tracking error of a hedge. Use<br />
the data collected in the previous question to obtain a best tracking error hedge using<br />
regression.<br />
Answer: To answer this question, one must first define “tracking error” (the question<br />
deliberately leaves this undefined). The intuitive definition of tracking error is also the<br />
most commonly used one: tracking error is the standard deviation (or the variance) of<br />
the difference between a target performance and the actual performance.<br />
Suppose we run a regression to determine the hedge ratio to be implemented:<br />
δS = a + bδF + ɛ
<strong>Chapter</strong> 6
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<strong>Chapter</strong> 6. Interest Rate Forwards & Futures<br />
1. Explain the difference between the following terms:<br />
(a) Payoff to an FRA.<br />
(b) Price of an FRA.<br />
(c) Value of an FRA.<br />
Answer: FRA terminology:<br />
(a) The payoff from an FRA is the dollar amount received at maturity of the FRA. For<br />
example, if we are long an FRA at a strike interest rate of 10% and the rate at<br />
maturity of the FRA is 11%, our payoff will be based on the interest difference of<br />
1% applied to the notional principal of the contract for the borrowing period.<br />
(b) The “price” of an FRA refers to the fixed rate locked in using the FRA. At inception<br />
of the FRA, this fixed rate is chosen so that the FRA has zero value to both parties.<br />
“Pricing” an FRA refers to the identification of this fixed rate.<br />
(c) Value is the net payment that would have to be made if the FRA were to be closed<br />
out today. At inception, the FRA has (by construction) zero value to both parties.<br />
But as time progresses, the fixed rate in the FRA will generally differ from the price<br />
of a new FRA (i.e., from the fixed rate that makes an FRA with the same maturity<br />
as the original one have zero value to both parties), so the FRA can have positive<br />
or negative value.<br />
2. What characteristic of the eurodollar futures contract enabled it to overcome the settlement<br />
obstacles with its predecessors?<br />
Answer: Cash settlement. Earlier attempts at developing an interest-rate futures contract<br />
based on commercial borrowing rates had floundered because they required physical<br />
settlement at maturity, but the deliverable instruments in these contracts lacked homogeneity<br />
because of perceived differences in the credit risk of the issuing entities. The<br />
eurodollar futures contract solved this by using cash settlement, an idea that was rapidly<br />
adopted in other contracts which had difficulties with physical settlement (e.g., stock<br />
index futures).<br />
3. How are eurodollar futures quoted?<br />
Answer: Eurodollar futures contracts are instruments that enable traders to lock in<br />
a Libor rate for a three-month period beginning on the expiry date of the contract.<br />
However, eurodollar futures are quoted not as rates but as prices. The price quoted is
Sundaram & Das: Derivatives - Problems and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70<br />
100 minus the three-month Libor rate, with the rate being expressed as a percentage<br />
(not in decimal form). So if the Libor rate is 3.18%, the futures price quoted is 96.82.<br />
4. It is currently May. What is the relation between the observed eurodollar futures price<br />
of 96.32 for the November maturity and the rate of interest that is locked-in using the<br />
contract? Over what period does this rate apply?<br />
Answer: The relation between the futures price and rate of interest that gets locked in<br />
via the contract is<br />
100 − 96.32 = 3.68%<br />
The interest rate applies to a 90-day borrowing or investment beginning at maturity of<br />
the futures contract, i.e., beginning in November.<br />
5. What is the price tick in the eurodollar futures contract? To what price move does this<br />
correspond?<br />
Answer: The price tick in the eurodollar futures contract is 0.01 (which corresponds<br />
to a move in the implied interest rate of 1 basis point). The price tick has a dollar<br />
value of $25. The minimum price move on the expiring eurodollar futures contract (the<br />
one currently nearest to maturity) is 1/4 tick or a dollar value of $6.25. On all other<br />
eurodollar futures contracts, it is 1/2 tick (or $12.50).<br />
6. What are the gains or losses to a short position in a eurodollar futures contract from a<br />
0.01 increase in the futures price?<br />
Answer: There will be a loss of $25.<br />
7. You enter into a long eurodollar futures contract at a price of 94.59 and exit the contract<br />
a week later at a price of 94.23. What is your dollar gain or loss on this position?<br />
Answer: A increase of 0.01 in the price corresponds to a margin account change of<br />
$25 (gain for the long, loss for the short). In this case, the price falls by 0.36, which<br />
corresponds to a loss of (36 × $25) = $900 for the long position.<br />
8. What is the cheapest to deliver in a Treasury bond futures contract? Are there other<br />
delivery options in this contract?<br />
Answer: The standard bond in a Treasury contract is one with a coupon of 6% and at<br />
least 15 years to maturity or first call. The main delivery option in the Treasury bond
Sundaram & Das: Derivatives - Problems and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75<br />
However, this rate is negative, and is hence not reasonable for a nominal interest rate.<br />
It is likely that rates such as these occur when there is an imbalance between demand<br />
and supply for money, or the bid/ask spreads are too large to allow someone to arbitrage<br />
the negative forward rate because there are no lenders for the forward period.<br />
18. If you expect interest rates to rise over the next three months and then fall over the<br />
three months succeeding that, what positions in FRAs would be appropriate to take?<br />
Would your answer change depending on the current shape of the forward curve?<br />
Answer: If rates are going to rise over the next three months and you wish to speculate<br />
on this view, then you can lock-in a rate today using a FRA for borrowing in three<br />
months and in three months’ time, you can invest the borrowed amount at the higher<br />
interest rates that prevail then. Similarly, if, in three months’ time, rates are going to<br />
fall over the next three months, you can enter into a short FRA at that time to speculate<br />
on your views.<br />
19. A firm plans to borrow money over the next two half-year periods, and is able to obtain<br />
a fixed-rate loan at 6% per annum. It can also borrow money at the floating rate of<br />
Libor + 0.5%. Libor is currently at 4%. If the 6 × 12 FRA is at a rate of 6%, find the<br />
cheapest financing cost for the firm.<br />
Answer: For simplicity, we treat each six-month period as exactly half a year. If the<br />
fixed rate loan is taken, the cost of financing is 3% each half year. That is, the firm<br />
receives 100 today, pays 3 in six months and 103 in one year at maturity. It is easy to<br />
see that the internal rate of return of this sequence of cash-flows is exactly 6%.<br />
Suppose the firm elects to go for the second alternative, i.e., takes a floating-rate loan<br />
and simultaneously enters into a long 6×12 FRA. Then, the cost of financing is 4.5% for<br />
the first six months and 6.5% for the next six months . (Note that in the second period,<br />
the cost of financing to the firm is ℓ + 0.5% plus the payoff to the FRA is 6 − ℓ. Hence,<br />
net this is 6.5%). So the cash-flows in this second financing are: {−100, 2.25, 103.25}<br />
at times 0, 0.5 and 1 years. The internal rate of return of this sequence of cash-flows is<br />
5.48%.<br />
A comparison of the internal rates of return indicates that the second option is cheaper.<br />
20. You enter into an FRA of notional 6 million to borrow on the three-month underlying<br />
Libor rate six months from now and lock in the rate of 6%. At the end of six months,<br />
if the underlying three-month rate is 6.6% over an actual period of 91 days, what is<br />
your payoff given that the payment is made right away? Recall that the Actual/360<br />
convention applies.
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3<br />
2<br />
1<br />
-1<br />
-2<br />
-3<br />
FRA Settlement Value<br />
0<br />
0 5 10 15 20<br />
Libor (%<br />
The plot looks almost linear, but it is actually concave (shaped like an inverted bowl).<br />
Concavity is equivalent to having a negative second derivative and it is easily checked<br />
that is, in fact, the case:<br />
∂2s < 0.<br />
∂ℓ2 23. You anticipate a need to borrow USD 10 million in six-months’ time for a period of three<br />
months. You decide to hedge the risk of interest-rate changes using eurodollar futures<br />
contracts. Describe the hedging strategy you would follow. What if you decided to use<br />
an FRA instead?
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Answer: A simple way to hedge interest rate changes over the next six months is to<br />
enter into a long 6 × 9 FRA. This is a clean and exact hedge. However, we may also<br />
use eurodollar futures. To hedge borrowing costs, we need to make money on the hedge<br />
when interest rates rise (since we will be paying more on the borrowing in this event)<br />
and vice versa. When interest rates rise, eurodollar futures prices fall, so to make money<br />
when interest rates rise, we need to short eurodollar futures contracts. Finally, since one<br />
eurodollar futures contract is for a notional value of 1 million, we need to short 10 of<br />
these contracts.<br />
As noted in the text, however, eurodollar futures are settled in undiscounted form, so<br />
unlike using an FRA, the hedge obtained with eurodollar futures will not be perfect even<br />
in theory.<br />
24. In Question 23, suppose that the underlying three-month Libor rate after six months (as<br />
implied by the price of the eurodollar futures contract expiring in 6 months) is currently<br />
at 4%. Assume that the three-month period has 90 days in it. Using the same numbers<br />
from Question 23 and adjusting for tailing the hedge, how many futures contracts are<br />
needed? Assume fractional contracts are permitted.<br />
Answer: As noted, without tailing the hedge, we need a short position in 10 contracts.<br />
If the hedge is tailed using the 4% rate reflected in current eurodollar prices, then the<br />
number of contracts needed is<br />
10<br />
= 9.901.<br />
1 + 0.04(90/360)<br />
25. Using the same numbers as in the previous two questions, compute the payoff after six<br />
months (i.e., at maturity) under (a) an FRA and (b) a tailed eurodollar futures contract<br />
if the Libor rate at maturity is 5%, and the locked-in rate in both cases is 4%. Also<br />
compute the payoffs if the Libor rate ends up at 3%. Comment on the difference in<br />
payoffs of the FRA versus the eurodollar futures.<br />
Answer: First suppose that the Libor rate at maturity is 5%.<br />
(a) For the FRA, the payoff is:<br />
10, 000, 000 ×<br />
(0.05 − 0.04) × (90/360)<br />
1 + 0.05(90/360)<br />
(b) For the tailed eurodollar futures, the payoff is<br />
= 24, 752.50.<br />
9.901 × (0.05 − 0.04) × 10, 000 × 25 = 25, 024.75<br />
Now suppose the Libor rate at maturity is 3%.
Sundaram & Das: Derivatives - Problems and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79<br />
(a) For the FRA, the payoff is:<br />
10, 000, 000 ×<br />
(0.03 − 0.04) × (90/360)<br />
1 + 0.03(90/360)<br />
(b) For the eurodollar futures, the payoff is:<br />
= −24, 813.90<br />
9.901 × (0.03 − 0.04) × 10000 × 25 = −24, 752.50<br />
In either case, the eurodollar futures contract does better than the FRA: in the first<br />
case, it results in a larger cash inflow, and in the second case in a smaller cash outflow.<br />
This is the “convexity bias” discussed in the chapter.<br />
26. The “standard bond” in the Treasury bond futures contract has a coupon of 6%. If,<br />
instead, delivery is made of a 5% bond of maturity 18 years, what is the conversion<br />
factor for settlement of the contract? Assume that the last coupon on the bond was<br />
just paid.<br />
Answer: Assume a principal amount of $100. Then, the bond pays 2.50 every six months<br />
and also repays the principal after 18 years. The present value of these cash flows when<br />
discounted at the 6% standard rate is<br />
2.5<br />
1.03<br />
+ 2.5<br />
1.03<br />
2.5 102.5<br />
+ · · · + + = 89.946<br />
2 1.0335 1.0336 Hence, the conversion factor is 0.89946.<br />
27. Suppose we have a flat yield curve of 3%. What is the price of a Treasury bond of<br />
remaining maturity seven years that pays a coupon of 4%? (Coupons are paid semiannually.)<br />
What is the price of a six-month Treasury bond futures contract? Make any<br />
assumption you require concerning the maturity of the delivered bond to find this price.<br />
Answer: Assuming the last coupon was just paid, the price of a the remaining seven-year<br />
maturity bond with coupon of 4% is equal to the cash flows from the bond discounted<br />
at a flat 3% rate (i.e., at 1.5% every six months):<br />
2 2<br />
2 102<br />
+ + · · · + + = 106.27.<br />
1.015 1.0152 1.01513 1.01514 A Treasury bond futures contract requires the delivery of a Treasury bond with coupon<br />
of 6% and any maturity of at least 15 years. To price the six-month futures contract,<br />
we (i) treat it as a forward contract, and (ii) assume that the maturity of the delivered