Chapter_9_Solutions_9e

CHAPTER 9: PRINCIPLES OF PRICING FORWARDS, FUTURES, AND OPTIONS ON FUTURES END OF CHAPTER QUESTIONS AND PROBLEMS 1. (Price of a Forward Contract) The price of a forward contract is the spot price compounded to expiration at the risk-free rate: F(0,T) = S0(1 + r)T Here, S0 = $45, r = 0.0601, T = 3/12. Thus: F(0,T) = 45(1.0601)0.25 = $45.661399 2. (Forward Versus Futures Prices) If interest rates are positively correlated with futures prices, an investor holding a long position will prefer futures contracts because futures contracts will generate mark to market profits during periods of rising interest rates. This means that gains will be reinvested at higher rates and losses will be incurred when the opportunity cost is falling. Futures contracts would, therefore, carry higher prices than forward contracts. 3. (Commodities and Storage Costs) In order to calculate the futures price given the initial risk-free rate of 6.83 percent (where S0 = $49.90, r = 0.0683, T = 8/12, and s = $5.60): F(0,T) = S0(1 + r)T + s = $49.90(1.0683)8/12 + 5.60 = $57.747004 If the risk-free rate were 6.60 percent instead, the futures price would be: F(0,T) = $49.90(1.066)8/12 + 5.60 = $57.672131 Therefore, the net effect of a change in the risk-free rate is given by: $57.747004 - $57.672131 = $.074873 4. (Futures Prices and Risk Premia) The characteristics of contango, backwardation, normal contango, and normal backwardation markets are given below: Contango market: here the futures price exceeds the current spot price. Thus, S0 < f0(T). Backwardation: this is a market in which the futures price lies below the current spot price. Thus, f0(T) < S0. Normal Contango: this is a market where the futures price is above the expected future spot price. Thus, E(ST) < f0(T). Normal Backwardation: this is a market in which the futures price is below the expected future spot price. Thus, f0(T) < E(ST). 5. (Forward/Futures Pricing Revisited) The futures price will not be the expected spot price in September because the dominance of the long hedgers will induce a risk premium. Thus, the futures price of $2.76 is biased low. Without information on the magnitude of the risk premium, it is impossible to come up with a precise estimate. The expected spot price in September, however, is no less than $2.76. Of course, the actual spot price in September could be far less. 6. (Early Exercise of Call and Put Option on Futures) In Chapters 3, 4 and 5 we covered American call options on the spot and explained that in the absence of dividends they will not be exercised early. They will always sell for at least the lower bound, which is higher than the intrinsic value, and usually more. Call options on the futures, however, might be exercised early. If the price of the underlying instrument is extremely high, the call will begin to behave like the underlying instrument. For an option on a futures, this means that the call will behave like the futures, changing almost dollar-for-dollar with the futures price. For an option on the spot, the call will behave like the spot, changing almost one-for-one with the price of the spot. Exercise of the futures call will release funds tied up in the call and provide a position in the futures. Exercise of the call on the spot does not, however, release funds, since the investor has to purchase the spot instrument. 7. (Black Futures Option Pricing Model) The Black model is not an American option on futures pricing model and Eurodollar options on futures are American. Also the Black model assumes constant interest rates. Since Eurodollars are interest-sensitive instruments, they violate an assumption of the model. 8. (Black Futures Option Pricing Model) A spot option pricing model such as Black-Scholes-Merton is a model for pricing options on instruments with a cost of carry of rc (or rc – δ if there is a dividend yield). Since a futures requires no outlay of funds nor does it incur a storage cost, it has no cost of carry. The cost of carry relevant to the futures price is the cost of carry of the underlying spot instrument. An option on a futures is, therefore, an option on an instrument with a zero cost of carry. The futures option pricing model is the same as the spot option pricing model where the spot price is replaced with the futures price and the cost of carry is zero. The latter is established by assuming a dividend yield equal to the interest rate. 9. (Value of a Forward Contract) The value of the forward contract can be found by subtracting the present value of the forward price from the current spot price. Thus, the value of the contract is $52 – $45(1.10)-0.5 = $9.09. This is the correct value of the contract at this point, six months into the life of the contract, because it is the value of a portfolio that could be constructed at this time to produce the same result six months later. That is, you could buy the asset costing $52 and take out a loan, promising to pay $45 in six months. This combination would guarantee that you would receive at time T, six months later, the value of the asset ST minus the $45 loan repayment, which is the value of the forward contract when it expires. 10. (Value of a Futures Contract) The value at the opening is 899.70 – 899.30 = 0.40 In dollars, this is 0.40($250) = $100 An instant before the close, the value is (899.10 – 899.30) = –0.20 In dollars, that is –0.20($250) = –$50 After the market has closed, the contract is marked-to-market, the gain or loss is distributed, and the value is zero. 11. (Price of a Futures Contract) Consider a futures contract on a stock. Say you sell short the stock and buy the futures. You receive S0 up front and during the holding period you earn interest of iS0. If the stock pays dividends, a short seller has to make them up so you would incur a cost of DT where DT is the compound future value of the dividends. At expiration, you accept delivery of the stock and pay f. Thus your profit is S0 – f0(T) + iS0 – DT Since this must equal zero, f0(T) = S0 + iS0 – DT 12. (Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets) (a) In a market with risk premiums, the futures price underestimates the spot price at expiration by the amount of the risk premium. Therefore, the expected spot price in December is $3.64 + $0.035 = $3.675. (b) Arbitrage assures us that whether or not a risk premium exists, the futures price equals the spot price plus the cost of carry. This is confirmed by noting that the spot price of $3.5225 plus the cost of carry of $.1175 equals the futures price of $3.64. (c) The answer is apparent in part (a). The expected price of wheat in December exceeds the futures price by the risk premium. (d) If there is a risk premium, holders of long futures contracts expect to sell them for a profit equal to the risk premium. Thus, the expected futures price at expiration is $3.64 + $.035 = $3.675, which is also the expected spot price at expiration. (e) Speculators who take long positions in futures earn the risk premium. They do so because they are supplying insurance to the hedgers and, therefore, expect to receive a return in compensation for their willingness to take the risk. 13. (Stock Indices and Dividends) (a) T = 73/365 = 0.2 f0(T) = 956.49e(0.0596 - 0.0275)(0.2) = 962.65 At 960.50, it is underpriced. (b) f0(T) = 956.49(1 + 0.0596)0.2 – 5.27= 962.36 At 960.50, it is underpriced. The main difference is compounding of interest. Annual compounding results in lower proceeds than continuous, hence the annual compounded carrying cost is lower than the continuous compounding. 14. (Futures Prices and Risk Premia) E(ST) = 60, E(() = 4, ( = 5.50 E(ST) = E(fT(T)) = 60 f0(T) = E(fT(T)) – E(() = 60 – 4 = 56 15. (Foreign Currencies and Foreign Interest Rates: Interest Rate Parity) (a) S0 = $0.009313, F = $0.010475, r = 0.07, ( = 0.01 With annual compounding, the forward rate should be So the forward rate should be $0.01045 but is actually $0.010475. Thus, the forward contract is overpriced. You should buy the yen in the spot market and sell it in the forward market. (b) With continuous compounding, the forward rate should be So the forward rate should be $0.01050 but is actually $0.010475. Thus, the forward contract is underpriced. You should sell the yen in the spot market and buy it in the forward market. 16. (Put-Call Parity of Options on Futures) 100 days between September 12 and December 21, T = 100/365 = 0.2740 C(f0(T),T,X) – P(f0(T),T,X) = 26.25 – 3.25 = 23 (f0(T) – X)(1 + r)-T = (423.70 – 400)(1.0275)-0.2740 = 23.52 We can view the futures as overpriced and assume the call and put are correctly priced. We sell the futures, buy a call and sell a put. Payoffs at Expiration ST ( X ST > X Short futures –(ST – f0(T)) –(ST – f0(T)) Long call 0 ST – X Short put –(X – ST) 0 f0(T) – X f0(T) – X f0(T) – X = 423.70 – 400 = 23.70. The present value of this is 23.70(1.0275)-0.2740 = 23.52. The portfolio will cost 26.25 – 3.25 = 23.00. Thus, you will earn a present value of 23.52 – 23 = 0.52. 17. (Pricing Options on Futures) The option’s life is January 31 to March 18, so T = 46/365 = 0.1260 a. Intrinsic Value = Max(0, f0 – X) = Max(0, 483.10 – 480) = 3.10 b. Time Value = Call Price – Intrinsic Value = 6.95 – 3.10 = 3.85 c. Lower bound = Max[0, (f0 – X)(1 + r)-T] = Max[0, (483.10 – 480)(1.0284)-0.1260] = 3.09 d. Intrinsic Value = Max(0, X – f0) = Max(0, 480 – 483.10) = 0 e. Time Value = Put Price – Intrinsic Value = 5.25 – 0 = 5.25 f. Lower bound = Max[0, (X – f0)(1 + r)-T] = Max[(0, (480 – 483.10)(1.0284)-0.1260] = 0 (Note: the lower bound applies only to European puts.) g. C = P + (f0 – X)(1 + r)-T = 5.25 + (483.10 – 480)(1.0284)-0.1260 = 8.34 The actual call price is 6.95, so put-call parity does not hold. 18. (Put-Call Parity of Options on Futures) (f0 – X)(1 + r)-T = (102 – 100)(1.10)-0.25 = 1.95 C – P = 4 – 1.75 = 2.25 C – P is too high so the call is overpriced and/or the put is underpriced (or we could assume the futures is underpriced). So sell the call, buy the put, and buy the futures. At expiration the payoffs will be fT ( X fT > X Short call 0 –(fT – X) Long put X – fT 0 Long futures fT – f0 fT – f0 X – f0 X – f0 This is equivalent to a risk-free loan, as a lender if X > f0 or as a borrower if f0 > X. Here f0 > X so you are a borrower. The present value should be (X – f0)((1 + r)–T = (102 – 100)(1.10)-0.25 = –1.95. Thus you sell the call for 4 and buy the put for –1.75 for a net inflow of 2.25. At expiration, you pay back 2.00. 19. (Black Futures Option Pricing Model) First find the continuously compounded risk-free rate: rc = ln(1.0284) = 0.0280. Then price the option: The option appears to be underpriced. You could sell futures and buy one call, adjusting the hedge ratio through time and earn an arbitrage profit. 20. (Black Futures Option Pricing Model) We already know that N(0.24) = 0.5948 and N(0.21) = 0.5832. Then P = 480e-0.0280(0.1260)[1 – 0.5832] – 483.10e-0.0280(0.1260)[1 – 0.5948] = 4.30 21. (Foreign Currencies and Foreign Interest Rates: Interest Rate Parity) The correct forward price is given by = 1.665(1.015)/(1.02)= 1.6568 Because the forward price is higher than the model price, we will sell the forward contract. If transaction costs could be covered, you would buy the foreign currency in the spot market at $1.665 and sell it in the forward market at $1.664. You would earn interest at the foreign interest rate of 2 percent. By selling it forward, you could then convert back to dollars at the rate of $1.664. In other words, $1.665 would be used to buy 1 unit of the foreign currency, which would grow to 1.02 units (the 2 percent foreign rate). Then 1.02 units would be converted back to 1.02($1.664) = $1.69728. This would be a return of $1.69728/$1.665 – 1 = 0.019387 or 1.9 percent, which is better than the U. S. rate. 22. (Lower Bound of a European Option on Futures) f0(T) = $100, X = 90, and r = 5%. The lower bound on a futures option is Ce(f0(T), T, X) ≥ Max[0, (f0(T) – X)(1 + r)–T] = Max[0, (100 – 90)(1 + 0.05)-1] = 9.5238 The quoted price of $9.40 violates the lower bound and the quoted price is low. Therefore, we would buy the futures call option and hedge the resulting risk as illustrated in the following cash flow table. Strategy Today (t=0) Expiration (fT(T) X) Expiration (fT(T)>X) Buy futures call option –C = –$9.40 $0 fT(T) – X = fT(T) – 90 Sell futures contract $0 (only margin required) +f0(T) – fT(T) = 100 – fT(T) +f0(T) – fT(T) = 100 – fT(T) Borrow +(f0(T) – X)(1 + r)–T = +$9.5238 –(f0(T) – X) *(1 + r)–T(1 + r)T = –$10 –(f0(T) – X) *(1 + r)–T(1 + r)T = –$10 NET CASH FLOW $0.1238 X – fT(T) (non-negative because fT(T) X) $0 23. (Put-Call-Forward/Futures Parity) We now have three versions of put-call parity. Put-call parity with options on the underlying: Put-call parity with options on futures: Put-call futures parity: 24. (Black Futures Option Pricing Model) Recall the standard Black-Scholes-Merton option pricing model: The generic carry formula for forward contracts is Solving for S0, Substituting this result into the standard Black-Scholes-Merton option pricing formula results in the Black forward option pricing formula. 25. (Stock Indices and Dividends) (a) Find the future value of the dividends 0.75(1.12)(60/365) + 0.85(1.12)(30/365) + 0.90 = 2.522 f0(T) = 100(1.12)(90/365) – 2.522 = 100.312 (b) Since f0(T) = S0 + (, then ( = f0(T) – S0 so 100.312 – 100 = 0.312 This is the compound future value of the interest lost minus the compound future value of the dividends. 26. (Forward/Futures Pricing Revisited) Let the spot price be S0, the futures price be f0(T), and the margin requirement be M. Consider the position of someone who buys the asset and sells a futures contract to form a risk-free hedge. Today: Buy the asset, paying S0, and sell the futures by depositing M dollars in a margin account that earns the rate q where q < r. At expiration: The accumulated costs of storage and the interest lost on S0 dollars add up to θ. When the trader delivers the asset, he receives f0(T). The total amount of cash will be f0(T) – θ + M + interest on M at the rate q. Since the transaction is still risk-free, the amount initially invested must grow at the risk-free rate to equal this future value; however, the spot price does not have to be compounded because the interest on it is already included in the cost of carry. Thus, 1 Solving for f0(T) gives 2 The bracketed term is the difference in interest between the rate q and rate r. If q is less than r, the whole bracketed term is greater than zero so the futures price will be greater than the spot price plus the cost of carry. In other words if the margin account pays interest at less than the risk-free rate, the futures price will be greater than the spot price plus the cost of carry. The higher futures price compensates for the loss of interest. PAGE 86 9th Edition: Chapter 9 End-of-Chapter Solutions © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. _1288091281.unknown _1288091728.unknown _1288091854.unknown _1305021462.unknown _1305021463.unknown _1305021461.unknown _1288091743.unknown _1288091690.unknown _1288091715.unknown _1288091370.unknown _1288090935.unknown _1288091216.unknown _1250946776.unknown _1216554367.unknown _1216554374.unknown _1198305858.unknown