Solutions Ch1 6

Chapter 1

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

5. Define a forward contract. Explain at what time are cash flows generated for thiscontract. How is settlement determined?

Answer: A forward contract is an agreement to buy or sell an asset at a future date(denoted T ), at a specified price called the delivery price (denoted F ). Denote the initialdate (the inception date or the date of the agreement) by t = 0. At inception there areno cash flows on a forward contract. At maturity, if the then-prevailing spot price ST ofthe underlying asset is greater than F , then the buyer (the long position) has gainedST F via the forward while the seller (the short position) has correspondingly lostST F . Depending on contract specifications, the settlement may either be in cash(the seller pays the buyer ST F ) or physical (the seller delivers the asset and receivesF ). If ST < F , the buyer loses F ST and the seller gains this quantity.

6. Explain who bears default risk in a forward contract.

Answer: Default arises if, at maturity, one of the parties fails to fulfill their obligationsunder the contract. Default risk only matters for the party that is in the money atmaturity, that is, that stands to profit at the locked-in price in the contract. (If thespot price at maturity is such that a party would lose from performing on the obligationin the contract, counterparty default is not a problem.) Prior to maturity, since eitherparty may finish in-the-money, both parties are exposed to default risk.

7. What risks are being managed by trading derivatives on exchanges?

Answer: An important one is counterparty default risk. In a typical futures exchange,the exchange interposes itself between buyer and seller and guarantees performanceon the contract. This reduces significantly the default risk exposure of both parties.Further, daily settling of marked-to-market gains and losses ensures that the loss to theexchange from an investors default is limited to at most one days settlement amount(and because of maintenance margins is usually less than even this; see Chapter 2 for adescription of the margining process).

8. Explain the difference between a forward contract and an option.

Answer: A forward contract is an agreement to buy or sell an asset at a future date(denoted T ), at a specified delivery price (denoted F ). The agreement is made at timet = 0 for settlement at maturity 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 beforea specified maturity date T . In comparing a long forward contract to a call option, themain difference lies in the fact that the forward buyer has to buy the stock at the forwardprice at maturity, whereas in a call option, the buyer is not required to carry out the


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

9. What is the difference between value and payoff in the context of derivative securities.

Answer: The value of a derivative is its current fair price or its worth. The payoff (orpayoffs) refers to the cash flows generated by the derivative at various times during itslife. For example, the value of a forward contract at inception is zero: neither party paysanything to enter into the contract. But the payoffs from the contract at maturity toeither party could be positive, negative, or zero depending on where the spot price ofthe asset is at that point relative to the locked-in delivery price.

10. What is a short position in a forward contract? Draw the payoff diagram for a shortposition at a forward price of $103, if the possible range of the underlying stock price is$50-150.

Answer: A short position in a forward is where you are the seller of the forward contract.In this case, you gain when the price of the underlying asset at maturity is below thelocked-in delivery price. The payoff diagram for this contract is as shown in the followingpicture. When the price of the stock at maturity is the delivery price of $103, there areneither gains nor losses.

A Short Forward Contract's Payoff

-60

-40

-20

0

20

40

60

50 60 70 80 90 100 110 120 130 140 150

Stock Price

Payo

ff

11. Forward prices may be derived using the notion of absence of arbitrage, and marketefficiency is not necessary. What is the difference between these two concepts?

Answer: Absence of arbitrage means that a trading strategy cannot be found that createscash inflows without any cash outflows, i.e., creates something out of nothing. Efficiency,


as that term is used by financial economists, implies more: not only the absence ofarbitrage but that asset prices reflect all relevant information.

12. Suppose you are holding a stock position, and wish to hedge it. What forward contractwould you use, a long or a short? What option contract might you use? Comparethe forward versus the option on the following three criteria: (a) uncertainty of hedgedposition cash-flow, (b) Up-front cash-flow and (c) maturity-time regret.

Answer: If a forward contract is to be used, then a short forward is required. Alterna-tively, a put option may also be used. The following describes the three criteria for thechoice 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 un-desirable, the option need not be exercised. With the futures contract there is apossible downside.

13. What derivatives strategy might you implement if you expected a bullish trend in stockprices? Would your strategy be different if you also forecast that the volatility of stockprices will drop?

Answer: If you expect prices to rise, there are several different strategies you couldfollow: you could go long a forward and lock in a price today for the future 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 ifprices trend upward as you expect.

The volatility issue is a bit trickier. As we explain in Chapter 7, both call and put optionsincrease in value with volatility, so if you expect volatility to decrease, you do not wantto buy a call: when volatility drops, what you have bought automatically becomes lessvaluable.

14. What are the underlyings in the following derivative contracts?

(a) A life insurance contract.

(b) A home mortgage.

(c) Employee stock options.

(d) A rate lock in a home loan.


Answer: The underlyings are as follows:

(a) A life insurance contract: the event of ones demise.

(b) A home mortgage: mortgage interest rate.

(c) Employee stock options: equity price of the firm.

(d) A rate lock in a home loan: mortgage interest rate.

15. Assume you have a portfolio that contains stocks that track the market index. You nowwant to change this portfolio to be 20% in commodities and only 80% in the marketindex. How would you use derivatives to implement your strategy?

Answer: One would use futures to do so. We would short market index futures for20% of the portfolios value, and go long 20% in commodity futures. A collection ofcommodity futures adding up to the 20% would be required.

16. In the previous question, how do you implement the same trading idea without usingfutures contracts?

Answer: Futures contracts are traded on exchanges and are known as exchange-tradedsecurities. An alternative approach to achieving the goal would be to use an over-the-counter or OTC product, for example, an index swap that exchanges the return on themarket 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 indexpositive or negative? If the nominal value of the contract is $100,000 and you arerequired to post $10,000 as margin, how much leverage do you have?

Answer: The futures contract is positively correlated with the stock index. The leverageis 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. What is the closing out of a position in futures markets? Why is closing out ofcontracts permitted in futures markets? Why is unilateral transfer or sale of the contracttypically not allowed in forward markets?

Answer: To close out a position in a futures market, an investor must take an offsettingopposite position in the same contract. (For example, to close out a long position in 10S&P 500 index futures contracts with expiry in March, an investor must take a shortposition in 10 S&P 500 index futures contracts with expiry in March.) Once a positionis closed out, the investor no longer has any obligations remaining.

Credit risk is key to allowing investors to close out contracts. In a futures exchange, theexchange interposes itself between buyer and seller as the guarantor of all trades; thus,there is little credit risk involved. In forward markets, allowing investors to unilaterallytransfer 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) thecontract may be unwound with the same counterparty or (b) the contract may be offsetby entering into an equal and opposite contract with a third party. The latter is theanalog of the unilateral close-out of futures contracts. However, while close-out of thefutures contract leaves the investor with no net obligations, offset of a forward contractleaves the investor with obligations on both contracts.

17. An investor enters into a long position in 10 silver futures contracts at a futures price of$4.52/oz and closes out the position at a price of $4.46/oz. If one silver futures contractis for 5,000 ounces, what are the investors gains or losses?

Answer: Effectively, the investor buys at $4.52 per oz and sells at $4.46 per oz, so takesa loss of $0.06 per oz. Per contract, this amounts to a loss of (5000 0.06) = $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 istraded at the begining of a trading session. The closing price is the last price at whichthe contract is traded at the close of a trading session. The settlement price is a pricechosen 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 tosettle gains and losses from futures trading and to invoice deliveries.

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


(a) What is the initial size of the margin account?

(b) Suppose the futures settlement price on the first day is $278.00 per oz. What isthe new balance in the margin account? Does a margin call occur? If so, assumethat the account is topped back to its original level.

(c) The futures settlement price on the second day is $281.00 per oz. What is the newbalance in the margin account? Does a margin call occur? If so, assume that theaccount 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 interest costs, what are his total gains or losses?

Answer: Futures position: short 10 contractsSize of one contract: 100 ozInitial margin per contract: $1,500Maintenance margin per contract: $1,100Initial futures price: $276.50 per oz

(a) Initial size of margin account = 1, 500 10 = 15, 000.(b) If the settlement price is $278 per oz, the short position has effectively lost $1.50

per oz. This is a loss of 1.50 100 = 150 per contract. Since the position has10 contracts, the overall loss is 150 10 = 1, 500. Thus, the new balance in themargin account is 15, 000 1, 500 = 13, 500. A margin call does not occur sincethis new balance is larger than the maintenance margin of $11,000.

(c) When the settlement price moves to $281 per oz, the short position effectivelyloses another $3 per oz. The loss per contract is 3 100 = 300, so the overallloss is 300 10 = 3, 000. Thus, the balance in the margin account is reducedto 13, 500 3, 000 = 10, 500. Since this is less than the maintenance margin, amargin call occurs. Assume the account is topped back to $15,000.

(d) When the position is closed out at $276 per oz, the short position makes a gain of281276 = 5 per oz. This translates to a gain of 500 per contract, and, therefore,to an overall gain of 5,000. Thus, the closing balance in the margin account is15, 000 + 5, 000 = 20, 000.

(e) The investor began with a margin account of $15,000, and deposited another$4,500 to meet the margin call, for a total outlay of $19,500. Since the marginaccount balance at time of close out is $20,000, his overall gain (ignoring interestcosts) is $500.

20. The current price of gold is $642 per troy ounce. Assume that you initiate a long positionin 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 initialprice. Assume the following evolution of gold prices over the next five days, and computethe margin account assuming that you meet all margin calls.

Date Price per Ounce

7-Jul-06 642

8-Jul-06 640

9-Jul-06 635

10-Jul-06 632

11-Jul-06 620

12-Jul-06 625

Answer: The initial margin is $321, and the maintenance margin is $193. The followingis the evolution of the margin account. Note that there is one margin call that takesplace on 11-July-2006.

Initiation Price = 642Initial Margin (5%) = 321Maintenance Margin (3%) = 192.6Number of contracts = 10

Margin AccountOpening Daily Profit Adjusted Margin Call Closing

Date Gold Price Balance and Loss Balance Deposit Balance

7-Jul-06 6428-Jul-06 640 321 -20 301 0 3019-Jul-06 635 301 -50 251 0 251

10-Jul-06 632 251 -30 221 0 22111-Jul-06 620 221 -120 101 220 32112-Jul-06 625 321 50 371 0 371

21. When is a futures market in backwardation? When is it in contango?

Answer: A futures market is said to be in backwardation if the futures price is less thanthe spot price. It is in contango if futures price is above spot.

22. Suppose there are three deliverable bonds in a Treasury Bond futures contract whosecurrent cash prices (for a face value of $100,000) and conversion factors are as follows:

(a) Bond 1: Price $98,750. Conversion factor 0.9814.


(b) Bond 2: Price $102,575. Conversion factor 1.018.

(c) Bond 3: Price $101,150. Conversion factor 1.004.

The futures price is $100,625. Which bond is currently the cheapest-to-deliver?

Answer: Since the long position will pay the futures price of 100,625 times the conversionfactor in settlement, the short position prefers to deliver the bond on which the ratio ofthe sale price to the purchase price is highest. Essentially, this means the bond deliveredis cheapest relative to the sale price. We compute this ratio for all three bonds as follows:

100, 625 0.981498, 750

= 1.0000

100, 625 1.018102, 575

= 0.99865

100, 625 1.004101, 150

= 0.99879

Hence, the first bond is the cheapest to deliver.

23. You enter into a short crude oil futures contract at $43 per barrel. The initial margin is$3,375 and the maintenence margin is $2,500. One contract is for 1,000 barrels of oil.By how much do oil prices have to change before you receive a margin call?

Answer: If the margin account falls to a value of $2500 then a call will occur. Therefore,the loss on the position must be equal to $3375-$2500=$875 for a margin call. Solvingthe following equation

1000 (P 43) = 875

gives P = 43.875, 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 isfor 250 the index. Ignoring interest, what are your losses/gains?Answer: The gain is

250(1131.75 1107.40) = $6087.50


25. An investor enters into 10 short futures contract on the Dow Jones Index at a futuresprice of 10,106. Each contract is for 10 the index. The investor closes out fivecontracts 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 investors 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 priceof $3.52/bushel. It closes out the contracts at maturity. The spot price at this time is$3.59/ bushel. Ignoring interest, what are the bakerys gains or losses from its futuresposition?

Answer: Each CBoT Wheat contract is for 50,000 bushels and so the settlement gainis

50 50, 000 (3.59 3.52) = $175, 000

27. An oil refining company enters into 1,000 long one-month crude oil futures contracts onNYMEX at a futures price of $43 per barrel. At maturity of the contract, the companyrolls half of its position forward into new one-month futures and closes the remaininghalf. At this point, the spot price of oil is $44 per barrel, and the new one-month futuresprice is $43.50 per barrel. At maturity of this second contract, the company closes outits remaining position. Assume the spot price at this point is $46 per barrel. Ignoringinterest, what are the companys gains or losses from its futures positions?

Answer: Exercise for the reader.

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

Answer: See section 2.3 of the book.

29. Distinguish between market-if-touched orders and stop 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 monthstime at some specified price and are considering hedging the price risk that you face. Ineach of the following scenarios, describe the kind of order (market, limit, etc.) that youwould use.


will be sustained. Thus, unless there is high volatility and a reversal of direction, thisapproach 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 this spread to widen to at least $0.10 per bushel. How would you use aspread order to bet on your view?

Answer: If the price differential between September and May futures is currently $0.06and is expected to widen to $0.10, then we should enter into a long position in theSeptember contract and a short position in the May contract. When the spreads widenswe close out both contracts.

33. The spread between one-month and three-month crude oil futures is $3 per barrel. Youexpect this spread to narrow sharply. Explain how you would use a spread order giventhis outlook.

Answer: Assuming the three-month minus one-month spread will narrow, we should golong the one-month contract and short the three-month contract. When the spreadnarrows, we buy back the short three-month contract and sell back the long one-monthcontract. We capture (ignoring interest) the difference between $3 and the new spread.

34. Suppose you anticipate a need for corn in three months time and are using corn futuresto hedge the price risk that you face. How is the value of your position affected by astrengthening of the basis at maturity?

Answer: The basis is the futures price minus the spot price. A strengthening of thebasis occurs if the basis increases. If this occurs, the position in the question is positivelyaffected since you are long futures. In notational terms, you go long futures today (atprice F0, say) and close it out at T (at price FT , say) for a net cash flow on the futuresposition of FT F0. In addition, you buy the corn you need at the time-T spot price ST ,leading to a total net cash flow of (FT F0) ST = (FT ST ) F0. A strengtheningof the basis FT ST at maturity improves this cash flow.

35. A short hedger is one who is short futures in order to hedge a spot cash flow risk. Along hedger is similarly one who goes long futures to hedge an existing risk. How doesa weakening of the basis affect the positions of short and long hedgers?

Answer: The short hedger is short futures and long spot, so gains if the basis weakens.The long hedger is long futures and short spot, so loses in this case.

Chapter 3


Answer: The spot price of wheat is $3.60. Since there are no storage costs, we computethe theoretical forward price of wheat as 3.60 exp(0.08 3/12) = 3.6727 which is lessthan 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, the net cash-flow is zero. At maturity, we deliver the wheat we own andreceive the forward price of $3.90. We return the borrowed amount with interest fora cash outflow of 3.60 exp(0.08 0.25) = 3.6727.This results in a net cash inflow of0.2273. The following table summarizes:

Cash FlowsSource Initial FinalShort Forward - +3.9000Long spot 3.6000 -Borrowing +3.6000 3.6727Net - +0.2273

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

5. A security is currently trading at $97. It will pay a coupon of $5 in two months. Noother payouts are expected in the next six months.

(a) If the term structure is flat at 12%, what should the be forward price on the securityfor delivery in six months?

(b) If the actual forward price is $92, explain how an arbitrage may be created.

Answer: We have that S = 97, and the PV of holding benefits is 5 exp(0.12 (2/12)) = 4.9010. Thus, the forward price should be

(97 4.9010) exp(0.12 (6/12)) = 97.794.

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 PV (5) = 4.901 for three months at 12%. Invest 97 PV (5) = 92.099 for six months at 12%. In three months, use the cash inflow of 5 from the investment to pay the coupon

due on the shorted security.

In six months, receive the cash from the six-month investment. Pay the deliveryprice of 97 on the forward and receive unit of the security, Use this to close theshort spot position.

The initial and interim cash flows are zero, and the final cash flow is positive as thefollowing table shows:

Cash FlowsSource Initial Interim FinalLong forward 0 - 92.00Short spot +97.00 - -3-month investment 4.901 +5 -6-month investment 92.099 - +97.794Net - - +5.794

6. Suppose that the current price of gold is $365 per oz and that gold may be storedcostlessly. Suppose also that the term structure is flat with a continuously compoundedrate of interest of 6% for all maturities.

(a) Calculate the forward price of gold for delivery in three months.

(b) Now suppose it costs $1 per oz per month to store gold (payable monthly inadvance). What is the new forward price?

(c) Assume storage costs are as in part (b). If the forward price is given to be $385per 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 forward price of gold is

365 exp(0.06 0.25) = 370.52.(b) With storage costs we need to first find the present value of the holding costs (M).

These are:

1 + 1 exp(0.06 1/12) + 1 exp(0.06 2/12) = 2.9851.


The forward price then is

(S +M) exp(rT ) = (365 + 2.9851) exp(0.06 0.25) = 373.55.This is higher because the storage costs have been factored in.

(c) If the forward price is 385, there is an arbitrage which is exploited as follows. Atinception:

Sell forward at 385. Buy spot at 365. Pay storage costs = 1. Invest 1 exp(0.06 1/12) = 0.9950 for one month. Invest 1 exp(0.06 2/12) = 0.9900 for two months. Borrow 365 + 1 + 0.9950 + 0.9900 = 367.985 for three months.

The net cash flow at inception is zero.

At the end of one month, realize $1 from the investment of 0.995 made at timezero, 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 timezero, 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 physicaldelivery. Cash flows at maturity are:

385 367.985 exp(0.06 3/12) = 11.454.This is positive irrespective of the time-T spot price of gold.

The following table summarizes all the cash-flows.

cash-flowsSource Initial Month 1 Month 2 Month 3Sell forward 0 - - 385Buy spot 365 - - -Month 1 storage cost 1 - - -Month 2 storage cost - 1 - -Month 3 storage cost - 1 -Borrow +367.985 - - 373.5464Invest 0.995 for one month 0.995 +1 - -Invest 0.99 for two months 0.99 - +1 -Net 0 0 0 +11.454

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


(a) Calculate the arbitrage-free price of (i) a three-month forward contract on the stockand (ii) a six-month forward contract on the stock.

(b) Suppose the six-month forward contract is quoted at 100. Identify the arbitrageopportunities, if any, that exist, and explain how to exploit them.

Answer: We are given: S = 90; r = 0.12 for all maturities; and that dividends of $1and $2 will be paid in one and four months, respectively.

(a) First, consider the case of a three-month horizon. There is only one dividend tobe considered, viz. the payment of $1 in one month. The present value of thisdividend is

exp{(0.12)( 112

)} 1 = 0.99.

Since the dividend represents a cash inflow, we have M = 0.99. Therefore, thearbitrage-free forward price is

F = (S +M)erT = (90 0.99)e(0.12)(0.25) = 91.72.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 presentvalue of the second dividend is

exp{(0.12)(13

)} 2 = 1.92.

Therefore, the present value of the dividends combined is 0.99+1.92 = 2.91. Sincethe 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

F = (S +M)erT = (90 2.91)e(0.12)(0.50) = 92.475.(b) The six-month forward is quoted at 100, so it is overvalued relative to spot. To

make an arbitrage profit, one should sell forward, buy spot, and borrow. Specifi-cally:

i. At time 0: Buy one unit of spot; borrow 87.09 for repayment in six months;borrow 1.92 for repayment in four months; and borrow 0.99 for repayment inone 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: receive dividend of $2; use this to repay the four-month loan.Net cash flow: 0.


iv. In six months: Use the unit of spot to settle the short forward position; receive100 from the forward position; use 92.475 of this to repay the six-month loan.Net cash flow: 100 92.475 = 7.525.

8. A bond will pay a coupon of $4 in two months time. The bonds current price is$99.75. The two-month interest rate is 5% and the three-month interest rate is 6%,both in continuously compounded terms.

(a) What is the arbitrage-free three-month forward price for the bond?

(b) Suppose the forward price is given to be $97. Identify if there is an arbitrageopportunity and, if so, how to exploit it.

Answer: The coupons represent a cash inflow, so the forward price is

[99.75 4 exp(0.05 2/12)] exp(0.06 3/12) = 97.231.

If the forward price is 97, then the forward is underpriced relative to spot. An arbitrageexists and is exploited with the following strategy:

Buy forward at 97. Sell spot at 99.75. Invest the present value of the coupon for two months. The PV of the coupon is

4 exp(0.05 2/12) = 3.9668. Invest the remaining proceeds for three months , i.e., invest 99.75 3.9668 =

95.7832.

The cash-flow at inception is zero.

After two months, realize $4 from the investment of 3.9688 and use it to pay the couponon 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 shortspot position.

Receive principal plus interest on the investment: 95.7832 exp(0.06 3/12).The net cash-flow is 0.2308, which is positive.

9. Suppose that the three-month interest rates in 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 price for delivery in three months.

(b) If the actual forward price is given to be $0.156, examine if there is an arbitrageopportunity.

Answer: The forward price of the Kroner is

0.155 exp((0.04 0.08) 3/12) = 0.15346.

Since the forward price is actually 0.156, it is overpriced. We may exploit this by thefollowing strategy:

Sell 1 Kroner forward at $0.156. Buy PV(1 Kroner)= e0.083/12 = 0.9802 spot at 0.155 e0.083/12 = $0.1519. Invest PV(1 Kroner) for three months. Amount received at maturity = 1 Kroner. Borrow $0.1519 for three months at 4%. After three months, deliver Kroner and receive $0.156 from the forward. Repay

dollar borrowing with interest for a total of $0.1534.

The resulting cash flows are summarized in the following table:

Cash flow in Kroner Cash flow in $

At inception +0.9802 (purchase) 0.1519 (sale)0.9802 (investment) +0.1519 (borrowing)

Net: 0 Net: 0

At T +1.00 (from investment) 0.1534 (repay borrowing)1.00 (deliver to forward) +0.156 (receive from forward)

Net: 0 Net: +0.0026

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


10. Consider a three-month forward contract on pound sterling. Suppose the spot exchangerate is $1.40/, the three-month interest rate on the dollar is 5%, and the three-monthinterest rate on the pound is 5.5%. If the forward price is given to be $1.41/, identifywhether there are any arbitrage opportunities and how you would take advantage ofthem.

Answer: We are given the information that S = 1.40, r = 0.05 and d = 0.055. Fromthis data, the arbitrage-free forward price of a three-month forward contract should be

F = e(rd)T S = e(0.050.055)(1/4) (1.40) = 1.3983.

Thus, at the given forward price of $1.41/, the forward contract is overvalued relativeto spot. To take advantage of the opportunity, we should sell forward, buy spot, andborrow to finance the spot purchase. Specifically:

Enter into a short forward contract to deliver pounds in three months at $1.41/. Buy edT = 0.9863 pounds spot at the spot price of $1.40/.

Cost: $(1.40)(0.9863) = $1.3809.

Invest the 0.9863 for three months at 5.5%.Amount received after three months: 1.

Borrow $1.3809 for three months at 5%.Amount due in three months: $(e(0.05)(1/4) (1.3809) = $1.3983.

The resulting cash flows are summarized in the following table:

Cash flow in Cash flow in $

At inception +0.9863 (from purchase) 1.3809 (to purchase )0.9863 (investment) +1.3809 (borrowing)

Net: 0 Net: 0

At T +1.00 (from investment) 1.3983 (repay borrowing)1.00 (deliver to forward) +1.41 (receive from forward)

Net: 0 Net: +0.0117

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


11. Three months ago, an investor entered into a six-month forward contract to sell a stock.The delivery price agreed to was $55. Today, the stock is trading at $45. Suppose thethree-month interest rate is 4.80% in continuously compounded terms.

(a) Assuming the stock is not expected to pay any dividends over the next threemonths, what is the current forward price of the stock?

(b) What is the value of the contract held by the investor?

(c) Suppose the stock is expected to pay a dividend of $2 in one month, and theone-month rate of interest is 4.70%. What are the current forward price and thevalue of the contract held by the investor?

Answer: The answers to the three parts are:

(a) The current forward price is

45 exp(0.048 3/12) = 45.543.(b) The value of the contract is PV (KF ). Since KF = 55.00045.543 = 9.457,

the contract value is 9.457 exp(0.048 3/12) = 9.3442.(c) Now suppose the stock is expected to pay a dividend of $2 in one month. The

present value of this dividend payment is

e(0.047)(1/12) 2 = 1.992.

Since the dividend payment represents a cash inflow, we have M = 1.992. Thus,the arbitrage-free forward price is now

F = erT (S +M) = e(0.048)(1/4) (45 1.992) = 43.527.The value of holding a short position in this forward contract with a delivery priceof K = 55 is now

PV(K F ) = e(0.048)(1/4) (55 43.527) = 11.336.

12. An investor enters into a forward contract to sell a bond in three months time at $100.After one month, the bond price is $101.50. Suppose the term-structure of interestrates is flat at 3% for all maturities.

(a) Assuming no coupons are due on the bond over the next two months, what is theforward price on the bond?

(b) What is the marked-to-market value of the investors short position?

(c) How would your answers change if the bond will pay a coupon of $3 in one monthstime?


Answer: The answers to the three parts are:

(a) The forward price at the end of one month is

101.50 exp(0.03 2/12) = 102.01.(b) The marked-to-market value of the original contract is

PV (F K) = (100 102.01) exp(0.03 2/12) = 2.00.(c) If there is a coupon one month from now, then the re-estimated forward price is:

(101.50 3 exp(0.03 1/12)) exp(0.03 2/12) = 99.001.In this case, the value of the contract to sell forward at 100 is

(100 99.001) exp(0.03 2/12) = 0.994.

13. A stock is trading at $24.50. The market consensus expectation is that it will pay adividend of $0.50 in two months time. No other payouts are expected on the stock overthe next three months. Assume interest rates are constant at 6% for all maturities. Youenter into a long position to buy 10,000 shares of stock in three months time.

(a) What is the arbitrage-free price of the three-month forward contract?

(b) After one month, the stock is trading at $23.50. What is the marked-to-marketvalue of your contract?

(c) Now suppose that at this point, the company unexpectedly announces that divi-dends will be $1.00 per share due to larger-than-expected earnings. Buoyed by thegood news, the share price jumps up to $24.50. What is now the marked-to-marketvalue 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 thethree-month forward is obtained by using the formula PV (F ) = S + M . Substi-tuting for the various input values, this gives us the original forward price as

[24.50 0.50 e0.062/12] e0.063/12 = 24.368.Denote this locked-in delivery price by K.


(b) After one month, the contract has two remaining months of life. Given that thenew spot price is S = 23.50 (and that interest rates are unchanged), the forwardprice of the same contract is

F = [23.50 0.50 e0.061/12] e0.062/12 = 23.234.Hence, the marked-to-market value of the original contract is

PV (F K) = (23.234 24.368) exp(0.06 2/12) = 1.1227i.e., a loss of $1.1227 per share. On 10,000 shares, the value of the position is$10, 0001.1227 = $11, 227.

(c) If the dividends change to 1.00, we need to rework the forward price and re-assessthe position. The new forward price will be:

[24.50 1.00 e0.061/12] e0.062/12 = 23.741.At this forward price, the value of the original contract is

(23.741 24.368) e0.062/12 = 0.6208or a loss of $0.6208 per share, for a total loss of $6,208 on the position.

14. Suppose you are given the 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-compoundedand 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:10,000 lbs of copper in one month, and 10,000 lbs in two months. The common deliveryprice in the contract for both deliveries is P , that is, the short position receives P uponmaking the one-month delivery and P upon making the two-month delivery. What isthe arbitrage-free value of P?

Answer: Let Q denote the quantity delivered each month (i.e., Q = 10, 000 lbs). Toreplicate this contract, we need to buy 2Q units of copper today and store it. After onemonth, we deliver the first Q units, and after one more month, the second Q units. Thecost of this replication strategy is the current spot price of 2Q units, which is

2 100 83.55


This must equal the present value of the cash outflows on the forward strategy, which is

Pe0.051/12 + Pe0.052/12 = P (e0.051/12 + e0.052/12)

Equating these, we can solve for P :

P =2 8, 355

exp(0.05 1/12) + exp(0.05 2/12) = 8, 407.40

This is the arbitrage free value of P .

15. This question generalizes the previous one from two deliveries to many. Consider acontract that requires the short position to make deliveries of one unit of an underlyingat time points t1, t2, . . . , tN . The common delivery price for all deliveries is F . Assumethe interest rates for these horizons are, respectively, r1, r2, . . . , rN in continuously-compounded annualized terms. What is the arbitrage-free value of F given a spot priceof S?

Answer: The answer follows the same logic as we had in the previous question, i.e.,

F =N SN

i=1 exp(rtiti)

Such a contract (one which calls for multiple deliveries at a fixed price F ) is a commod-ity swap. Commodity swaps are usually settled in cash, rather than by physical deliveryas weve assumed here, though this does not change the arguments. Commodity swapsare discussed in Chapter 25.

16. In the absence of interest-rate uncertainty and delivery options, futures and forwardprices must be the same. Does this mean the two contracts have identical cash-flowimplications? (Hint: Suppose you expected a steady increase in prices. Would you prefera futures contract with its daily mark-to-market or a forward with its single mark-to-market at maturity of the contract? What if you expected a steady decrease in prices?)

Answer: Evidently not. For example, if prices ares steadily trending upward, a futurescontract with its daily mark-to-market will result in earlier cash inflows to the long andcash outflows for the short. So if you were a long investor, you would prefer the futuresto the forward (vice versa if you were a short investor).


17. Consider a forward contract on a non-dividend-paying stock. If the term-structure ofinterest rates is flat (that is, interest rates for all maturities are the same), then thearbitrage-free forward price is obviously increasing in the maturity of the forward contract(i.e., a longer-dated forward contract will have a higher forward price than a shorter-dated one). Is this statement true even if the term-structure is not flat?

Answer: Consider two dates t1 and t2 where t1 < t2. Assume that the spot rates forthese two dates are respectively, r1 and r2. Given a spot price of S, the two correspondingforward prices are F1 = Se

r1t1 and F2 = Ser2t2 . When the term structure is flat, r1 = r2,

and hence, it is easy to see that F1 < F2.

For general curves of spot rates, i.e., when the term structure is not flat, suppose wewant that F2 < F1. Then it must be that

Ser2t2 < Ser1t1

r2t2 < r1t1

r2 < r1(t1/t2)

Is this feasible? Mathematically, yes, we may find parameter values for which thiscondition holds, implying that when term structures are not flat, we may have longerterm forward prices lower than shorter term ones. For example, r2 = 0.02, r1 = 0.06,t1 = 0.25 and t2 = 0.50, satisfies the condition.

But economically, does this make sense? The answer is no. The condition that r1t1 >r2t2 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 t2 and investshort-term for t1.

18. The spot price of copper is $1.47 per lb, and the forward price for delivery in threemonths is $1.51 per lb. Suppose you can borrow and lend for three months at aninterest rate of 6% (in annualized and continuously-compounded terms).

(a) First, suppose there are no holding costs (i.e., no storage costs, no holding benefits).Is there an arbitrage opportunity for you given these prices? If so, provide detailsof 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,payable in advance. How would your answer to Part (a) change? (Note thatstorage 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 there are no holding costs, the forward price is

1.47e0.063/12 = 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 copperspot at 1.47, and borrow 1.47 to finance the copper spot purchase. This leaves azero net cash flow at inception and has a cash inflow of 1.51 1.47e0.063/12 =0.0178 at maturity.

(b) When there are storage costs and these are asymmetric, the problem is trickier. Ifwe treat the storage costs as a cost of carry, we arrive at the forward price

(1.47 + 0.03)e0.063/12 = 1.5227

This makes it appear that the the given price of 1.51 is too low, i.e., that the forwardis now underpriced, but this is illusory. If we try implementing the arbitrage strategy(long forward, short spot, invest), then we will have a cash outflow at maturitybecause 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, inthis case because we now have to pay storage costs.

Thus, the asymmetric nature of storage costs wipes out any perceived arbitrageopportunity. Put differently, it is as if there are two correct theoretical arbitrage-free forward prices: the price is 1.4922 if you plan to be long copper (and shortspot) 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 (theDow Jones EuroSTOXX Index of 50 stocks, referred to from here on as STOXX) istrading at e3,671. The dollar interest rate is 3% , and the euro interest rate is 5%.The exchange rate is $1.28/euro. The six-month futures on the STOXX is quoted ate3,782. All interest rates are continuously compounded. There are no borrowing costsfor securities.

(a) Compute the correct six-month forward futures prices of the SPX, STOXX, andthe currency exchange rate between the dollar and the euro.

(b) Is the futures on the STOXX correctly priced? If not, show how to undertake anarbitrage strategy assuming you are not allowed to undertake borrowing or lendingtransactions in either currency.

Answer:


(a) The required forward rates are:

SPX forward price: 1265e0.031/2 = 1284.1

STOXX forward price: 3671e0.051/2 = 3763.9

Currency ($/e) forward: 1.28e(0.030.05)1/2 = 1.2673

(b) The STOXX forward contract is quoted at e3782, whereas its correct quote ascomputed above should be e3763.9. To exploit this error we undertake the fol-lowing arbitrage strategy (sequence of trades), without using borrowing or lendingin either currency:

At 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 of the SPX spot for $1,265. Do this for 3.7145 contracts(we will see why soon).

iv. Convert the $1,265 (for 3.7145 contracts) into euros at the spot exchange rateof $1.28/euro. This results in 1, 265 3.7145/1.28 =e3, 671 which is exactlywhat is needed to buy the components of the STOXX above.

v. Buy SPX forward at $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 transactionsis zero. We now move forward to maturity at the end of six months and examinethe net cash-flow that is generated. We denote the spot value of the U.S. stockindex as SPX and that of the euro stock index as STOXX. Below we describe thecash flows from each of the six components of the trading strategy we presentedabove:

i. Close out the STOXX forward contract by delivery of the spot position: thecash-flow is 3,782 euros.

ii. Buy back the components of the SPX index using the forward: cash-flow is$3.7145 1, 284.10 and close out the short SPX position.

iii. Sell the 3,782 euros forward, cash-flow is $3, 782 1.2673 = 4, 792.9.Thus, after all transactions netted off, we are left with a guaranteed gain of $23.1,representing the arbitrage profits.

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

Chapter 4


Chapter 4. Pricing Forwards & Futures II

1. What is meant by the term convenience yield? How does it affect futures prices?

Answer: Commodities are used in production and gets consumed in the process. In-ventories of commodities are held by producers because this provides them with theflexibility to alter production schedules or as insurance against a stock-out that couldcause business disruptions. The value of these options to consume the commodity outof storage ias needed is referred to as the commodity convenience yield. As a holdingbenefit, the convenience yield reduces the price of forwards and futures on the underlying.

2. True or false: An arbitrage-free forward market can be in backwardation only if thebenefits of carrying spot (dividends, convenience yields, etc.) exceed the costs (storage,insurance, etc.).

Answer: True. Backwardation occurs when the present value of the benefits of carryingthe physical commodity (including the convenience yield) outweigh the carrying costs.

3. Suppose an active lease market exists for a commodity with a lease rate ` expressedin annualized continuously-compounded terms. Short-sellers can borrow the asset atthis rate and investors who are long the asset can lend it out at this rate. Assume thecommodity has no other cost of carry. Modify the arguments in the appendix to thechapter to show that the theoretical futures price is F = e(r`)TS.

Answer: Suppose the forward price tat prevails (denoted, say, F ) is not equal to F .Assume first that F < F . Consider the following strategy:

Take a long forward position at F . This involves no current cash-flow. Borrow e`T units of gold for T years; sell the borrowed gold at the spot price ofS. Cash inflow today: e`TS. Note that given the lease rate `, the amount of golddue at maturity T is one unit.

Invest the cash of e`TS for maturity at T at the interest rate r.

The net cash-flow at inception is zero. At date T , pay F on the forward, receive oneunit of gold, and use this to close the gold lease. Receive erT e`TS = e(r`)TS fromthe investment. Net cash flow at T :

e(r`)TS F .

By hypothesis, this amount is positive, so we have an arbitrage profit. Similarly, ifF > e(r`)TS, reversing the above strategy would result in an arbitrage profit.


Re-arranging we have that

r = `+ (1/t) ln(F/S)

So we can see that the new repo rates will be the old repo rates plus 0.005, which givesthe two-month and five-month rates as:

r2 = 0.034402, r5 = 0.036789.

10. Copper is currently trading at $1.28/lb. Suppose three-month interest rates are 4% andthe convenience yield on copper is c = 3%.

(a) What is the range of arbitrage-free forward prices possible using

S0e(rc)T F S0erT ? (1)

(b) What is the lowest value of c that will create the possibility of the market being inbackwardation?

Answer:

(a) Plug values into the equation above, i.e.,

1.28e(0.040.03)3/12 F 1.28e0.043/12

or

1.2832 F 1.2929(b) The lowest value of c to create backwardation is r.

11. You are given the following information on forward prices (gold and silver prices are peroz, copper prices are per lb):

Commodity Spot One-month Two-month Three-month Six-month

Gold 436.4 437.3 438.8 440.0 444.5Silver 7.096 7.125 7.077 7.160 7.220Copper 1.610 1.600 1.587 1.565 1.492


iv. Lease the gold out for 3 months at 1% lease rate. Amount received at end ofthe lease: 1 oz.

v. Borrow 359.10 for 3 months at 4%. Amount owed at maturity: 362.71

At maturity, deliver the 1 oz. of gold received from the lessee to the forward contractand receive F = 366. Repay 362.71 on the borrowing. Net cash flow: +3.29.

14. A three-month forward contract on a non-dividend-paying asset is trading at 90, whilethe spot price is 84.

(a) Calculate the implied repo rate.

(b) Suppose it is possible for you to borrow at 8% for three months. Does this giverise to any arbitrage opportunities? Why or why not?

Answer: The implied repo rate is

r =1

T[lnF lnS] = 1

0.25[ln 90 ln 84] = 0.27957,

or 27.98%. Since we can borrow at 8% for three months, there is a clear arbitrageopportunity:

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 ST ) 84 exp(0.08 3/12) + ST = 4.3031

which is the difference between the repo rate and market borrowing rate on a base priceof 84. To see this, note that

84[exp(r 3/12) exp(0.08 3/12)] = 4.3031.


Reinvest all dividends into buying more of the index. Amount of index held in3 months: 1 unit.

Borrow 580.613 for three months at 6%. Take a short position in the forward contract at F = 600.

At maturity, deliver the unit of the index on the forward contact, and receive 600.Repay the borrowing: the cash outflow is 580.613 e0.061/4 = 589.39. The netcash flow of +10.61 represents arbitrage profits.

17. A three month-forward contract on an index is trading at 756 while the index itself is at750. The three-month interest rate is 6%.

(a) What is the implied dividend yield on the index?

(b) You estimate the dividend yield to be 1% over the next three months. Is there anarbitrage opportunity from your perspective?

Answer: The forward to spot relationship for this contract is as follows:

750 exp[(0.06 d)(0.25)] = 756

Deriving the dividend yield from this results in d = 0.028127.

Now if the dividend is only expected to be 1%, then the current forward price is toolow. The arbitrage strategy is to buy forward, and sell spot. The details are left as anexercise. Be careful to account for the dividends in the strategy that you create for therisk-less arbitrage.

18. The spot US dollar-euro exchange rate is $1.10/euro. The one-year forward exchangerate is $1.0782/euro. If the one-year dollar interest rate is 3%, then what must be theone-year rate on the euro?

Answer: We exploit the following relationship:

F = S exp[rUSD rEuro]

noting that time is one year. The equation to be solved is:

1.0782 = 1.1000 exp[0.03 rEuro]

which means that rEuro = 0.05.


19. You are given information that the spot price of an asset is trading at a bid-ask quoteof 80 80.5, and the six-month interest rate is 6%. What is the bid-ask quote for thesix-month forward on the asset if there are no dividends?

Answer: There are two possible forward-spot arbitrage strategies: one where we buyforward (at F a) and sell spot (at Sb), and the other where we sell forward (at F b) andbuy spot (at Sa), where the superscripts a and b refer to ask and bid, respectively.

For the first strategy to not admit arbitrage profits, we must have

F a e0.061/2 Sb = 82.436

For the second strategy not to be an arbitrage, we must have

F b e0.061/2 Sa = 82.952

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

20. Redo the previous question if the interest rate for borrowing and lending are not equal,i.e., there is a bid-ask spread for the interest rate, which is 6.006.25%.

Answer: Here, the first arbitrage strategy involves investing the short sale proceeds ofSb at the lending rate of 6.00%, while the second strategy involves borrowing the spotpurchase price of Sa at the borrowing rate of 6.25%. These interest rates should beused in computing the inequalities. Carrying out the computations is left as an exercise.

21. In the previous question, what is the maximum bid-ask spread in the interest rate marketthat 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 isalso 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

F (correct) = 40e0.040.02 = 40.808

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


will invest the proceeds (40) at 4%. Therein lies the source of the arbitrage profits. Allthese transactions at inception are net zero in cash-flow.

We now examine the cash-flow at the end of the year. The forward position results ina cash-flow of 40 on purchase. The short spot position is closed out by delivering thestock to the lender. The net flow from the cost of borrowing the stock and the gainsfrom lending the sales proceeds of the stock is 2% of $40. Hence, we gain a net amountof 0.808 (= 40 + 40e0.040.02).

23. You are given two stocks, A and B. Stock A has a beta of 1.5, and stock B has a beta of0.25. The one-year risk-free rate is 2%. Both stocks currently trade at $10. Assumea CAPM model where the expected return on the stock market portfolio is 10%. StockA 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:

E(rA) = 0.02 + 1.5[0.10 0.02] = 0.14

Stock B:

E(rB) = 0.02 0.25[0.10 0.02] = 0.0

(b) The forward price for stock A is

FA = 10e0.020.01 = 10.101

The forward price for stock B is

FB = 10e0.020.0 = 10.202

(c) There is no arbitrage even though stock B has a forward price greater than thatof stock A even though its expected return and dividend is zero. The forward price isbased on a mathematical relationship between spot prices and interest rates, and doesnot 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, i.e., a hedge ratio of 1, always betterthan not hedging at all?

Answer: Basis risk arises from the fact that the hedge and the underlying spot values arenot perfectly correlated. Depending on this correlation, it may be better to not hedgethan to hedge one-for-one. For example, this is certainly the case if price changes in theunderlying spot asset and the hedge have no correlation with each other at all: in sucha situation, hedging only results in additional uncertainty. For a more precise statementon 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 = 0.8, what fraction ofcash-flow uncertainty is removed by minimum-variance hedging?

Answer: As shown in Section 5.5, the fraction of unhedged cash flow variance removedby the minimum-variance hedge is 2, which in this case is 0.64 or 64%.

7. The correlation between changes in the price of the underlying and a futures contract is+80%. The same underlying is correlated with another futures contract with a (negative)correlation of 85%. Which of the two contracts would you prefer for the minimum-variance hedge?

Answer:

The second one. As shown in Section 5.5, the fraction of unhedged cash flow varianceremoved by the minimum-variance hedge is 2, where is the correlation of the spotprice changes and price changes in the futures contract used for hedging. Since 2

increases as || increases, we should use a futures contract with the highest value of ||.The only impact of the negative correlation is that the sign of the futures position getsreversed, i.e., we hedge a long spot exposure (a commitment to buy spot at maturityT ) with a short futures position and a short spot exposure (a commitment to sell spotat date T ) with a short futures position.

8. Given the following information on the statistical properties of the spot and futures,compute the minimum-variance hedge ratio: S = 0.2, F = 0.25, = 0.96.

Answer: The minimum-variance hedge ratio is

h = (S)

(F )= 0.96 0.2

0.25= 0.768.

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


9. Assume that the spot position comprises 1,000,000 units in the stock index. If the hedgeratio 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, where H is the number ofunits of the hedge, and Q is the number of units of the spot position. Hence, therequired number of units in futures is

H = h Q = 1.09 1, 000, 000 = 1, 090, 000.

In words, we enter into a futures contract that calls for the delivery of 1,090,000 unitsof the asset underlying the futures contract.

10. You have a position in 200 shares of a technology stock with an annualized standarddeviation of changes in the price of the stock being 30. Say that you want to hedge thisposition over a one-year horizon with a technology stock index. Suppose that the indexvalue has an annual standard deviation of 20. The correlation between the two annualchanges is 0.8. How many units of the index should you hold to have the best hedge?

Answer: In the notation of the chapter, we are given that (S) = 30, (F ) = 20,and = 0.8. So the minimum-variance hedge ratio is

h = (S)

(F )= 0.8(30/20) = 1.20

Hence, you need to short 1.2 200 = 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 Spot Futures0 80.000 81.0001 79.635 80.8692 77.880 79.0923 76.400 77.7164 75.567 77.0745 77.287 78.8416 77.599 79.3157 78.147 80.0678 77.041 79.2169 76.853 79.204

10 77.034 79.63811 75.960 78.65912 75.599 78.54913 77.225 80.51214 77.119 80.40515 77.762 81.22416 77.082 80.65417 76.497 80.23318 75.691 79.60519 75.264 79.27820 76.504 80.76721 76.835 81.28022 78.031 82.58023 79.185 84.03024 77.524 82.33725 76.982 82.04526 76.216 81.25227 76.764 81.88228 79.293 84.62329 78.861 84.20530 76.192 81.429

Carry out the following analyses using Excel:

(a) Compute (S), (F ), and .

(b) Using the results from (a), compute the hedge ratio you would use.

(c) Using this hedge ratio, calculate the daily change in value of the hedged portfolio.

(d) What is the standard deviation of changes in value of the hedged portfolio? Howdoes this compare to the standard deviation of changes in the unhedged spotposition?


Answer: The results are presented in the following tables. The first step is to computethe covariance matrix of the changes in spot and futures, as follows. Using Excel, weobtain:

Covariance MatrixS F

S 1.276F 1.308 1.415

Using these numbers to compute the correlation and the hedge ratio h, we obtain:

= 0.9732

h = 0.9732 1.2761.308

= 0.9244

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

(79.635 80) [0.9244 (80.869 81)] = 0.243

The following table summarizes these numbers:


Day Spot Futures S F P&L0 80.000 81.0001 79.635 80.869 -0.365 -0.131 -0.2432 77.880 79.092 -1.756 -1.777 -0.1133 76.400 77.716 -1.479 -1.376 -0.2074 75.567 77.074 -0.834 -0.642 -0.2415 77.287 78.841 1.721 1.768 0.0876 77.599 79.315 0.312 0.474 -0.1267 78.147 80.067 0.547 0.752 -0.1488 77.041 79.216 -1.106 -0.851 -0.3199 76.853 79.204 -0.188 -0.012 -0.17610 77.034 79.638 0.180 0.434 -0.22111 75.960 78.659 -1.074 -0.979 -0.16912 75.599 78.549 -0.361 -0.111 -0.25813 77.225 80.512 1.626 1.964 -0.18914 77.119 80.405 -0.106 -0.107 -0.00715 77.762 81.224 0.643 0.820 -0.11416 77.082 80.654 -0.681 -0.571 -0.15317 76.497 80.233 -0.585 -0.420 -0.19618 75.691 79.605 -0.805 -0.629 -0.22419 75.264 79.278 -0.427 -0.327 -0.12520 76.504 80.767 1.240 1.488 -0.13621 76.835 81.280 0.330 0.513 -0.14422 78.031 82.580 1.196 1.300 -0.00523 79.185 84.030 1.153 1.450 -0.18724 77.524 82.337 -1.661 -1.693 -0.09625 76.982 82.045 -0.541 -0.292 -0.27126 76.216 81.252 -0.766 -0.793 -0.03327 76.764 81.882 0.548 0.629 -0.03428 79.293 84.623 2.529 2.742 -0.00629 78.861 84.205 -0.432 -0.419 -0.04530 76.192 81.429 -2.669 -2.776 -0.103

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

12. Use the same data as presented above to compute the hedge ratio using regressionanalysis, again using Excel. Explain why the values are different from what you obtainedabove.

Answer: The regression in Excel of daily changes S on F produces the following results.

S = 0.14 + 0.947 F


Finally, the current USD/SEK spot rate is 0.104, the current three-month USD/EURforward rate is 1.071, and the current three-month USD/CHF forward rate is 0.602.

(a) Which currency should the company use for hedging purposes?

(b) What is the minimum-variance hedge position? Indicate if this is to be a long orshort position.

Answer:

(a) Since the correlation of changes in the spot USD/SEK is higher with changes inforward USD/EUR than with changes in forward USD/CHF, the hedge will bebetter if the USD/EUR forward is used for hedging.

(b) The optimal hedge ratio is

h = SF

= 0.90 0.0070.018

= 0.35

Since the correlation of USD/SEK and USD/EUR is positive, appreciation in theSEK should mostly be offset by appreciation in the EUR. Hence, the hedge positionshould be a long USD/EUR forward contract calling for the delivery of EUR (0.35100) million = EUR 35 million.

14. You use silver wire in manufacturing. You are looking to buy 100,000 oz of silver inthree months time and need to hedge silver price changes over these three months. OneCOMEX silver futures contract is for 5,000 oz. You run a regression of daily silver spotprice changes on silver futures price changes and find that

s = 0.03 + 0.89F +

What should be the size (number of contracts) of your optimal futures position. Shouldthis be long or short?

Answer: From the regression, the optimal hedge ratio is 0.89, so the size of the requiredfutures position is 89,000 oz or 89, 000/5, 000 = 17.8 contracts. This should be a longposition.

15. Suppose you have the following information: = 0.95, S = 24, F = 26, K = 90,R = 1.00018. What is the minimum-variance tailed hedge?


(a) Ambiguous. The question does not indicate if profits on the hedge position aremore likely if the term-structure is upward sloping.

(b) Again, ambiguous. The forward price is convex in the interest rate, but the spotprice too may be correlated with interest rates, so without more information onthe nature of the underlying, it is not clear how the forward/futures price behaveswhen interest rates become more volatile.

(c) Ambiguous. The performance of the hedge depends only on the correlation betweenprice changes in the instrument used for hedging and price changes in the exposurebeing hedged, and not on the variances of price changes of the instruments usedfor hedging.

However, if we add some more conditions, we can provide a qualified answer. Forexample, the size of the optimal hedge depends on the standard deviation of pricechanges of the contract used for hedging, and decreases as this standard deviationincreases. So if the correlations are the same for both hedging instruments, thecontract with greater volatility may be preferable since fewer positions are neededin an optimal hedge and this may reduce transactions costs.

(d) Here, the answer is unambiguous. You want the correlation of spot to futures tobe higher,as the hedged position will have a lower variance.

23. You are trying to hedge the sale of a forward contract on a security A. Suggest a frame-work you might use for making a choice between the following two hedging schemes:

(a) Buy a futures contract B that is highly correlated with security A but trades veryinfrequently. Hence, the hedge may not be immediately available.

(b) Buy a futures contract C that is poorly correlated with A but trades more fre-quently.

Answer: The question requires you to use your imagination to develop a model totrade off liquidity risk against basis risk. Here is one possible way to approach theproblem (obviously not the only one). Suppose we denote the probability of being ableto implement the hedge B by p. The probability of A remaining unhedged is then 1p.The variance of the unhedged position is 2(A). The variance of the hedged positionis 2(A)(12AB), where AB is the correlation between changes in A and B. Hence,the expected variance of the hedged position when B is used is

p 2(A)(1 2AB) + (1 p)2(A).

If C is used to hedge, the variance of the hedged position is

2(A)(1 2AC)


If we care about only the expected variance of the hedged portfolio, then, dependingon which of these values is computed to be lower, the hedge instrument may be chosenaccordingly. If the latter is lower, choose C, and if the former is lower, choose B. Inparticular, the first alternative is preferred when

p 2(A)(1 2AB) + (1 p)2(A)2(A)(1 2AC)

24. Download data from the Web as instructed below and answer the questions below:

(a) Extract one years data on the S&P 500 index from finance.yahoo.com. Alsodownload corresponding period data for the S&P 100 index.

(b) Download, for the same period, data on the three-month Treasury Bill rate (con-stant maturity) from the Federal Reserves Web page on historical data:

www.federalreserve.gov/releases/h15/data.htm.

(c) Create a data series of three-month forwards on the S&P 500 index using the indexdata and the interest rates you have already extracted. Call this synthetic forwarddata series F .

(d) How would you use this synthetic forwards data to determine the tracking error ofa hedge of three-month maturity positions in the S&P 100 index? You need tothink (a) about how to set up the time lags of the data and (b) how to representtracking error.

Answer: This exercise is left to the reader. For the definition of tracking error, seethe answer to the next question.

25. Explain the relationship between regression R2 and tracking error of a hedge. Usethe data collected in the previous question to obtain a best tracking error hedge usingregression.

Answer: To answer this question, one must first define tracking error (the questiondeliberately leaves this undefined). The intuitive definition of tracking error is also themost commonly used one: tracking error is the standard deviation (or the variance) ofthe difference between a target performance and the actual performance.

Suppose we run a regression to determine the hedge ratio to be implemented:

S = a+ bF +

Chapter 6


Chapter 6. Interest Rate Forwards & Futures

1. Explain the difference between the following terms:

(a) Payoff to an FRA.

(b) Price of an FRA.

(c) Value of an FRA.

Answer: FRA terminology:

(a) The payoff from an FRA is the dollar amount received at maturity of the FRA. Forexample, if we are long an FRA at a strike interest rate of 10% and the rate atmaturity of the FRA is 11%, our payoff will be based on the interest difference of1% applied to the notional principal of the contract for the borrowing period.

(b) The price of an FRA refers to the fixed rate locked in using the FRA. At inceptionof 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 is the net payment that would have to be made if the FRA were to be closedout 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 priceof a new FRA (i.e., from the fixed rate that makes an FRA with the same maturityas the original one have zero value to both parties), so the FRA can have positiveor negative value.

2. What characteristic of the eurodollar futures contract enabled it to overcome the set-tlement obstacles with its predecessors?

Answer: Cash settlement. Earlier attempts at developing an interest-rate futures con-tract based on commercial borrowing rates had floundered because they required physicalsettlement at maturity, but the deliverable instruments in these contracts lacked homo-geneity because of perceived differences in the credit risk of the issuing entities. Theeurodollar futures contract solved this by using cash settlement, an idea that was rapidlyadopted in other contracts which had difficulties with physical settlement (e.g., stockindex futures).

3. How are eurodollar futures quoted?

Answer: Eurodollar futures contracts are instruments that enable traders to lock ina Libor rate for a three-month period beginning on the expiry date of the contract.However, eurodollar futures are quoted not as rates but as prices. The price 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 May. What is the relation between the observed eurodollar futures priceof 96.32 for the November maturity and the rate of interest that is locked-in using thecontract? Over what period does this rate apply?

Answer: The relation between the futures price and rate of interest that gets locked invia the contract is

100 96.32 = 3.68%

The interest rate applies to a 90-day borrowing or investment beginning at maturity ofthe futures contract, i.e., beginning in November.

5. What is the price tick in the eurodollar futures contract? To what price move does thiscorrespond?

Answer: The price tick in the eurodollar futures contract is 0.01 (which correspondsto a move in the implied interest rate of 1 basis point). The price tick has a dollarvalue of $25. The minimum price move on the expiring eurodollar futures contract (theone currently nearest to maturity) is 1/4 tick or a dollar value of $6.25. On all othereurodollar 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 a0.01 increase in the futures price?

Answer: There will be a loss of $25.

7. You enter into a long eurodollar futures contract at a price of 94.59 and exit the contracta 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 for the long, loss for the short). In this case, the price falls by 0.36, whichcorresponds to a loss of (36 $25) = $900 for the long position.

8. What is the cheapest to deliver in a Treasury bond futures contract? Are there otherdelivery options in this contract?

Answer: The standard bond in a Treasury contract is one with a coupon of 6% and atleast 15 years to maturity or first call. The main delivery option in the Treasury 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 demandand supply for money, or the bid/ask spreads are too large to allow someone to arbitragethe 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 thethree 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?

Answer: If rates are going to rise over the next three months and you wish to speculateon this view, then you can lock-in a rate today using a FRA for borrowing in threemonths and in three months time, you can invest the borrowed amount at the higherinterest rates that prevail then. Similarly, if, in three months time, rates are going tofall over the next three months, you can enter into a short FRA at that time to speculateon your views.

19. A firm plans to borrow money over the next two half-year periods, and is able to obtaina fixed-rate loan at 6% per annum. It can also borrow money at the floating rate ofLibor + 0.5%. Libor is currently at 4%. If the 6 12 FRA is at a rate of 6%, find thecheapest financing cost for the firm.

Answer: For simplicity, we treat each six-month period as exactly half a year. If thefixed rate loan is taken, the cost of financing is 3% each half year. That is, the firmreceives 100 today, pays 3 in six months and 103 in one year at maturity. It is easy tosee 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 loanand simultaneously enters into a long 612 FRA. Then, the cost of financing is 4.5% forthe first six months and 6.5% for the next six months . (Note that in the second period,the cost of financing to the firm is `+ 0.5% plus the payoff to the FRA is 6 `. Hence,net this is 6.5%). So the cash-flows in this second financing are: {100, 2.25, 103.25}at times 0, 0.5 and 1 years. The internal rate of return of this sequence of cash-flows is5.48%.

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 underlyingLibor rate six months from now and lock in the rate of 6%. At the end of six months,if the underlying three-month rate is 6.6% over an actual period of 91 days, what isyour payoff given that the payment is made right away? Recall that the Actual/360convention applies.


FRA Settlement Value

-3

-2

-1

0

1

2

3

0 5 10 15 20

Libor (%

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 checkedthat is, in fact, the case:

2s

`2< 0.

23. You anticipate a need to borrow USD 10 million in six-months time for a period of threemonths. You decide to hedge the risk of interest-rate changes using eurodollar futurescontracts. Describe the hedging strategy you would follow. What if you decided to usean FRA instead?


Answer: A simple way to hedge interest rate changes over the next six months is toenter into a long 6 9 FRA. This is a clean and exact hedge. However, we may alsouse eurodollar futures. To hedge borrowing costs, we need to make money on the hedgewhen 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 moneywhen interest rates rise, we need to short eurodollar futures contracts. Finally, since oneeurodollar futures contract is for a notional value of 1 million, we need to short 10 ofthese contracts.

As noted in the text, however, eurodollar futures are settled in undiscounted form, sounlike using an FRA, the hedge obtained with eurodollar futures will not be perfect evenin theory.

24. In Question 23, suppose that the underlying three-month Libor rate after six months (asimplied by the price of the eurodollar futures contract expiring in 6 months) is currentlyat 4%. Assume that the three-month period has 90 days in it. Using the same numbersfrom Question 23 and adjusting for tailing the hedge, how many futures contracts areneeded? Assume fractional contracts are permitted.

Answer: As noted, without tailing the hedge, we need a short position in 10 contracts.If the hedge is tailed using the 4% rate reflected in current eurodollar prices, then thenumber of contracts needed is

10

1 + 0.04(90/360)= 9.901.

25. Using the same numbers as in the previous two questions, compute the payoff after sixmonths (i.e., at maturity) under (a) an FRA and (b) a tailed eurodollar futures contractif the Libor rate at maturity is 5%, and the locked-in rate in both cases is 4%. Alsocompute the payoffs if the Libor rate ends up at 3%. Comment on the difference inpayoffs of the FRA versus the eurodollar futures.

Answer: First suppose that the Libor rate at maturity is 5%.

(a) For the FRA, the payoff is:

10, 000, 000 (0.05 0.04) (90/360)1 + 0.05(90/360)

= 24, 752.50.

(b) For the tailed eurodollar futures, the payoff is

9.901 (0.05 0.04) 10, 000 25 = 25, 024.75

Now suppose the Libor rate at maturity is 3%.


(a) For the FRA, the payoff is:

10, 000, 000 (0.03 0.04) (90/360)1 + 0.03(90/360)

= 24, 813.90

(b) For the eurodollar futures, the payoff is:

9.901 (0.03 0.04) 10000 25 = 24, 752.50

In either case, the eurodollar futures contract does better than the FRA: in the firstcase, 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 standard bond in the Treasury bond futures contract has a coupon of 6%. If,instead, delivery is made of a 5% bond of maturity 18 years, what is the conversionfactor for settlement of the contract? Assume that the last coupon on the bond wasjust paid.

Answer: Assume a principal amount of $100. Then, the bond pays 2.50 every six monthsand also repays the principal after 18 years. The present value of these cash flows whendiscounted at the 6% standard rate is

2.5

1.03+

2.5

1.032+ + 2.5

1.0335+

102.5

1.0336= 89.946

Hence, the conversion factor is 0.89946.

27. Suppose we have a flat yield curve of 3%. What is the price of a Treasury bond ofremaining maturity seven years that pays a coupon of 4%? (Coupons are paid semian-nually.) What is the price of a six-month Treasury bond futures contract? Make anyassumption you require concerning the maturity of the delivered bond to find this price.

Answer: Assuming the last coupon was just paid, the price of a the remaining seven-yearmaturity bond with coupon of 4% is equal to the cash flows from the bond discountedat a flat 3% rate (i.e., at 1.5% every six months):

2

1.015+

2

1.0152+ + 2

1.01513+

102

1.01514= 106.27.

A Treasury bond futures contract requires the delivery of a Treasury bond with couponof 6% and any maturity of at least 15 years. To price the six-month futures contract,we (i) treat it as a forward contract, and (ii) assume that the maturity of the delivered

Solutions Ch1 6

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