Derivateives Solutions Ch1-6

Chapter 1

Sundaram & Das: Derivatives - Problems and Solutions.................................. 4

5. Define Define a forw forwaard contrac contract. t. Expla Explain in at what what time time are are cash cash flows flows genera generated ted for this contract. How is settlement determined? Answer: A forward contract is an agreement to buy or sell an asset at a future date

(denoted T (denoted T ), ), at a specified price called the delivery price (denoted F ). Denote the initial initial F ). Denote date (the inception date or the date of the agreement) by t = t = 0. At inception there are no cash flows on a forward contract. At maturity, if the then-prevailing spot price S T T of the underlying asset is greater than F , F , then the buyer (the “long position”) has gained S T F via the forward while the seller (the “short position”) has correspondingly lost T − F via Depending on contract contract specifications, specifications, the settlement settlement may either either be in cash S T F . Depending T − F . (the seller pays the buyer S T F ) or physical (the seller delivers the asset and receives T − F ) F ). F ). If S S T F , the buyer loses F − S T T < F , T and the seller gains this quantity.

6. Explain Explain who bears default default risk in a forwar forward d contract. contract. Default arises if, at maturity maturity, one of the parties parties fails to fulfill fulfill their obligations obligations Answer: Default under under the contract contract.. Default Default risk only matters matters for the party party that is ”in the money” money” at matur maturit ityy, that that is, that that stands stands to profit profit at the locked-i locked-in n price price in the contra contract. ct. (If the spot price at maturity is such that a party would lose from performing on the obligation in the contract, contract, counterpa counterparty rty default default is not a problem problem.) .) Prior Prior to maturit maturityy, since either party may finish in-the-money, both parties are exposed to default risk.

7. What What risks are being managed by trading trading derivative derivativess on exchanges? exchanges? important one is counterpa counterparty rty default default risk. In a typical typical futures futures exchang exchange, e, Answer: An important the exchange interposes itself between buyer and seller and guarantees performance on the contract contract.. This reduces reduces significantly significantly the default risk exposure of both parties. parties. Further, daily settling of marked-to-market gains and losses ensures that the loss to the exchange from an investor’s default is limited to at most one day’s settlement amount (and because of maintenance margins is usually less than even this; see Chapter 2 for a description of the margining process).

8. Explain the difference between between a forward forward contract and an option. Answer: A forward contract is an agreement to buy or sell an asset at a future date

(denoted T ), T ), at a specified delivery price (denoted F ). F ). The agreement is made at time t = 0 for settlement at maturity T . T . An option is the right but not the obligation to buy (a “call” option) or sell (a “put” option) an asset at a specified strike price on or before a specified specified maturity maturity date T date T .. In comparing a long forward contract to a call option, the main difference lies in the fact that the forward buyer has to buy the stock at the forward price at maturity, whereas in a call option, the buyer is not required to carry out the


purchase purchase if it is not in his interest interest to do so. The forwa forward rd contract contract confers confers the obligation obligation to buy, whereas the option contract provides this right with no attendant obligation.

9. What What is the difference difference between between value value and payoff payoff in the context context of derivative derivative securities. securities. of a derivativ derivativee is its current current fair price or its worth. worth. The payoff (or Answer: The value of payoffs) refers to the cash flows generated generated by the derivative at various times during its life. For example, the value of a forward contract at inception is zero: neither party pays anything anything to enter enter into the contract. contract. But the payo payoffs ffs from the contract contract at maturity maturity to either party could be positive, negative, or zero depending on where the spot price of the asset is at that point relative to the locked-in delivery price.

10. What What is a short position position in a forwar forward d contract? contract? Draw Draw the payo payoff ff diagram for for a short short position at a forward price of $103, if the possible range of the underlying stock price is $50-150. forward is where you are the seller of the forward forward contract. Answer: A short position in a forward In this case, you gain when the price of the underlying asset at maturity is below the locked-in delivery price. The payoff diagram for this contract is as shown in the following picture. When the price of the stock at maturity is the delivery price of $103, there are neither gains nor losses. 60

A Short Forward Contract's Payoff

40

20

f f o y a P

Stock Price 0 50

60

70

80

90

100

110

120

13 0

14 0

1 50

-20

-40

-60

11. Forward prices may be derived using the notion of absence of arbitrage, and market efficiency is not necessary. What is the difference between these two concepts? arbitrage means that a trading strategy cannot be found that creates Answer: Absence of arbitrage cash inflows without any cash outflows, i.e., creates something out of nothing. Efficiency, Efficiency,


as that that term term is used used by financia financiall econo economis mists, ts, implie impliess more more:: not not only only the absence absence of arbitrage but that asset prices reflect all relevant information.

12. Suppose Suppose you are holding holding a stock position, position, and wish to hedge hedge it. What What forwa forward rd contract contract would wo uld you you use, a long or a short? short? What What option option contra contract ct might might you you use? Co Comp mpar aree the forward versus the option on the following three criteria: (a) uncertainty of hedged position cash-flow, (b) Up-front cash-flow and (c) maturity-time regret. forward rd contract contract is to be used, used, then a short short forwar forward d is required required.. AlternaAlternaAnswer: If a forwa tively, a put option may also be used. The following describes the three criteria for the choice of the forward versus the option.

• Cash-flow uncertainty is lower for the futures contract. • The futures contract has no up-front cash-flow, whereas the option contract has an initial premium to be paid. • There is no maturity-time regret with the option, because if the outcome is undesirable desirable,, the option need not be exercised exercised.. With the futures contract contract there is a possible downside.

13. What What derivativ derivatives es strategy might might you implemen implementt if you expected a bullish bullish trend in stock prices? Would your strategy be different if you also forecast that the volatility of stock prices will drop? Answer: If you expect prices to rise, there are several different strategies you could

follow: follow: you you could could go long a forwa forward rd and lock in a price price today for the future purchase; purchase; you could buy a call which gives you the right to buy the stock at a fixed strike price; or you could sell a put today, receive a premium, and keep the premium as your profit if prices trend upward as you expect. The volatility issue is a bit trickier. As we explain in Chapter 7, both call and put options in value with volatility, so if you expect volatility to decrease, you do not want increase in to buy a a call: when volatility drops, what you have bought automatically becomes less valuable.

14. What are the underlyings underlyings in the following derivative derivative contracts? (a) A life insurance insurance contract. contract. (b) A home mortgag mortgage. e. (c) Employee Employee stock options. (d) A rate lock in a home loan.

Sundaram & Das: Derivatives - Problems and Solutions.................................. 7 Answer: The underlyings are as follows:

(a) A life insurance insurance contract: contract: the event of one’s demise. demise. (b) A home mortgage: mortgage: mortga mortgage ge interest interest rate. rate. (c) Employee Employee stock options: options: equity equity price price of the firm. (d) A rate lock in a home loan: mortgage interest rate.

15. Assume you have a portfolio portfolio that contains stocks that track the market market index. You now want to change this portfolio to be 20% in commodities and only 80% in the market index. How would you use derivatives to implement your strategy? would use futur futures es to do so. We would would short short market market index future futuress for for Answer: One would 20% of the portfoli portfolio’s o’s value, and go long 20% in commodity commodity futures. futures. A collection collection of commodity futures adding up to the 20% would be required.

16. In the previous question, how do you implement the same trading idea without using futures contracts? Answer: Futures contracts are traded on exchanges and are known as “exchange-traded”

securitie securities. s. An alternative alternative approac approach h to achieving achieving the goal would would be to use an over-theover-thecounter or OTC product, for example, an index swap that exchanges the return on the market index for the return on a broadly defined commodity index.

17. You buy a futures contract on the S&P 500. Is the correlation with the S&P 500 index positiv positivee or negati negative? ve? If the nomin nominal al value value of the contra contract ct is $100 $100,00 ,0000 and you are are required to post $10,000 as margin, how much leverage do you have? contract is positively correlated correlated with the stock index. The leverage Answer: The futures contract is 10 times. That is, for every $1 invested in margin, you get access to $10 in exposure.

Chapter 2

Sundaram & Das: Derivatives - Problems and Solutions ................................ 13

16. 16. What What is the the “clo “closin singg out” out” of a posit positio ion n in futu future ress mark market ets? s? Why Why is clos closin ingg out out of contracts permitted in futures markets? Why is unilateral transfer or sale of the contract typically not allowed in forward markets? market, an investor must take an offsetting Answer: To close out a position in a futures market, opposite opposite position in the same contract. contract. (For (For example, example, to close out a long positi p osition on in 10 S&P 500 index futures contracts with expiry in March, an investor must take a short position in 10 S&P 500 index futures contracts with expiry in March.) Once a position is closed out, the investor no longer has any obligations remaining. is key to allowing investors to close out contracts. In a futures exchange, the Credit risk is exchange interposes itself between buyer and seller as the guarantor of all trades; thus, there there is little credit risk involved. involved. In forwa forward rd marke markets, ts, allowing allowing investors investors to unilatera unilaterally lly transfer their obligations could exacerbate credit risk, so it is typically disallowed. An obligation under a forward contract may be eliminated in one of two ways: (a) the contract may be unwound with the same counterparty or (b) the contract may be offset by enterin enteringg into into an equal equal and opposit oppositee contr contract act with a third third party party.. The latter latter is the analog analog of the unilateral unilateral close-out close-out of futures futures contracts. contracts. Howev However, er, while close-ou close-outt of the futures contract leaves the investor with no net obligations, offset of a forward contract leaves the investor with obligations on both contracts.

17. An investor investor enters into a long position position in 10 silver futures futures contracts contracts at a futures price price of $4.52/oz and closes out the position at a price of $4.46/oz. If one silver futures contract is for 5,000 ounces, what are the investor’s gains or losses? Effectively, the investor buys at $4.52 per oz and sells at $4.46 per oz, so takes Answer: Effectively, a loss of $0.06 $0.06 per oz. Per Per contrac contract, t, this this amoun amounts ts to a loss loss of (5000 of (5000 × 0. 0.06) = $300. $300. Over 10 contracts, this results in a total loss of $3,000.00.

18. What is the settlement price? The opening and closing price? Answer: The opening price for a futures contract is the price at which the contract is

traded at the begining of a trading session. The closing price is the last price at which the contract is traded at the close of a trading session. The settlement price is a price chosen by the exchange as a representative price from the prices at the end of a session. The settlement price is the official closing price of the exchange; it is the price used to settle gains and losses from futures trading and to invoice deliveries.

19. An investor investor enters into a short short futures position position in 10 contracts contracts in gold at a futures futures price of $276.50 $276.50 per oz. The size of one futures futures contra contract ct is 100 100 oz. The initial initial margin margin per contract is $1,500, and the maintenance margin is $1,100.


(a) What What is the initial size of the margin accoun account? t? (b) Suppose Suppose the futures futures settlement settlement price on the first day is $278.00 $278.00 per oz. What What is the new balance in the margin account? Does a margin call occur? If so, assume that the account is topped back to its original level. (c) The futures futures settlement settlement price price on the second day day is $281.00 $281.00 per oz. What is the new balance in the margin account? Does a margin call occur? If so, assume that the account is topped back to its original level. (d) On the third day, the investor closes out the short position at a futures price of $276.00. What is the final balance in his margin account? (e) Ignoring Ignoring interest interest costs, what are his total total gains gains or losses? losses? Answer: Futures position: short 10 contracts

Size of one contract: 100 oz Initial margin per contract: $1,500 Maintenance margin per contract: $1,100 Initial futures price: $276.50 per oz (a) Initial Initial size of margin margin account account = 1, 500 × 10 10 = 15, 15, 000. 000. (b) If the settlement settlement price price is $278 per oz, the short short position position has effectively effectively lost $1.50 $1.50 150 per contract per oz. oz. This This is a loss loss of 1.50 × 100 = 150 contract.. Since the position position has 10 contracts, the overall loss is 150 × 10 = 1, 1, 500. 500. Thus, Thus, the new balanc balancee in the margin account is 15, 15, 000 − 1, 1, 500 = 13, 13, 500. 500. A margin margin call does not occur since since this new balance is larger than the maintenance margin of $11,000. (c) When the settlement price moves to $281 per oz, the short position effectively loses loses another another $3 per oz. The loss per contract contract is 3 × 100 = 300, 300, so the overall loss is 300 × 10 10 = 3, 3 , 000. 000. Thus, Thus, the balance balance in the marg margin in account account is reduc reduced ed to 13, 13, 500 − 3, 3, 000 = 10, 10, 500. 500. Since Since this is less than the maintena maintenance nce margin margin,, a margin call occurs. Assume the account is topped back to $15,000. (d) When When the positio p osition n is closed closed out at $276 $276 per oz, the short short positio p osition n makes a gain of 281 − 276 = 5 per 5 per oz. This translates to a gain of 500 per contract, and, therefore, to an overal overalll gain gain of 5,000. 5,000. Thus, Thus, the closing closing balance balance in the margin margin account account is 15, 15, 000 + 5, 5, 000 = 20, 20, 000. 000. (e) The invest investor or began began with with a margi margin n accoun accountt of $15,0 $15,000 00,, and deposit deposited ed anothe anotherr $4,500 $4,500 to meet the margin margin call, for a total outlay outlay of $19,500 $19,500.. Since the margin margin account balance at time of close out is $20,000, his overall gain (ignoring interest costs) is $500.

20. The current price price of gold is $642 per troy troy ounce. Assume that you initiate initiate a long position in 10 COMEX gold futures contracts at this price on 7-July-2006. The initial margin is


5% of the initial price of the futures, and the maintenance margin is 3% of the initial price. Assume the following evolution of gold prices over the next five days, and compute the margin account assuming that you meet all margin calls. Date Price per Ounce 7-Ju 7-Jull-06 06 642 642 8-Ju 8-Jull-06 06 640 640 9-Ju 9-Jull-06 06 635 635 10-J 10-Jul ul-0 -06 6 632 632 11-J 11-Jul ul-0 -06 6 620 620 12-J 12-Jul ul-0 -06 6 625 625 Answer: The initial margin is $321, $321, and the maintenance maintenance margin is $193. $193. The following

is the evolution evolution of the margin margin account. account. Note that there is one margin margin call that takes place on 11-July-2006. Initiation Price = 642 Initial Margin (5%) = 321 Maintenance Margin (3%) = 192.6 Number of contracts = 10

Date 7-Jul-06 8-Jul-06 9-Jul-06 10-Jul-06 11-Jul-06 12-Jul-06

Margin Account Openi Opening ng Daily Daily Profi Profitt Adju Adjust sted ed Marg Margin in Call Call Clos Closin ingg Gold Price Balance and Loss Balance Deposit Balance 642 640 635 632 620 62 6 25

321 301 251 221 321

-20 -50 -30 -120 50

301 251 221 101 371

0 0 0 220 0

301 251 221 321 371

21. When When is a futures futures market market in “backwar “backwardatio dation”? n”? When is it in “contango” “contango”?? Answer: A futures market is said to be in backwardation if the futures price is less than the spot price. It is in contango if if futures price is above spot.

22. Suppose there are three deliverable bonds in a Treasury Bond futures contract whose current cash prices (for a face value of $100,000) and conversion factors are as follows: (a) Bond Bond 1: Price Price $98,750. $98,750. Convers Conversion ion factor factor 0.9814. 0.9814.


(b) Bond Bond 2: Price Price $102,575. $102,575. Convers Conversion ion factor factor 1.018. (c) Bond Bond 3: Price Price $101,150. $101,150. Convers Conversion ion factor factor 1.004. The futures price is $100,625. Which bond is currently the cheapest-to-deliver? 100,625 times the conversion Answer: Since the long position will pay the futures price of 100,625 factor factor in settlement settlement,, the short position prefers prefers to deliver deliver the bond on which the ratio of the sale price to the purchase price is highest. Essentially, this means the bond delivered is cheapest relative to the sale price. We compute this ratio for all three bonds as follows: 100, 100, 625 × 0. 0.9814 = 1.0000 98, 98, 750 100, 100, 625 × 1. 1.018 = 0.99865 102, 102, 575 100, 100, 625 × 1. 1.004 = 0.99879 101, 101, 150 Hence, the first bond is the cheapest to deliver.

23. You enter into a short short crude oil futures contract contract at $43 per barrel. barrel. The initial margin margin is $3,375 $3,375 and the maintenence maintenence margin margin is $2,500. $2,500. One contract contract is for for 1,000 1,000 barrel barrelss of oil. By how much do oil prices have to change before you receive a margin call? account falls to a value of $2500 then a call will occur. Therefore, Therefore, Answer: If the margin account the loss on the position position must be equal equal to $3375-$ $3375-$2500 2500=$8 =$875 75 for a margin margin call. Solving Solving the following equation 1000 (P (P − 43) = 875 gives P gives P = 43 4 3.875, 875, which is the price at which a margin call will take place.

24. You take a long futures contract on the S&P 500 when the futures price is 1,107.40, and close it out three days later at a futures price of 1,131.75. One futures contract is for 250 for 250 × the index. Ignoring interest, what are your losses/gains? Answer: The gain is

250(1131. 250(1131.75 − 1107. 1107.40) = $6087. $6087.50


25. An investor enters into 10 short futures contract on the Dow Jones Index at a futures price price of 10,106. 10,106. Each Each contr contrac actt is for 10× the the index. index. The invest investor or closes closes out five contracts when the futures price is 10,201, and the remaining five when it is 10,074. Ignoring interest on the margin account, what are the investor’s net profits or losses? Answer: Exercise for the reader.

26. A bakery enters into 50 long wheat futures contracts on the CBoT at a futures price of $3.52/bushel. It closes out the contracts at maturity. The spot price at this time is $3.59/ $3.59/ bushel. Ignoring Ignoring interest, interest, what are the bakery’ bakery’ss gains gains or losses from its futures futures position? Answer: Each CBoT Wheat contract is for 50,000 bushels and so the settlement gain

is 50 × 50, 50, 000 × (3. (3.59 − 3. 3.52) = $175, $175, 000

27. An oil refining company enters into 1,000 long one-month one-month crude oil futures contracts on NYMEX at a futures price of $43 per barrel. At maturity of the contract, the company rolls half of its position forward into new one-month futures and closes the remaining half. At this point, the spot price of oil is $44 per barrel, and the new one-month futures price is $43.50 per barrel. At maturity of this second contract, the company closes out its remaining remaining position. position. Assume Assume the spot price price at this point is $46 per barrel. barrel. Ignoring Ignoring interest, what are the company’s gains or losses from its futures positions? Answer: Exercise for the reader.

28. Define the following terms terms in the context of futures markets: markets: market market orders, limit orders, orders, spread orders, one-cancels-the-other orders. Answer: See section 2.3 of the book.

29. Distinguish between market-if-touched market-if-touched orders orders and stop orders. orders. Answer: See section 2.3 of the book.

30. You have a commitment to supply 10,000 oz of gold to a customer in three months’ time at some specified price and are considering hedging the price risk that you face. In each of the following scenarios, describe the kind of order (market, limit, etc.) that you would use.


will be sustained sustained.. Thus, Thus, unless unless there is high volatilit volatilityy and a reversal reversal of direction direction,, this approach may not be profitable and might turn out to be loss-making.

32. The spread between May and September wheat futures is currently $0.06 per bushel. You expect expect this this sprea spread d to widen to at least $0.10 $0.10 per bushel. bushel. How How wo would uld you you use a spread order to bet on your view? Answer: If the price differential between September and May futures is currently $0.06

and is expected to widen to $0.10, then we should enter into a long position in the September contract and a short position in the May contract. When the spreads widens we close out both contracts.

33. The spread between between one-month and three-month three-month crude oil futures is $3 per p er barrel. You expect expect this spread spread to narro narrow w sharply sharply.. Explain Explain how you would use a spread spread order order given this outlook. narrow, we should go Answer: Assuming the three-month minus one-month spread will narrow, long the one-month one-month contract contract and short the three-mo three-month nth contract. contract. When the spread narrows, we buy back the short three-month contract and sell back the long one-month contract. We capture (ignoring interest) the difference between $3 and the new spread.

34. Suppose you anticipate a need for corn corn in three months’ time and are using corn futures futures to hedge the price price risk that you you face. How is the value of your your position affected affected by a strengthening of the basis at maturity? basis is the futures futures price price minus minus the spot price. price. A streng strengthe thenin ningg of the Answer: The basis basis occurs if the basis increases. If this occurs, the position in the question is positively affected affected since you are long futures. futures. In notational notational terms, you go long futures futures today (at price F price F 0, say) and close it out at T (at price F T T (at price F T , say) for a net cash flow on the futures position of F spot price S price S T F T − F 0 . In addition, you buy the corn you need at the time- T spot T − T , leading to a total net cash flow of ( of (F F T T − F 0 ) − S T T = (F T T − S T T ) − F 0 . A strengthening of the basis F basis F T T − S T T at maturity improves this cash flow.

35. A short hedger is one who is short short futures futures in order order to hedge hedge a spot cash cash flow flow risk. risk. A is similarly one who goes long futures to hedge an existing risk. How does long hedger is a weakening of the basis affect the positions of short and long hedgers? short hedger hedger is short short futures and long spot, so gains if the basis weakens. weakens. Answer: The short The long hedger is long futures and short spot, so loses in this case.

Chapter 3


storage costs, we compute compute Answer: The spot price of wheat is $3.60. Since there are no storage the theoretical forward price of wheat as 3.60 exp(0 exp(0..08 × 3/ 3/12) = 3. 3.6727 which 6727 which is less than the forward price. Hence, there is an arbitrage opportunity. In order to arbitrage this situation, we would undertake the following strategy:

• Sell wheat forward at 3.90. • Buy wheat spot at 3.60. • Borrow 3.60 for three months . At inception, inception, the net cash-flow cash-flow is zero. zero. At maturity maturity, we deliver deliver the wheat we own and receive receive the forwa forward rd price of $3.90. $3.90. We return the borrow borrowed ed amount with interest interest for a cash outflow of 3.60 exp(0 exp(0..08 × 0. 0.25) = 3. 3.6727.This 6727.This results in a net cash inflow of 0.2273. 2273. The following table summarizes:

Source Short Forward Long spot Borrowing Net

Cash Flows Initial Final +3. +3.9000 −3.6000 +3. +3.6000 −3.6727 +0. +0.2273

Note that it makes no difference if the contract is cash-settled instead of settled by physical delivery. If it is cash-settled, letting W T T denote the spot price of wheat at date T , T , we receive 3.90 − W T T on the forward contract, sell the spot wheat we own for W T T , and repay the borrowing, for exactly the same final cash flow.

5. A securi security ty is current currently ly trading trading at $97. It will pay a coupo coupon n of $5 in two months months.. No other payouts are expected in the next six months. (a) If the term structure is flat at 12%, what should s hould the be forward forward price on the security security for delivery in six months? (b) If the actual actual forwar forward d price price is $92, $92, explain explain how an arbitra arbitrage ge may be created. created. 97, and the PV of holding benefits is 5exp(−0.12 × Answer: We have that S = 97, (2/ (2/12)) = 4. 4.9010. 9010. Thus, the forward price should be (97 − 4. 4.9010) exp(0 exp(0.12 × (6/ (6/12)) = 97. 97.794. 794. Since the forward price is 92, it is mispriced (under-priced). The arbitrage is as follows. At inception:

• Buy forward at 92.


• Sell short spot at 97 Invest P V (5) 4.901 for 901 for three months at 12%. • Invest P V (5) = 4. Invest 97 − P V (5) 92.099 for 099 for six months at 12%. V (5) = 92. • Invest 97

• In three months, use the cash inflow of 5 from the investment to pay the coupon due on the shorted security. receive the cash from the six-month six-month investmen investment. t. Pay Pay the delivery delivery • In six months, receive price of 97 on the forward and receive unit of the security, Use this to close the short spot position. The initial and interim cash flows are zero, and the final cash flow is positive as the following table shows:

Source Long forward Short spot 3-month investment 6-month investment Net

Cash Flows Initial Interim Final 0 92.00 −92. +97. +97.00 +5 −4.901 92.099 +97. +97.794 −92. +5. +5.794

6. Suppose that the current price of gold is $365 per oz and that gold may be stored costlessly. Suppose also that the term structure is flat with a continuously compounded rate of interest of 6% for all maturities. (a) Calculate the forward forward price price of gold for delivery in three months. (b) Now suppose it costs $1 per oz per month to store gold (payable monthly in advance). What is the new forward price? (c) Assume Assume storage storage costs are as in part (b). If the forwar forward d price is given to be $385 per oz, explain whether there is an arbitrage opportunity and how to exploit it. Answer: The answers to the three parts are given below:

(a) The forwa forward rd price price of gold is 365 exp(0 exp(0..06 × 0. 0.25) = 370. 370.52. 52. (b) With storage costs costs we need to first find the present value of the holding costs ( M ). ). These are: 1 + 1exp(−0.06 × 1/ 1/12) + 1 exp( exp(−0.06 × 2/ 2/12) = 2. 2.9851. 9851.


The forward price then is (S + + M )exp( 2 .9851) exp(0 exp(0.06 × 0. 0.25) = 373. 373.55. 55. M )exp(rT rT )) = (365 + 2. This is higher because the storage costs have been factored in. (c) If the forwar forward d price is 385, there is an arbitrag arbitragee which is exploited exploited as follows. follows. At inception:

• Sell forward at 385. • Buy spot at 365. • Pay storage costs = 1. Invest 1 exp( exp(−0.06 × 1/ 1/12) = 0. 0.9950 for 9950 for one month. • Invest 1 Invest 1 exp( exp(−0.06 × 2/ 2/12) = 0. 0.9900 for 9900 for two months. • Invest 1 Borrow 365 + 1 + 0. 0.9950 + 0. 0.9900 = 367. 367.985 for 985 for three months. • Borrow 365 The net cash flow at inception is zero. At the end of one month, realize $1 from the investment of 0.995 made at time zero, and use this to pay off the storage costs. There are no net cash flows. At the end of two months, realize $1 from the investment of 0.99 made at time zero, and use this to pay off the storage costs. Again, there are no net cash flows. On maturity, deliver the spot holding to close out the forward contract by physical delivery. Cash flows at maturity are: 385 − 367. 367.985 exp(0 exp(0..06 × 3/ 3/12) = 11. 11.454. 454. This is positive irrespective of the time- T spot T spot price of gold. The following table summarizes all the cash-flows. Source Sell forward Buy spot Month 1 storage cost Month 2 storage cost Month 3 storage cost Borrow Invest 0.995 for one month Invest 0.99 for two months Net

Initial 0 −365 −1 – +367.985 −0.995 −0.99 0

cash-flows Month 1 Month 2 −1 −1 +1 +1 0 0

Mo Month 3 385 373.5464 −373. +11.454

7. A stock stock will pay pay a divide dividend nd of $1 in one month month and $2 in four months months.. The risk-fr risk-free ee rate of interest for all maturities is 12%. The current price of the stock is $90.


(a) Calculate the arbitrage-free arbitrage-free price price of (i) a three-month forward forward contract contract on the stock sto ck and (ii) a six-month forward contract on the stock. (b) Suppose Suppose the six-month six-month forwa forward rd contract contract is quoted quoted at 100. 100. Identify Identify the arbitra arbitrage ge opportunities, if any, that exist, and explain how to exploit them. 90; r = 0.12 for 12 for all maturities; and that dividends of $1 Answer: We are given: S = 90; and $2 will be paid in one and four months, respectively. (a) First, First, consider consider the case of a three-month three-month horizon horizon.. There There is only one dividend dividend to be conside considered red,, viz. viz. the payme payment nt of $1 in one one month month.. The prese present nt value value of this dividend is exp{−(0. (0.12)(

1 )} × 1 = 0.99. 99. 12

Since the dividend represents a cash inflow, we have M = − 0.99. 99. Therefo Therefore, re, the arbitrage-free forward price is (0.12)(0. 12)(0.25) + M ) 0.99)e 99)e(0. = 91. 91.72. 72. F = (S + M )erT = (90 − 0.

Now, consider the six-month horizon. There are two dividend payments that occur. The present value of the first dividend is 0.99, as we have seen above. The present value of the second dividend is 1 exp{−(0. (0.12)( )} × 2 = 1.92. 92. 3 Therefore, the present value of the dividends combined is 0. 0.99+1. 99+1.92 = 2. 2.91. 91. Since 91. the dividends represent a cash inflow, we must have M = − 2.91. It follows that the arbitrage-free forward price for a six-month horizon is (0.12)(0. 12)(0.50) + M ) 2.91)e 91)e(0. = 92. 92.475. 475. F = (S + M )erT = (90 − 2.

(b) The six-month forward is quoted at 100, so it is overvalued relative to spot . To make make an arbitrage arbitrage profit, profit, one should sell forwa forward, rd, buy spot, and borrow. borrow. Specifically: i. At time 0: Buy one unit of spot; borrow borrow 87.09 87.09 for repaymen repaymentt in six months; months; borrow 1 borrow 1..92 for 92 for repayment in four months; and borrow 0.99 for repayment in one month. Net cash flow: 0. ii. In one month: receive dividend of $1; use this to repay the one-month loan. Net cash flow: 0. iii. In four months: months: receive receive dividend dividend of $2; use this to repay repay the four-mon four-month th loan. Net cash flow: 0.


iv. In six months: Use the unit of spot to settle settle the short short forward forward position; receive receive 100 from the forward position; use 92.475 of this to repay the six-month loan. Net cash flow: 100 − 92. 92.475 = 7. 7.525. 525.

8. A bond bond will will pay a coupo coupon n of $4 in two two mont months hs’’ time time.. The The bond’ bond’ss curr curren entt price rice is $99.75. $99.75. The two-mon two-month th interest rate is 5% and the three-mon three-month th interest rate is 6%, both in continuously compounded terms. (a) What is the arbitrage-free arbitrage-free three-month three-month forward forward price for the bond? (b) Suppose Suppose the forwa forward rd price price is given given to be $97. $97. Ident Identify ify if there there is an arbitr arbitrag agee opportunity and, if so, how to exploit it. Answer: The coupons represent a cash inflow, so the forward price is

[99. [99.75 − 4 exp( exp(−0.05 × 2/ 2/12)] 12)] exp(0 exp(0..06 × 3/ 3/12) = 97. 97.231. 231. If the forward price is 97, then the forward is under priced priced relative to spot. An arbitrage exists and is exploited with the following strategy:

• Buy forward at 97. • Sell spot at 99.75. Invest the present present value of the coupon coupon for two two months. The PV of the coupon is • Invest 4exp(−0.05 × 2/ 2/12) = 3. 3.9668. 9668. 99.75 − 3. 3 .9668 = • Invest the remaining proceeds for three months , i.e., invest 99. 95. 95.7832. 7832. The cash-flow at inception is zero. After two months, realize $4 from the investment of 3.9688 and use it to pay the coupon on the shorted bond. Net cash flow: zero. At maturity we have the following cash-flows:

• Pay 97 on the forward contract and receive the bond. Use it to close out the short spot position. position. 95.7832 7832 exp(0 exp(0..06 × 3/ 3/12). 12). • Receive principal plus interest on the investment: 95. The net cash-flow is 0 is 0..2308, 2308, which is positive.

9. Suppose that the three-month interest rates in Norway Norway and the US are, respectively, 8% and 4%. Suppose that the spot price of the Norwegian Kroner is $0.155.


(a) Calculate the forward forward price price for delivery in three months. (b) If the actual forward price is given to be $0.156, examine if there is an arbitrage opportunity. Answer: The forward price of the Kroner is

0.155 exp(( exp((00.04 − 0. 0.08) × 3/ 3/12) = 0. 0.15346. 15346. Since the forwa forward rd price is actually 0.156, 0.156, it is overpr overpriced iced.. We may exploit exploit this by the following strategy:

• Sell 1 Kroner forward at $0.156. • Buy PV(1 Kroner)= e

0.08×3/12

−

= 0.9802 spot at 0. 0 .155 × e

0.08×3/12

−

= $0.1519.

• Invest PV(1 Kroner) for three months. Amount received at maturity = 1 Kroner. • Borrow $0.1519 for three months at 4%. months, deliver deliver Kroner and receive receive $0.156 from the forwa forward. rd. Repay Repay • After three months, dollar borrowing with interest for a total of $0.1534. The resulting cash flows are summarized in the following table:

At inception

At T At T

Cash flow in Kroner

Cash flow in $

+0. +0.9802 9802 (purchase) 9802 (investment) −0.9802 (investment)

1519 (sale) −0.1519 (sale) +0. +0.1519 (borrowing) 1519 (borrowing)

Net: 0

Net: 0

+1. +1.00 (from 00 (from investment) 00 (deliver to forward) −1.00 (deliver

1534 (repay borrowing) −0.1534 (repay +0. +0.156 (receive 156 (receive from forward)

Net: 0

Net: +0. +0.0026

Since all cash flows are zero or positive, we have the required arbitrage.


10. Consider a three-month three-month forwa forward rd contract on pound sterling. sterling. Suppose the spot exchange rate is $1.40/ £, the three-month interest rate on the dollar is 5%, and the three-month interest rate on the pound is 5.5%. If the forward price is given to be $1.41/ £, identify whether there are any arbitrage opportunities and how you would take advantage of them. 40, r = 0.05 and d = 0.055. 055. From rom Answer: We are given the information that S = 1.40, this data, the arbitrage-free forward price of a three-month forward contract should be F = e(r

d)T

−

(0.05 S = e(0.

0.055)(1/ 055)(1/4)

−

(1. (1.40) = 1. 1.3983. 3983.

Thus, at the given forward price of $1.41/ £, the forward contract is overvalued relative to spot. To take take advantag advantagee of the opportunit opportunityy, we should sell forwar forward, d, buy spot, and borrow borrow to finance finance the spot purchas purchase. e. Specifically: Specifically:

• Enter into a short forward contract to deliver pounds in three months at $1.41/ Buy e dT = 0.9863 pounds 9863 pounds spot at the spot price of $1.40/ £. • Buy e Cost: $(1. $(1.40)(0. 40)(0.9863) = $1. $1.3809. 3809. • Invest the £0.9863 for three months at 5.5%. Amount received after three months: £1. • Borrow $1.3809 for three months at 5%. (0.05)(1/ 05)(1/4) Amount due in three months: $(e $(e(0. (1. (1.3809) = $1. $1.3983. 3983. −

The resulting cash flows are summarized in the following table:

Cash flow in At inception

At T

£

Cash flow in $

+0. +0.9863 (from 9863 (from purchase) 9863 (investment) −0.9863 (investment)

3809 (to purchase £) −1.3809 (to +1. +1.3809 (borrowing) 3809 (borrowing)

Net: 0

Net: 0

+1. +1.00 (from 00 (from investment) 00 (deliver to forward) −1.00 (deliver

3983 (repay borrowing) −1.3983 (repay +1. +1.41 (receive 41 (receive from forward)

Net: 0

Net: +0. +0.0117

Since all cash flows are zero or positive, we have the required arbitrage.

£.


11. Three months ago, an investor entered entered into a six-month forward forward contract to sell a stock. sto ck. The delivery price agreed to was $55. Today, the stock is trading at $45. Suppose the three-month interest rate is 4.80% in continuously compounded terms. (a) Assuming the stock is not expected to pay any dividends over the next three months, what is the current forward price of the stock? (b) What What is the value of the contract contract held by the investor? investor? (c) Suppose the stock is expected to pay a dividend of $2 in one month, and the one-mon one-month th rate of interest is 4.70%. 4.70%. What What are are the current current forw forward ard price price and the value of the contract held by the investor? Answer: The answers to the three parts are:

(a) The current forward forward price price is 45 exp(0 exp(0..048 × 3/ 3/12) = 45. 45.543. 543. (b) The value of the contract contract is P is P V ( Since K − F = 55. 55 .000 − 45. 45.543 = 9. 9.457, 457, V (K − F ) F ). Since K the contract value is 9. 9 .457exp(−0.048 × 3/ 3/12) = 9. 9.3442. 3442. (c) Now Now suppose suppose the stock stock is expected expected to pay a divide dividend nd of $2 in one month. month. The present value of this dividend payment is (0. (0.047)(1/ 047)(1/12)

−

e

2 = 1.992. 992.

Since the dividend payment represents a cash inflow, we have M = − 1.992. 992. Thus, the arbitrage-free forward price is now (0.048)(1/ 048)(1/4) + M ) (45 − 1. 1.992) = 43. 43.527. 527. F = erT (S + M ) = e(0.

The value of holding a short position in this forward contract with a delivery price of K 55 is now K = 55 is (0. (0.048)(1/ 048)(1/4)

−

PV( PV(K − − F ) F ) = e

(55 − 43. 43.527) = 11. 11.336. 336.

12. An investor investor enters into a forwa forward rd contract contract to sell a bond in three three months’ time at $100. $100. After After one one month month,, the bond price price is $101.5 $101.50. 0. Suppose Suppose the term-s term-stru tructu cture re of inter interest est rates is flat at 3% for all maturities. (a) Assuming Assuming no coupons are are due on the bond over the next two two months, months, what is the forward price on the bond? (b) What is the marked-to-mark marked-to-market et value of the investor’s short position? (c) How would would your answers answers change if the bond will pay a coupon of $3 in one month’s time?

Sundaram & Das: Derivatives - Problems and Solutions ................................ 30 Answer: The answers to the three parts are:

(a) The forwa forward rd price price at the end of one month is 101. 101.50 exp(0 exp(0..03 × 2/ 2/12) = 102. 102.01. 01. (b) The marked-to-ma marked-to-market rket value of the original contract is 102.01) exp( exp(−0.03 × 2/ 2/12) = −2.00. 00. P V ( V (F − K ) = (100 − 102. (c) If there is a coupon one month from now, then the re-estima re-estimated ted forward forward price price is: (101. (101.50 − 3 exp( exp(−0.03 × 1/ 1/12)) 12)) exp(0 exp(0..03 × 2/ 2/12) = 99. 99.001. 001. In this case, the value of the contract to sell forward at 100 is (100 − 99. 99.001) 001) exp( exp(−0.03 × 2/ 2/12) = 0. 0.994. 994.

13. A stock stock is tradin tradingg at $24.50. $24.50. The mark market et consens consensus us expectat expectation ion is that that it will will pay pay a dividend of $0.50 in two months’ time. No other payouts are expected on the stock over the next three months. Assume interest rates are constant at 6% for all maturities. You enter into a long position to buy 10,000 shares of stock in three months’ time. (a) What is the arbitrage-free arbitrage-free price of the three-month forward forward contract? (b) After After one month, the stock is trading at $23.50. $23.50. What What is the marke marked-tod-to-mar market ket value of your contract? (c) Now suppose that at this point, the company unexpectedly announces that dividends will be $1.00 per share due to larger-than-expected earnings. Buoyed by the good news, the share price jumps up to $24.50. What is now the marked-to-market value of your position? Answer: The answers to the three parts are as follows:

(a) The dividends represent a cash inflow, so the arbitrage free forward price of the three-month forward is obtained by using the formula P V ( + M .. SubstiV (F ) F ) = S + M tuting for the various input values, this gives us the original forward price as 0.06×2/12

−

[24. [24.50 − 0. 0.50 e

] × e0.06

3/12

×

Denote this locked-in delivery price by K K ..

= 24. 24.368. 368.


(b) After After one month, the contract contract has two remaining remaining months months of life. Given that that the new spot price is S = 23. 23.50 (and 50 (and that interest rates are unchanged), the forward price of the same contract is 0.06×1/12

−

[23.50 − 0. 0.50 e F F = [23.

] × e0.06

2/12

×

= 23. 23.234. 234.

Hence, the marked-to-market value of the original contract is (23.234 − 24. 24.368) 368) exp( exp(−0.06 × 2/ 2/12) = − 1.1227 P V ( V (F − K ) = (23. i.e., i.e., a loss loss of $1.122 $1.12277 per share share.. On 10,000 10,000 shares shares,, the value value of the positio position n is $10, $10, 000 × −1.1227 = −$11, $11, 227. 227. (c) If the dividends change to 1.00, we need to rework the forward forward price price and re-assess the position. The new forward price will be: 0.06×1/12

−

[24. [24.50 − 1. 1.00 e

] × e0.06

2/12

×

= 23. 23.741. 741.

At this forward price, the value of the original contract is 0.06×2/12

−

(23. (23.741 − 24. 24.368) × e

= −0.6208

or a loss of $0.6208 per share, for a total loss of $6,208 on the position.

14. Suppose you are given the following following information:

• The current price of copper is $83.55 per 100 lbs. • The term-structure of interest rates is flat at 5%, i.e., that the risk-free interest rate for borrowing/investment is 5% for all maturities in continuously-compounded and annualized terms. • You can take long and short positions in copper costlessly. • There are no costs of storing or holding copper. Consider a forward contract in which the short position has to make two deliveries: deliveries: 10,000 lbs of copper in one month, and 10,000 lbs in two months. The common delivery price in the contract for both deliveries is P , P , that is, the short position receives P P upon making the one-month delivery and P making the two-mo two-month nth delivery delivery.. What is P upon making the arbitrage-free value of P of P ?? Answer: Let Q denote the quantity delivered each month (i.e., Q = 10, 10, 000 000 lbs). lbs). To

replicate this contract, we need to buy 2Q 2 Q units of copper today and store it. After one month, we deliver the first Q units, Q units, and after one more month, the second Q units. Q units. The cost of this replication strategy is the current spot price of 2Q units, which is 2 × 100 × 83. 83.55


This must equal the present value of the cash outflows on the forward strategy, which is 0.05×1/12

−

Pe

+ P e0.05

2/12

×

0.05×1/12

−

= P × (e (e

0.05×2/12

−

+e

)

Equating these, we can solve for P : P : P =

2 × 8, 8, 355 = 8, 407. 407.40 exp(−0.05 × 1/ 1/12) + exp(−0.05 × 2/ 2/12)

This is the arbitrage free value of P of P ..

15. This This quest question ion general generalize izess the previ previou ouss one one from from two two delive deliverie riess to many many.. Co Consi nside derr a contract that requires the short position to make deliveries of one unit of an underlying at time points t points t 1 , t2 , . . . , tN . The common delivery price for all deliveries is F . F . Assume the interest interest rates rates for for these these horizon horizonss are, are, respectivel respectivelyy, r1 , r2 , . . . , rN in continuo continuouslyuslycompounded annualized terms. What is the arbitrage-free value of F given F given a spot price of S ? S ? Answer: The answer answer follows follows the same logic as we had in the previous previous question, question, i.e.,

N × × S i=1 exp(−rt ti )

F = N

i

Such a contract (one which calls for multiple deliveries at a fixed price F ) F ) is a “commodity swap.” Commodity swaps are usually settled in cash, rather than by physical delivery as we’ve assumed here, though this does not change the arguments. Commodity swaps are discussed in Chapter 25.

16. In the absence absence of interest-r interest-rate ate uncertain uncertainty ty and delivery delivery options, futures futures and forw forward ard prices prices must be the same. Does this mean the two two contracts contracts have identical identical cash-flow cash-flow implications? (Hint : Suppose you expected a steady increase in prices. Would you prefer a futures contract with its daily mark-to-market or a forward with its single mark-tomarket at maturity of the contract? What if you expected a steady decrease in prices?) Evidently not. For For example, example, if prices prices ares steadily trending trending upwar upward, d, a futures futures Answer: Evidently contract with its daily mark-to-market will result in earlier cash inflows to the long and cash outflows for the short. So if you were a long investor, you would prefer the futures to the forward (vice versa if you were a short investor).


17. Consider Consider a forwa forward rd contract contract on a non-divid non-dividendend-pa paying ying stock. If the term-stru term-structur cturee of interest rates is flat (that is, interest rates for all maturities are the same), then the arbitrage-free arbitrage-free forward forward price is obviously increasing in the maturity of the forward forward contract (i.e., a longer-dated forward contract will have a higher forward price than a shorterdated one). Is this statement true even if the term-structure is not flat? Assumee that the spot rates rates for Answer: Consider two dates t1 and t2 where t1 < t2 . Assum these two dates are respectively, r respectively, r1 and r and r2 . Given a spot price of S of S , the two corresponding r t r t forward prices are F are F 1 = S = See and F and F 2 = S = See . When the term structure is flat, r1 = r = r 2 , and hence, it is easy to see that F 1 < F 2 . 1 1

2 2

For general curves of spot rates, i.e., when the term structure is not flat, suppose we want that F that F 2 < F 1 . Then it must be that Se r t < Se r t r2 t2 < r1 t1 r2 < r1 (t1 /t2 ) 2 2

1 1

Is this this feasib feasible? le? Mathe Mathemat matica ically lly,, yes, yes, we may find param paramete eterr values values for which which this condition holds, implying that when term structures are not flat, we may have longer term forward forward prices lower lower than shorter term ones. For For example, r2 = 0.02, 02, r1 = 0.06, 06, 25 and t 2 = 0.50, 50, satisfies the condition. t1 = 0.25 and But economically, does this make sense? The answer is no. The condition that r1 t1 > r2 t2 means that an investor can make more investing for t1 years than for t2 years. This gives rise to an arbitrage opportunity where you borrow long term for t 2 and invest short-term short-term for t for t 1 .

18. The spot price of copper is $1.47 per lb, and the forward price for delivery in three month monthss is $1.51 per lb. Suppose Suppose you you can borro borrow w and lend lend for for three three months months at an interest rate of 6% (in annualized and continuously-compounded terms). (a) First, suppose there are no holding costs (i.e., no storage storage costs, no holding benefits). Is there an arbitra arbitrage ge opportunity opportunity for you given these prices? prices? If so, provide provide details of the cash flows. If not, explain why not. (b) Suppose now that the cost of storing copper for three months is $0.03 per lb, paya payable ble in advan advance. ce. How How wo would uld your your answer answer to Part Part (a) change change?? (Note (Note that storage costs are asymmetric : you have to pay storage costs if you are long copper, but you do not receive the storage costs if you short copper.) Answer:


(a) When When there are no holding holding costs, the forwar forward d price is 1.47e 47e0.06

3/12

×

= 1.4922

which implies that the quote of 1.51 per lb is an overstatement of the price. To take advantage of the opportunity, we go short the forward at 1.51, buy copper spot at 1.47, and borrow 1.47 to finance the copper spot purchase. This leaves a 1.47e 47e0.06 3/12 = zero net cash flow at inception and has a cash inflow of 1.51 − 1. 0.0178 at 0178 at maturity. ×

(b) When When there there are storage storage costs and these are asymmetric, asymmetric, the problem problem is trickier. trickier. If we treat the storage costs as a cost of carry, we arrive at the forward price (1. (1.47 + 0. 0.03)e 03)e0.06

3/12

×

= 1. 1 .5227

This makes it appear that the the given price of 1.51 is ttoo oo low, i.e., that the forward forward is now underpriced , but this is illusory. illusory. If we try implementing the arbitrage arbitrage strategy (long forward, short spot, invest), then we will have a cash outflow at maturity because we do not receive the storage costs when we are short copper. On the other hand, we also cannot create an arbitrage by the opposite strategy (short forward, long spot, borrow): this too leads to a cash outflow at maturity, in this case because we now have to pay storage costs. Thus, the asymmetric nature of storage costs wipes out any perceived arbitrage opportunity. Put differently, it is as if there are two “correct” theoretical arbitragefree forwa forward rd prices: prices: the price is 1.4922 1.4922 if you plan to be long copper copper (and short short spot) and 1.5227 if you plan to be short copper (and long spot).

19. The SPX index is currently trading at a value of $1,265, and the FESX index (the Dow Jones EuroSTOXX Index of 50 stocks, referred to from here on as “STOXX”) is trading at e3,671 3,671.. The dollar dollar interes interestt rate rate is 3% , and and the euro euro intere interest st rate is 5%. The exchange exchange rate is $1.28/eu $1.28/euro. ro. The six-month six-month futures futures on the STOXX is quoted at e3,782. All interest rates are continuously compounded. There are no borrowing costs for securities. (a) Compute the correct six-month forward futures prices of the SPX, STOXX, and the currency exchange rate between the dollar and the euro. (b) Is the futures futures on the STOXX STOXX correctly correctly priced? priced? If not, show how to undertak undertakee an arbitrage strategy assuming you are not allowed to undertake borrowing or lending transactions in either currency. Answer:


(a) The required forwa forward rd rates are: SPX forward price: 1265e 1265e0.03

1/2

×

STOXX forward price: 3671e 3671e0.05

= 1284. 1284.1 1/2

×

(0.03 Currency ($/ e) forward: 1.28e 28e(0.

= 3763. 3763.9

0.05)×1/2

−

= 1. 1 .2673

(b) The STOXX forward contract is quoted at e3782, whereas its correct quote as computed above should be e3763.9. 3763.9. To exploit this error error we undertak undertakee the following arbitrage strategy (sequence of trades), without using borrowing or lending in either currency: At time t time t = = 0: i. Sell the STOXX forward at e3,782. ii. Buy the component stocks of the STOXX spot at e3,671. iii. Short the components components of the SPX spot for for $1,265. Do this for 3.7145 3.7145 contracts contracts (we will see why soon). iv. Convert the $1,265 (for (for 3.7145 contracts) contracts) into euros at the spot exchange rate of $1.28/eur $1.28/euro. o. This results results in 1, 1 , 265 × 3.7145/ 7145/1.28 =e3, 671 which 671 which is exactly what is needed to buy the components of the STOXX above. v. Buy SPX forwar forward d at $1284.1. $1284.1. vi. Book a currency forward to sell 3,782 euros forward at an exchange rate of $1.2673/euro (the fair forward currency rate computed above). Notice the the total cash-flow in both currencies as a result of these six transactions is zero. We now move forward to maturity at the end of six months and examine the net cash-flow cash-flow that is generated. generated. We denote denote the spot value of the U.S. stock index as SPX and that of the euro stock index as STOXX. Below we describe the cash flows from each of the six components of the trading strategy we presented above: i. Close Close out the STOXX forwa forward rd contract contract by delivery delivery of the spot position: position: the cash-flow is 3,782 euros. ii. Buy back the components components of the SPX index index using the forwar forward: d: cash-flo cash-flow w is $3.7145 × 1, 1, 284. 284.10 and 10 and close out the short SPX position. −$3. iii. Sell the 3,782 euros forward, cash-flow is $ 3, 782 × 1. 1.2673 = 4, 4, 792. 792.9. Thus, after all transactions netted off, we are left with a guaranteed gain of $23.1, representing the arbitrage profits.

20. The curre current nt level level of a stock index index is 450. 450. The divide dividend nd yield yield on the index index is 4% (in continuously compounded terms), and the risk-free rate of interest is 8% for six-month

Chapter 4


Chapte Chapterr 4. Pricing Pricing Forw Forwa ards & Futures utures II 1. What is meant by the term “convenience yield”? How does it affect futures prices? Commodities are used in product production ion and gets consumed consumed in the process. process. InAnswer: Commodities ventories of commodities are held by producers because this provides them with the flexibility to alter production schedules or as insurance against a stock-out that could cause cause business business disruptions disruptions.. The value of these options options to consume consume the commodit commodityy out of storage storage ias needed needed is referre referred d to as the commodity commodity convenien convenience ce yield. As a holding benefit, the convenience convenience yield reduces the price of forwards forwards and futures on the underlying.

2. True or false: An arbitrage-f arbitrage-free ree forwar forward d market market can be in backwa backwardat rdation ion only if the benefits of carrying spot (dividends, convenience yields, etc.) exceed the costs (storage, insurance, etc.). Backwardation occurs when the present value of the benefits of carrying carrying Answer: True. Backwardation the physical commodity (including the convenience yield) outweigh the carrying costs.

3. Suppose an active lease market exists for a commodity with a lease rate  expressed in annualize annualized d continuo continuouslyusly-compo compound unded ed terms. terms. Short-se Short-sellers llers can borrow borrow the asset at this rate and investor investorss who are long the asset can lend it out at this rate. Assume Assume the commodit commodityy has no other other cost of carry carry. Modify Modify the argument argumentss in the appendix appendix to the (r )T chapter to show that the theoretical futures price is F = e S . −

Answer: Suppose the forward price tat prevails (denoted, say, F ) is not equal to F . F .

Assume first that F that F < F . F . Consider the following strategy: 

• Take a long forward position at F . This involves no current cash-flow. 

T years; sell the borrowed gold at the spot price of • Borrow e T units of gold for T years; T that given the lease rate  rate ,, the amount of gold S . Cash inflow today: e S . Note that due at maturity T maturity T is is one unit. −

−

of e T S for for maturity at T at • Invest the cash of e T at the interest rate r. r . The net cash-flow cash-flow at inception inception is zero. zero. At date T , T , pay F on the forward, receive one = e (r )T S from unit of gold, and use this to close the gold lease. Receive e rT × e T S = e from the investment. Net cash flow at T : T : 

−

e(r

)T

−

−



− F . S −

By hypoth hypothesi esis, s, this this amoun amountt is positiv positive, e, so we have an arbit arbitrag ragee profit profit.. Simila Similarly rly,, if (r )T F > e S , reversing the above strategy would result in an arbitrage profit. 

−


Re-arranging we have that =  + (1/t (1/t)ln( )ln(F r =  F /S ) So we can see that the new repo rates will be the old repo rates plus 0.005, which gives the two-month and five-month rates as: 034402, r2 = 0.034402,

036789. r5 = 0.036789.

10. Copper is currently trading trading at $1.28/lb. $1.28/lb. Suppose three-month three-month interest rates are are 4% and the convenience yield on copper is c = 3%. c = 3%. (a) What is the range of arbitrage-free forwa forward rd prices possible using S 0 e(r

c)T

≤ F ≤ S 0 erT ?

−

(1)

(b) What What is the lowest lowest value of c c that will create the possibility of the market being in backwardation? Answer:

(a) Plug values values into the equation above, above, i.e., (0.04 1.28e 28e(0.

0.03)×3/12

−

1 .28e 28e0.04 ≤ F ≤ 1.

3/12

×

or 1.2832 ≤ F ≤ 1. 1 .2929 (b) The lowest lowest value of c c to create backwardation is r. r .

11. You are given the following information on forward forward prices prices (gold and silver prices are per oz, copper prices are per lb): Comm Co mmodi odity ty

Spot Spot

Gold Silver Copper

436.4 7.096 1.610

OneOne-mo mont nth h Tw Twoo-mo mont nth h Thre Threee-mo mont nth h SixSix-mo mont nth h 437.3 7.125 1.600

438.8 7.077 1.587

440.0 7.160 1.565

444.5 7.220 1.492


iv. Lease the gold out for 3 months at 1% lease rate. Amount received at end of the lease: 1 oz. v. Borrow Borrow 359. 359.10 for 10 for 3 months at 4%. Amount owed at maturity: 362.71 At maturity, maturity, deliver the 1 oz. of gold received from the lessee to the forward forward contract and receive F = 366. Repay 362.71 on the borrowing. Net cash flow: +3. +3.29. 29. F = 366.

14. A three-month forward contract on a non-dividend-paying asset is trading at 90, while the spot price is 84. (a) Calculate Calculate the implied implied repo rate. (b) Suppose Suppose it is possible possible for you to borro borrow w at 8% for for three three months. months. Does this give rise to any arbitrage opportunities? Why or why not? Answer: The implied repo rate is

r =

1 1 [ln F − ln S ] = [ln90 − ln 84] 84] = 0. 0.27957, 27957, 0.25 T

or 27.98 27.98%. %. Since Since we can borro borrow w at 8% for for three three months, months, there there is a clea clear arbit arbitra rage ge opportunity:

• Sell the forward at 90. • Borrow 84 at 8%. • Buy spot at 84. The net cash-flow at inception is zero. The net cash-flow at maturity is (90 − S T exp(0..08 × 3/ 3/12) + S T 4 .3031 T ) − 84 exp(0 T = 4. which is the difference between the repo rate and market borrowing rate on a base price of 84. To see this, note that 84[exp(r 84[exp(r × 3/ 3/12) − exp(0. exp(0.08 × 3/ 3/12)] = 4. 4.3031. 3031.


Reinvest all dividends dividends into buying buying more of the index. index. Amount Amount of index index held in • Reinvest 3 months: 1 unit. • Borrow 580.613 for three months at 6%. 600. • Take a short position in the forward contract at F = F = 600. At maturity, deliver the unit of the index on the forward contact, and receive 600. Repay Repay the borrowi borrowing: ng: the cash outflow outflow is 580. 580.613 × e0.06 1/4 = 589. 589.39. 39. The net cash flow of +10. +10.61 represents 61 represents arbitrage profits. ×

17. A three month-for month-forwa ward rd contract contract on an index is trading trading at 756 while the index itself is at 750. The three-month interest rate is 6%. (a) What What is the implied implied dividend dividend yield on the index? index? (b) You estimate the dividend yield to be 1% over the next three months. Is there an arbitrage opportunity from your perspective? forward d to spot relations relationship hip for this contract contract is as follows: follows: Answer: The forwar 750 exp[( exp[(00.06 − d)(0. )(0.25)] = 756 Deriving the dividend yield from this results in d = 0.028127. 028127. Now if the dividend is only expected to be 1%, then the current forward price is too low. low. The arbitrag arbitragee strategy is to buy forwa forward, rd, and sell spot. The details details are left as an exercise. Be careful to account for the dividends in the strategy that you create for the risk-less arbitrage.

18. The spot US dollar-eu dollar-euro ro exchange exchange rate is $1.10/eu $1.10/euro. ro. The one-yea one-yearr forwa forward rd exchange exchange rate is $1.0782 $1.0782/eur /euro. o. If the one-ye one-year ar dollar dollar interest interest rate is 3%, then what must be the one-year rate on the euro? Answer: We exploit the following relationship:

exp[rU SD − rEuro] F = S exp[r noting that time is one year. The equation to be solved is: 1.0782 = 1. 1.1000 exp[0 exp[0..03 − rEuro] which means that r Euro = 0.05. 05.


19. You are given information that the spot price of an asset is trading at a bid-ask quote of 80 − 80. 80.5, and the six-month interest rate is 6%. What is the bid-ask quote for the six-month forward on the asset if there are no dividends? There are are two two possible possible forw forward ard-spot -spot arbitrage arbitrage strategies: strategies: one where we buy Answer: There forward (at F a) and sell spot (at S b ), and the other where we sell forward (at F b ) and buy spot (at S (at S a), where the superscripts a and a and b refer to “ask” and “bid,” respectively. For the first strategy to not admit arbitrage profits, we must have F a ≥ e 0.06

1/2

×

82.436 × S b = 82.

For the second strategy not to be an arbitrage, we must have F b ≤ e 0.06

1/2

×

82.952 × S a = 82.

Any pair (F a , F b ) consistent with these inequalities (and, of course, F a ≥ F b ) can be an equilibrium bid-ask pair of forward prices.

20. Redo the previous previous question if the interest interest rate for borrowing borrowing and lending are not equal, equal, i.e., there is a bid-ask spread for the interest rate, which is 6.00–6.25%. Answer: Here, the first arbitrage strategy involves investing the short sale proceeds of

S b at the lending rate of 6.00%, while the second strategy involves borrowing the spot purchase price of S a at the borrowing rate of 6.25%. 25%. These These intere interest st rates rates should should be used in computing the inequalities. Carrying out the computations is left as an exercise.

21. In the previous previous question, what is the maximum bid-ask spread spread in the interest rate market market that is permissible to give acceptable forward prices? Answer: There is none.

22. Stock ABC is trading spot at a price of 40. The one-year forward quote for the stock is also 40. If the one-year interest rate is 4% and the borrowing cost for the stock is 2%, show how to construct a risk-less arbitrage in this stock. Answer: First, we note that the correct forward price should be

40e0.04 F ( F (correct) correct) = 40e

0.02

−

= 40. 40 .808

Since the actual forward price is less than this, it is cheap. Hence we should buy it, and short short the stock. To short short the stock we will need need to borrow borrow it at a cost cost of 2%, but we


will invest the proceeds (40) at 4%. Therein lies the source of the arbitrage profits. All these transactions at inception are net zero in cash-flow. We now examine examine the cash-flo cash-flow w at the end of the year. year. The forward forward position position results in a cash-flow of − 40 on purchase. The short spot position is closed out by delivering the − 40 on stock to the lender. The net flow from the cost of borrow borrowing ing the stock and the gains from lending the sales proceeds of the stock is 2% of $40. Hence, we gain a net amount of 0.808 (= ( = − 40 + 40e 40e0.04 0.02 ). −

23. You are are given two two stocks, A and B. Stock A has a beta of 1.5, and stock B has a beta of 25. The one-year risk-free rate is 2%. Both stocks currently trade at $10. Assume −0.25. a CAPM model where the expected return on the stock market portfolio is 10%. Stock A has an annual dividend yield of 1% and stock B does not pay a dividend. (a) What is the expected return on both stocks? (b) What is the one-year forward price for the two stocks? (c) Is there an arbitrage? Explain. Answer: (a) We may use the CAPM to determine the expected return on both stocks,

which are as follows. Stock A: 1.5[0. 5[0.10 − 0. 0.02] = 0. 0.14 E (rA ) = 0.02 + 1. Stock B: 0 .02 − 0. 0.25[0. 25[0.10 − 0. 0.02] = 0. 0.0 E (rB ) = 0. (b) The forward price for stock A is 10e0.02 F A = 10e

0.01

−

= 10. 10 .101

The forward price for stock B is 10e0.02 F B = 10e

0.0

−

= 10. 10 .202

(c) There is no arbitrage even though stock B has a forward price greater than that of stock A even though though its expected expected return and dividend dividend is zero. zero. The forwar forward d price is based on a mathematical relationship between spot prices and interest rates, and does not have any relation to the expected growth rate of the stock.

Chapter 5


5. In the presence of basis risk, is a one-for-one hedge, hedge, i.e., a hedge ratio of 1, always always better than not hedging at all? hedge and the underlying spot values are are Answer: Basis risk arises from the fact that the hedge not perfectly perfectly correl correlated ated.. Depending Depending on this correlatio correlation, n, it may may be better to not hedge than to hedge one-for-one. For example, this is certainly the case if price changes in the underlying spot asset and the hedge have no correlation with each other at all: in such a situation, hedging only results in additional uncertainty. For a more precise statement on when not hedging may be superior to hedging one-for-one, see Section 5.5.

6. If the correlation between spot and futures price changes is ρ = 0.8, what fraction of cash-flow uncertainty is removed by minimum-variance hedging? removed Answer: As shown in Section 5.5, the fraction of unhedged cash flow variance removed by the minimum-variance hedge is ρ 2 , which in this case is 0.64 or 64%. 7. The correlation correlation between changes changes in the price of the underlying and a futures contract is +80%. +80%. The same underlying is correlated correlated with another futures contract contract with a (negative) correlation of −85%. 85%. Which Which of the two two contract contractss would would you prefer prefer for the minimumminimumvariance variance hedge? Answer:

The second one. As shown in Section 5.5, the fraction of unhedged cash flow variance removed by the minimum-variance hedge is ρ2 , where ρ is the correlation of the spot price price chang changes es and and price price changes changes in the futures futures contr contrac actt used used for for hedgi hedging. ng. Since Since ρ2 increases as | ρ| increases, we should use a futures contract with the highest value of | ρ|. The only impact of the negative correlation correlation is that the sign of the futures position gets reversed, i.e., we hedge a long spot exposure (a commitment to buy spot at maturity T ) T ) with a short futures position and a short spot exposure (a commitment to sell spot at date T date T )) with a short futures position. 8. Given the following information on the statistical properties of the spot and futures, 25, ρ = 0.96. 96. compute the minimum-variance hedge ratio: σS = 0.2, σF = 0.25, Answer: The minimum-variance hedge ratio is ∗

h = ρ

0.2 σ(∆S ) = 0. 0 .96 × = 0. 0 .768. 768. 0.25 σ(∆F )

This means to hedge a long spot exposure of size Q (i.e., to hedge a commitment to buy Q buy Q units spot at date T ), T ), we use long futures contracts of size 0.768 Q units.


9. Assume that the spot position comprises comprises 1,000,000 1,000,000 units in the the stock index. If the hedge hedge ratio is 1.09, how many units of the futures contract are required to hedge this position? Answer: Note that the optimal hedge ratio is h∗ = H/Q, H/Q, where H is H is the number of

units of the hedge, and Q is the number number of units of the spot positio position. n. Hence Hence,, the required number of units in futures is ∗

1, 000, 000, 000 = 1, 1, 090, 090, 000. 000. H = h × Q = 1.09 × 1, In words, we enter into a futures contract that calls for the delivery of 1,090,000 units of the asset underlying the futures contract.

10. You have a position in 200 shares of a technology stock with an annualized standard deviation of changes in the price of the stock being 30. 30 . Say that you want to hedge this position position over a one-yea one-yearr horizo horizon n with a technolo technology gy stock index. index. Suppose Suppose that the index value has an annual standard deviation of 20. 20. The correlatio correlation n between between the two two annual annual changes is 0 is 0..8. How many units of the index should you hold to have the best hedge? 30, σ(∆F ) = 20, 20, Answer: In the notation of the chapter, we are given that σ (∆S ) = 30, and ρ and ρ = = 0.8. So the minimum-variance hedge ratio is ∗

h = ρ

σ(∆ σ (∆S ) = 0.8(30/ 8(30/20) = 1. 1.20 σ(∆F )

Hence, you need to short 1.2 × 200 = 240 units 240 units of the index to set up the hedge.

11. You are a portfolio manager looking to hedge a portfolio daily over a 30-day horizon. Here are the values of the spot portfolio and a hedging futures for 30 days.


Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Spot Futures 80.000 81.000 79.635 80.869 77.880 79.092 76.400 77.716 75.567 77.074 77.287 78.841 77.599 79.315 78.147 80.067 77.041 79.216 76.853 79.204 77.034 79.638 75.960 78.659 75.599 78.549 77.225 80.512 77.119 80.405 77.762 81.224 77.082 80.654 76.497 80.233 75.691 79.605 75.264 79.278 76.504 80.767 76.835 81.280 78.031 82.580 79.185 84.030 77.524 82.337 76.982 82.045 76.216 81.252 76.764 81.882 79.293 84.623 78.861 84.205 76.192 81.429

Carry out the following analyses using Excel: (a) Compute Compute σ (∆S ), σ(∆F ), and ρ and ρ.. σ(∆ (b) Using the results from (a), compute compute the hedge ratio you would would use. (c) Using this hedge ratio, ratio, calculate calculate the daily change change in value value of the hedged hedged portfolio. portfolio. (d) What What is the standard standard deviation deviation of changes in value of the hedged hedged portfolio? portfolio? How How does this compare to the standard deviation of changes in the unhedged spot position?

Sundaram & Das: Derivatives - Problems and Solutions ................................ 55 Answer: The results are presented in the following tables. The first step is to compute

the covarian covariance ce matrix of the changes changes in spot and futures, as follows. follows. Using Excel, Excel, we obtain: Covariance Matrix ∆S ∆F ∆S 1.276 ∆F 1.308 1.308 1.41 1.4155 Using these numbers to compute the correlation ρ and the hedge ratio h , we obtain: ∗

ρ = 0.9732 ∗

h = 0.9732 ×

1. 1 .276 = 0.9244 1.308

Using a hedge ratio of h , we can calculate the daily changes (“P&L”) in the value of the hedged portfolio. For example, on day 1, this P&L is ∗

(79. (79.635 − 80) − [0. [0.9244 × (80. (80.869 − 81)] = −0.243 The following table summarizes these numbers:

Sundaram & Das: Derivatives - Problems and Solutions ................................ 56 Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Spot 80.000 79.635 77.880 76.400 75.567 77.287 77.599 78.147 77.041 76.853 77.034 75.960 75.599 77.225 77.119 77.762 77.082 76.497 75.691 75.264 76.504 76.835 78.031 79.185 77.524 76.982 76.216 76.764 79.293 78.861 76.192

Futures 81.000 80.869 79.092 77.716 77.074 78.841 79.315 80.067 79.216 79.204 79.638 78.659 78.549 80.512 80.405 81.224 80.654 80.233 79.605 79.278 80.767 81.280 82.580 84.030 82.337 82.045 81.252 81.882 84.623 84.205 81.429

∆S

∆F

P&L

-0.365 -1.756 -1.479 -0.834 1.721 0.312 0.547 -1.106 -0.188 0.180 -1.074 -0.361 1.626 -0.106 0.643 -0.681 -0.585 -0.805 -0.427 1.240 0.330 1.196 1.153 -1.661 -0.541 -0.766 0.548 2.529 -0.432 -2.669

-0.131 -1.777 -1.376 -0.642 1.768 0.474 0.752 -0.851 -0.012 0.434 -0.979 -0.111 1.964 -0.107 0.820 -0.571 -0.420 -0.629 -0.327 1.488 0.513 1.300 1.450 -1.693 -0.292 -0.793 0.629 2.742 -0.419 -2.776

-0.243 -0.113 -0.207 -0.241 0.087 -0.126 -0.148 -0.319 -0.176 -0.221 -0.169 -0.258 -0.189 -0.007 -0.114 -0.153 -0.196 -0.224 -0.125 -0.136 -0.144 -0.005 -0.187 -0.096 -0.271 -0.033 -0.034 -0.006 -0.045 -0.103

The P&L has a variance of 0.009 which is less than 1% of the variance of the unhedged position of 1.276.

12. Use the same data as presented above to compute the hedge ratio using regression analysis, again using Excel. Explain why the values are different from what you obtained above. Excel of daily changes changes δ S on δ F following results. Answer: The regression in Excel S on δ F produces the following 0.947 δ 947 δ F δ S S = −0.14 + 0. F


Finally, the current USD/SEK spot rate is 0.104, the current three-month USD/EUR forward rate is 1.071, and the current three-month USD/CHF forward rate is 0.602. (a) Which currency should should the company use for hedging purposes? (b) What What is the minimum-v minimum-var arianc iancee hedge position? position? Indicate Indicate if this is to be a long or short position. Answer:

(a) Since the correlation of changes in the spot USD/SEK is higher with changes in forward USD/EUR than with changes in forward USD/CHF, the hedge will be better if the USD/EUR forward is used for hedging. (b) The optimal hedge ratio is ∗

h = ρ ×

0. 0 .007 σS = 0. 0 .90 × = 0.35 0.018 σF

Since the correlation of USD/SEK and USD/EUR is positive, appreciation in the SEK should mostly be offset by appreciation in the EUR. Hence, the hedge position should be a long USD/EUR forward contract calling for the delivery of EUR ( 0.35 × 100) 100) million = EUR 35 million.

14. You use silver silver wire wire in manuf manufact acturi uring. ng. You are looking looking to buy 100,00 100,0000 oz of silver silver in three months’ time and need to hedge silver price changes over these three months. One COMEX silver futures contract is for 5,000 oz. You run a regression of daily silver spot price changes on silver futures price changes and find that 0.89δ 89δ F δ s = 0.03 + 0. F +  What should be the size (number of contracts) of your optimal futures position. Should this be long or short? regression, the optimal hedge ratio is 0.89, so the size si ze of the required Answer: From the regression, futures position is 89,000 oz or 89 8 9, 000/ 000/5, 000 = 17. 17.8 contracts. This should be a long position.

15. Suppose you have the following information: ρ = 0.95, 95, σS = 24, 24, σF = 26, 26, K = 90, 90, 00018. What is the minimum-variance tailed hedge? R = 1.00018.


(a) Ambiguou Ambiguous. s. The question question does not indicate indicate if profits profits on the hedge position are are more likely if the term-structure is upward sloping. (b) Again, Again, ambiguous. ambiguous. The forwa forward rd price price is convex convex in the interest interest rate, but the spot price too may be correlated with interest rates, so without more information on the nature of the underlying, it is not clear how the forward/futures price behaves when interest rates become more volatile. (c) Ambiguous. The performance performance of the hedge depends depends only on the correlation correlation between between price changes in the instrument used for hedging and price changes in the exposure being hedged, and not on the variances of price changes of the instruments used for hedging. However, if we add some more conditions, we can provide a qualified answer. For example, the size of of the optimal hedge depends on the standard deviation of price changes of the contract used for hedging, and decreases as this standard deviation increase increases. s. So if the correlat correlations ions are the same for both hedging hedging instruments, instruments, the contract with greater volatility may be preferable since fewer positions are needed in an optimal hedge and this may reduce transactions costs. (d) Here, Here, the answer answer is unambigu unambiguous. ous. You want the correlatio correlation n of spot to futures futures to be higher,as the hedged position will have a lower variance.

23. You are trying to hedge the sale of a forward forward contract on a security A. A. Suggest a framework you might use for making a choice between the following two hedging schemes: (a) Buy a futures futures contract contract B that is highly correlated with security A but trades very infrequently. Hence, the hedge may not be immediately available. (b) Buy a futures contract C C that is poorly correlated with A but trades more frequently. Answer: The question requires you to use your imagination to develop a model to

trade trade off liquid liquidit ityy risk risk agains againstt basis basis risk. risk. Here Here is one possible possible way way to approa approach ch the problem (obviously not the only one). Suppose we denote the probability of being able to implement the hedge B by p by p.. The probability of A remaining A remaining unhedged is then 1 − p. p. The variance of the unhedged position is σ 2 (∆A (∆A). The variance of the hedged position 2 2 is σ is σ (∆A (∆A)(1 − ρAB ), where ρ where ρ AB is the correlation between changes in A and A and B B.. Hence, the expected variance of the hedged position when B is used is 2 (∆A)(1 − ρAB ) + (1 − p) (∆A). p σ 2 (∆A p)σ2 (∆A

If C is C is used to hedge, the variance of the hedged position is 2 (∆A)(1 − ρAC ) σ 2(∆A


If we care about only the expected variance of the hedged portfolio, then, depending on which of these values is computed to be lower, the hedge instrument may be chosen according accordingly ly.. If the latter is lower, lower, choose C , and if the former is lower, choose B . In particular, the first alternative is preferred when 2 2 (∆A)(1 − ρAB ) + (1 − p) (∆A)σ2 (∆A (∆A)(1 − ρAC ) p σ 2 (∆A p)σ2 (∆A

24. Dow Downloa nload d data from the Web Web as instructed instructed below below and answer answer the questions questions below: below: (a) Extract one year’s data on the S&P 500 index from finance.yahoo.com . Also Also download corresponding period data for the S&P 100 index. (b) Downloa Download, d, for the same period, data on the three-mont three-month h Treasury reasury Bill rate (constant maturity) from the Federal Reserve’s Web page on historical data: www.federalreserve.gov/releases/h15/data.htm . (c) Create a data series of three-month forwa forwards rds on the S&P 500 index using the index data and the interest rates you have already extracted. Call this synthetic forward data series F series F .. (d) How would would you use this synthetic forwards forwards data to determine the tracking error of a hedge hedge of threethree-mon month th maturi maturity ty position positionss in the S&P 100 index? index? You need to think (a) about how to set up the time lags of the data and (b) how to represent tracking error. exercise is left to the reader. reader. For For the definition definition of “tracking “tracking error,” error,” see Answer: This exercise the answer to the next question.

25. Explain Explain the relationsh relationship ip between between regressio regression n R2 and and trac tracki king ng erro error of a hedg hedge. e. Use Use the data collected in the previous question to obtain a best tracking error hedge using regression. Answer: To answer this question, one must first define “tracking error” (the question

deliberate deliberately ly leaves leaves this undefined undefined). ). The intuitive intuitive definition definition of tracking tracking error error is also the most commonly used one: tracking error is the standard deviation (or the variance) of the difference between a target performance and the actual performance. Suppose we run a regression to determine the hedge ratio to be implemented: δ S S = a + bδ F F + 

Chapter 6


Chapte Chapterr 6. Intere Interest st Rate Rate Forwa rwards & Futures utures 1. Explain the difference between between the following terms: terms: (a) Payoff to an FRA. (b) Price of an FRA. (c) Value of an FRA. Answer: FRA terminology:

(a) The payoff payoff from an FRA is the dollar amount received at maturity maturity of the FRA. For For example, if we are long an FRA at a strike interest rate of 10% and the rate at maturity of the FRA is 11%, our payoff will be based on the interest difference of 1% applied to the notional principal of the contract for the borrowing period. (b) The “price” “price” of an FRA refers to the fixed fixed rate locked in using the FRA. At At inception of the FRA, this fixed rate is chosen so that the FRA has zero value to both parties. “Pricing” an FRA refers to the identification of this fixed rate. (c) Value Value is the net payment payment that that would have have to be made if the FRA were were to be closed closed out today. At inception, the FRA has (by construction) zero value to both parties. But as time progresses, the fixed rate in the FRA will generally differ from the price of a new FRA (i.e., from the fixed rate that makes an FRA with the same maturity as the original one have zero value to both parties), so the FRA can have positive or negative value.

2. What characteristic of the eurodollar futures contract enabled it to overcome the settlement obstacles with its predecessors? settlement. Earlier Earlier attempts attempts at developin developingg an interest-rate interest-rate futures futures conAnswer: Cash settlement. tract based on commercial borro b orrowing wing rates had floundered because they required physical settlement at maturity, but the deliverable instruments in these contracts lacked homogeneity geneity because because of perceived perceived difference differencess in the credit risk of the issuing issuing entities. The eurodollar futures contract solved this by using cash settlement, an idea that was rapidly adopted in other contracts which had difficulties with physical settlement (e.g., stock index futures).

3. How are are eurodollar futures futures quoted? Eurodolla llarr futur futures es contr contract actss are are instru instrume ments nts that that enable enable trade traders rs to lock in Answer: Eurodo a Libor rate for a three-month period beginning on the expiry date of the contract. Howev However, er, eurodollar eurodollar futures futures are quoted not as rates rates but as prices. prices. The price quoted quoted is


100 minus the three-month Libor rate, with the rate being expressed as a percentage (not in decimal form). So if the Libor rate is 3.18%, the futures price quoted is 96.82.

4. It is currently currently May May. What What is the relation relation between between the observed observed eurodollar eurodollar futures futures price price of 96.32 for the November maturity and the rate of interest that is locked-in using the contract? Over what period does this rate apply? relation betw b etween een the futures price price and rate of interest interest that gets locked locked in Answer: The relation via the contract is 100 − 96. 96.32 = 3. 3.68% The interest rate applies to a 90-day borrowing or investment beginning at maturity of the futures contract, i.e., beginning in November.

5. What is the price tick in the eurodollar futures contract? To what price move does this correspond? Answer: The price tick in the eurodollar futures contract is 0.01 (which corresponds

to a move move in the implie implied d interest interest rate rate of 1 basis basis point). point). The price price tick tick has has a dolla dollarr value of $25. The minimum price move on the expiring eurodollar futures contract (the one currently currently nearest nearest to maturity) maturity) is 1/4 tick or a dollar dollar value of $6.25. $6.25. On all other other eurodollar futures contracts, it is 1/2 tick (or $12.50).

6. What are the gains or losses to a short position in a eurodollar futures contract from a 0.01 increase in the futures price? Answer: There will be a loss of $25.

7. You enter into a long eurodollar futures futures contract at a price of 94.59 94.59 and exit the contract a week later at a price of 94.23. What is your dollar gain or loss on this position? Answer: A increase of 0.01 in the price corresponds to a margin account change of

$25 (gain (gain for the long, loss for the short). short). In this case, the price falls by 0.36, 0.36, which corresponds to a loss of (36 × $25) = $900 for $900 for the long position.

8. What What is the cheapest cheapest to deliver deliver in a Treasur Treasuryy bond futures futures contract? contract? Are there other other delivery options in this contract? Answer: The standard bond in a Treasury contract is one with a coupon of 6% and at

least 15 years years to maturit maturityy or first call. The main delivery delivery option option in the Treasu Treasury ry bond


However, this rate is negative, and is hence not reasonable for a nominal interest rate. It is likely that rates such as these occur when there is an imbalance between demand and supply for money, or the bid/ask spreads are too large to allow someone to arbitrage the negative forward rate because there are no lenders for the forward period.

18. If you expect interest rates to rise over the next three months and then fall over the three months succeeding that, what positions in FRAs would be appropriate to take? Would your answer change depending on the current shape of the forward curve? three months and you wish to speculate speculate Answer: If rates are going to rise over the next three on this view, then you can lock-in a rate today using a FRA for borrowing in three months and in three months’ time, you can invest the borrowed amount at the higher interest interest rates that prevai prevaill then. then. Similarly Similarly,, if, in three three months’ months’ time, rates are going to fall over the next three months, you can enter into a short FRA at that time to speculate on your views.

19. A firm plans to borrow borrow money over the next two half-ye half-year ar periods, periods, and is able to obtain obtain a fixed-r fixed-rate ate loan at 6% per annum. annum. It can also also borro borrow w money money at the floating floating rate of Libor + 0.5%. Libor is currently at 4%. If the 6 × 12 FRA 12 FRA is at a rate of 6%, find the cheapest financing cost for the firm. simplicity,, we treat each six-month six-month period as exactly half a year. year. If the Answer: For simplicity fixed fixed rate rate loan loan is taken, taken, the cost cost of financin financingg is 3% each each half year. year. That That is, the firm receives receives 100 today, today, pays pays 3 in six months months and 103 in one year at maturity maturity. It is easy to see that the internal rate of return of this sequence of cash-flows is exactly 6%. Suppose the firm elects to go for the second alternative, i.e., takes a floating-rate loan and simultaneously enters into a long 6 × 12 FRA. 12 FRA. Then, the cost of financing is 4.5% for the first six months and 6.5% 6.5% for the next six months . (Note (Note that in the second second period, the cost of financing to the firm is  + 0.5% plus 5% plus the payoff to the FRA is 6 − . Hence, net this is 6.5%). So the cash-flows in this second financing are: {−100, 100, 2.25, 25, 103. 103.25} at times 0, 0.5 and 1 years. The internal rate of return of this sequence of cash-flows is 5.48%. A comparison of the internal rates of return indicates that the second option is cheaper.

20. You enter into an FRA of notional 6 million to borrow on the three-month underlying Libor Libor rate six months from now now and lock in the rate of 6%. At the end of six months, months, if the underlying three-month rate is 6.6% over an actual period of 91 days, what is your your payoff payoff given that the payment payment is made made right right away away?? Recall Recall that the Actual/36 Actual/3600 convention applies.


3

FRA Settlement Value 2

1

0 0

5

10

15

20

Libor (% -1

-2

-3

The plot looks almost linear, but it is actually concave (shaped like an inverted bowl). Concavity is equivalent to having a negative second derivative and it is easily checked that is, in fact, the case: ∂ 2s < 0. ∂ 2

23. You anticipate a need to borrow borrow USD 10 million in six-months’ time for a period of three months. months. You decide decide to hedge hedge the risk of interest-rate interest-rate changes changes using using eurodolla eurodollarr futures futures contracts. Describe the hedging strategy you would follow. What if you decided to use an FRA instead?

Sundaram & Das: Derivatives - Problems and Solutions ................................ 78 Answer: A simple way to hedge interest rate changes over the next six months is to

enter into a long 6 × 9 FRA. This is a clean and exact exact hedge. hedge. Howev However, er, we may also use eurodollar futures. To hedge borrowing costs, we need to make money on the hedge when interest rates rise (since we will be paying more on the borrowing in this event) and vice versa. When interest rates rise, eurodollar futures prices fall, so to make money when interest rates rise, we need to short eurodollar futures contracts. Finally, since one eurodollar futures contract is for a notional value of 1 million, we need to short 10 of these contracts. As noted in the text, however, eurodollar futures are settled in undiscounted form, so unlike using an FRA, the hedge obtained with eurodollar futures will not be perfect even in theory.

24. In Question 23, suppose that the underlying three-month three-month Libor rate after six months (as implied by the price of the eurodollar futures contract expiring in 6 months) is currently at 4%. Assume that the three-month period has 90 days in it. Using the same numbers from Question 23 and adjusting for tailing the hedge, how many futures contracts are needed? Assume fractional contracts are permitted. without tailing tailing the hedge, hedge, we need a short short position in 10 contracts. contracts. Answer: As noted, without If the hedge is tailed using the 4% rate reflected in current eurodollar prices, then the number of contracts needed is 10 = 9.901. 901. 1 + 0. 0 .04(90/ 04(90/360)

25. Using the same numbers as in the previous two questions, compute the payoff after six months (i.e., at maturity) under (a) an FRA and (b) a tailed eurodollar futures contract if the Libor rate at maturit maturityy is 5%, and the lockedlocked-in in rate in both both cases cases is 4%. Also compu compute te the payo payoffs ffs if the Libor Libor rate rate ends ends up at 3%. Co Comme mment nt on the differen difference ce in payoffs of the FRA versus the eurodollar futures. Answer: First suppose that the Libor rate at maturity is 5%.

(a) For For the FRA, the payoff payoff is: 10, 10, 000, 000, 000 ×

(0. (0 .05 − 0. 0.04) × (90/ (90/360) = 24, 24, 752. 752.50. 50. 1 + 0. 0 .05(90/ 05(90/360)

(b) For For the tailed eurodollar eurodollar futures, futures, the payoff payoff is 9.901 × (0. (0.05 − 0. 0.04) × 10, 10, 000 × 25 = 25, 25, 024. 024.75 Now suppose the Libor rate at maturity is 3%.


(a) For For the FRA, the payoff payoff is: 10, 10, 000, 000, 000 ×

(0. (0 .03 − 0. 0.04) × (90/ (90/360) = −24, 24, 813. 813.90 1 + 0. 0 .03(90/ 03(90/360)

(b) For For the eurodollar eurodollar futures, futures, the payo payoff ff is: 9.901 × (0. (0.03 − 0. 0.04) × 10000 × 25 = − 24, 24, 752. 752.50 In either case, the eurodollar futures contract does better than the FRA: in the first case, it results in a larger cash inflow, and in the second case in a smaller cash outflow. This is the “convexity bias” discussed in the chapter.

26. The “standa “standard rd bond” in the Treasur Treasuryy bond futures contract contract has a coupon of 6%. If, instead, delivery is made of a 5% bond of maturity 18 years, what is the conversion factor factor for for settlement settlement of the contract? contract? Assume Assume that the last coupon on the bond was just paid. principal amount of $100. $100. Then, the bond pays 2.50 2.50 every six months Answer: Assume a principal and also repays the principal after 18 years. The present value of these cash flows when discounted at the 6% standard rate is 2.5 2.5 2.5 102. 102.5 + + + + = 89. 89 .946 · · · 1.03 1.032 1.0335 1.0336 Hence, the conversion factor is 0.89946.

27. Suppose Suppose we have have a flat yield yield curve curve of 3%. What What is the price price of a Treasur reasuryy bond of remaining remaining maturity maturity seven years years that pays a coupon coupon of 4%? (Coupons (Coupons are paid semiansemiannually nually.) .) What What is the price price of a six-month six-month Treas Treasury ury bond futures futures contract? contract? Make Make any assumption you require concerning the maturity of the delivered bond to find this price. seven-year Answer: Assuming the last coupon was just paid, the price of a the remaining seven-year maturity bond with coupon of 4% is equal to the cash flows from the bond discounted at a flat 3% rate (i.e., at 1.5% every six months): 2 2 2 102 + + + + = 106. 106.27. 27. · · · 1.015 1.0152 1.01513 1.01514 A Treasury bond futures contract requires the delivery of a Treasury bond with coupon of 6% and any maturity maturity of at least 15 year years. s. To price the six-month six-month futures contract, contract, we (i) treat it as a forward contract, and (ii) assume that the maturity of the delivered

Derivateives Solutions Ch1-6

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