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CHAPTER 15
Options on Stock Indices and Currencies
Practice Questions
Problem 15.8. Show that the formula in equation (15.9) for a put option to sell one unit of currency A for currency B at strike price gives the same value as equation (15.8) for a call option to buy units of currency B for currency A at a strike price of. A put option to sell one unit of currency A for units of currency B is worth where and and are the risk-free rates in currencies A and B, respectively. The value of the option is measured in units of currency B. Defining and The put price is therefore where This shows that put option is equivalent to call options to buy 1 unit of currency A for units of currency B. In this case the value of the option is measured in units of currency A. To obtain the call option value in units of currency B (the same units as the value of the put option was measured in) we must divide by. This proves the result. Problem 15.9. A foreign currency is currently worth $1.50. The domestic and foreign risk-free interest rates are 5% and 9%, respectively. Calculate a lower bound for the value of a six-month call option on the currency with a strike price of $1.40 if it is (a) European and (b) American.
K
K^1 / K
K
Ke -^ r TB^ N - d - S e - r TA N - d 2 0 1 ln( S K ) ( rB rA 2) T d T
s
s
2 0 2 ln( S K ) ( rB rA 2) T d T
s
s
r A rB S 0 (^) 1 S 0
= / K *^ = 1 / K 2 0 1 ln( S K ) ( rA rB 2) T d T
- *^ / * - - - /
s
s
2 0 2 ln( S K ) ( rA rB 2) T d T
s
s
0 [^0 (^1 )^ (^2 )
S K S e *^ - r T^ B^ N d *^ - K e *^ - r TAN d * 2 0 1 2 ln( S K ) ( rA rB 2) T d d T
- (^) = - = /^ +^ -^ +^ s / s 2 0 2 1 ln( S K ) ( rA rB 2) T d d T
- (^) = - = /^ +^ -^ -^ s / s KS 0
1 / K
S 0
Lower bound for European option is Lower bound for American option is Problem 15.10. Consider a stock index currently standing at 250. The dividend yield on the index is 4% per annum, and the risk-free rate is 6% per annum. A three-month European call option on the index with a strike price of 245 is currently worth $10. What is the value of a three-month put option on the index with a strike price of 245? In this case , , , , , and. Using put–call parity or Substituting: The put price is 3.84. Problem 15.11. An index currently stands at 696 and has a volatility of 30% per annum. The risk-free rate of interest is 7% per annum and the index provides a dividend yield of 4% per annum. Calculate the value of a three-month European put with an exercise price of 700. In this case , , , , and. The option can be valued using equation (15.5). and The value of the put, , is given by: i.e., it is $40.6. Problem 15.12. Show that if is the price of an American call with exercise price and maturity on a stock paying a dividend yield of , and is the price of an American put on the same stock with the same strike price and exercise date, where is the stock price, is the risk-free rate, and. (Hint: To obtain the first half of the inequality, consider possible values of: Portfolio A; a European call option plus an amount invested at the risk-free rate 0 09 0 5 0 05 0 5
0 1 5^ 1 4^ 0 069
S e^ -^ r Tf^ - Ke -^ rT =. e -^.^ ´^.^ -. e -^.^ ´^. =.
S 0 - K = 0 10.
S 0 = 250 q = 0 04. r = 0 06. T = 0 25. K = 245 c = 10
0 rT qT c Ke p S e
= + - p = 10 + 245 e -^ 0 25 0 06^.^ ´^.^ - 250 e - 0 25 0 04^.^ ´^. = 3 84.
S 0 = 696 K = 700 r = 0 07. s = 0 3. T = 0 25. q = 0 04.
1 2 1 ln(696 700) (0 07 0 04 0 09 2) 0 25 0 0868 0 3 0 25 0 3 0 25 0 0632 d d d
N ( - d 1 (^) ) = 0 4654. , N ( - d 2 ) = 0 5252. p 0 07 0 25 0 04 0 25 p 700 e 0 5252 696 e 0 4654 40 6
-. ´. -. ´. = ´. - ´. =.
C K T
q P 0 0 S e -^^ qT^ - K < C - P < S - Ke - rT
S 0 r r > 0
K
Since portfolio D is worth at least as much as portfolio C in all circumstances: Since : or This proves the second part of the inequality. Hence: Problem 15.13. Show that a European call option on a currency has the same price as the corresponding European put option on the currency when the forward price equals the strike price. This follows from put–call parity and the relationship between the forward price, , and the spot price, and so that If this reduces to. The result that when is true for options on all underlying assets, not just options on currencies. An at-the-money option is sometimes defined as one where (or ) rather than one where. Problem 15.14. Would you expect the volatility of a stock index to be greater or less than the volatility of a typical stock? Explain your answer. The volatility of a stock index can be expected to be less than the volatility of a typical stock. This is because some risk (i.e., return uncertainty) is diversified away when a portfolio of stocks is created. In capital asset pricing model terminology, there exists systematic and unsystematic risk in the returns from an individual stock. However, in a stock index, unsystematic risk has been diversified away and only the systematic risk contributes to volatility. Problem 15.15. Does the cost of portfolio insurance increase or decrease as the beta of a portfolio increases? Explain your answer. The cost of portfolio insurance increases as the beta of the portfolio increases. This is because portfolio insurance involves the purchase of a put option on the portfolio. As beta increases, the volatility of the portfolio increases causing the cost of the put option to increase. When index options are used to provide portfolio insurance, both the number of options required and the strike price increase as beta increases. 0 rT C Ke p S
- £ + p £ P 0 C + Ke -^ rT £ P + S 0 rT C P S Ke
- £ - 0 0 qT rT S e K C P S Ke
- £ - £ - F 0 S 0 0
c + Ke - rT = p + S e - r T^ f
( ) 0 0
F = S e r^ - r^ f^ T
0 c + Ke -^^ rT^ = p + F e - rT K = F 0 c = p c = p K = F 0 K = F 0 c = p K = S 0
Problem 15.16. Suppose that a portfolio is worth $60 million and the S&P 500 is at 1200. If the value of the portfolio mirrors the value of the index, what options should be purchased to provide protection against the value of the portfolio falling below $54 million in one year’s time? If the value of the portfolio mirrors the value of the index, the index can be expected to have dropped by 10% when the value of the portfolio drops by 10%. Hence when the value of the portfolio drops to $54 million the value of the index can be expected to be 1080. This indicates that put options with an exercise price of 1080 should be purchased. The options should be on: times the index. Each option contract is for $100 times the index. Hence 500 contracts should be purchased. Problem 15.17. Consider again the situation in Problem 15.16. Suppose that the portfolio has a beta of 2.0, the risk-free interest rate is 5% per annum, and the dividend yield on both the portfolio and the index is 3% per annum. What options should be purchased to provide protection against the value of the portfolio falling below $54 million in one year’s time? When the value of the portfolio falls to $54 million the holder of the portfolio makes a capital loss of 10%. After dividends are taken into account the loss is 7% during the year. This is 12% below the risk-free interest rate. According to the capital asset pricing model, the expected excess return of the portfolio above the risk-free rate equals beta times the expected excess return of the market above the risk-free rate. Therefore, when the portfolio provides a return 12% below the risk-free interest rate, the market’s expected return is 6% below the risk-free interest rate. As the index can be assumed to have a beta of 1.0, this is also the excess expected return (including dividends) from the index. The expected return from the index is therefore 1% per annum. Since the index provides a 3% per annum dividend yield, the expected movement in the index is 4%. Thus when the portfolio’s value is $54 million the expected value of the index is 0.96×1,200 = 1,152. Hence European put options should be purchased with an exercise price of 1,152. Their maturity date should be in one year. The number of options required is twice the number required in Problem 15.16. This is because we wish to protect a portfolio which is twice as sensitive to changes in market conditions as the portfolio in Problem 15.16. Hence options on $100,000 (or 1,000 contracts) should be purchased. To check that the answer is correct consider what happens when the value of the portfolio declines by 20% to $48 million. The return including dividends is 17%. This is 22% less than the risk-free interest rate. The index can be expected to provide a return (including dividends) which is 11% less than the risk-free interest rate, i.e. a return of 6%. The index can therefore be expected to drop by 9% to 1,092. The payoff from the put options is (1,152-1,092)×100,000 = $6 million. This is exactly what is required to restore the value of the portfolio to $54 million. Problem 15.18. An index currently stands at 1,500. European call and put options with a strike price of 1, and time to maturity of six months have market prices of 154.00 and 34.25, respectively. The six-month risk-free rate is 5%.What is the implied dividend yield?
this must also be true today. Hence, or This proves equation (15.1) In portfolio C, the reinvestment of dividends means that the portfolio is one put option plus one share at time T. If , the put option is exercised at time and portfolio C is worth
. If the put option expires worthless and the portfolio is worth. Hence, at time T , portfolio C is worth Portfolio D is worth at time T. It follows that portfolio C is always worth as much as, and is sometimes worth more than, portfolio D at time T. In the absence of arbitrage opportunities, this must also be true today. Hence, or This proves equation (15.2) Portfolios A and C are both worth at time T. They must, therefore, be worth the same today, and the put–call parity result in equation (15.3) follows. Further Questions Problem 15.23. The Dow Jones Industrial Average on January 12, 2007 was 12,556 and the price of the March 126 call was $2.25. Use the DerivaGem software to calculate the implied volatility of this option. Assume that the risk-free rate was 5.3% and the dividend yield was 3%. The option expires on March 20, 2007. Estimate the price of a March 126 put. What is the volatility implied by the price you estimate for this option? (Note that options are on the Dow Jones index divided by 100. Options on the DJIA are European. There are 47 trading days between January 12, 2007 and March 20, 2007. Setting the time to maturity equal to 47/252 = 0.1865, DerivaGem gives the implied volatility as 10.23%. (If instead we use calendar days the time to maturity is 67/365=0.1836 and the implied volatility is 10.33%.) From put call parity (equation 15.3) the price of the put, , (using trading time) is given by so that. DerivaGem shows that the implied volatility is 10.23% (as for the call). (If calendar time is used the price of the put is 2.1597 and the implied volatility is 10.33% as for the call.) A European call has the same implied volatility as a European put when both have the same strike price and time to maturity. This is formally proved in the appendix to Chapter 19. Problem 15. A stock index currently stands at 300 and has a volatility of 20%. The risk-free interest rate is 8% and the dividend yield on the index is 3%. Use a three-step binomial tree to value a six- 0 rT qT c Ke S e
³ - S T < K T K ST > K , ST max ( ST , K ) K 0 p + S e -^^ qT^ ³ Ke - rT 0 p ³ Ke -^^ rT^ - S e - qT max ( ST , K ) p 2 25. + 126 e -^ 0 053 0 1865^.^ ´^.^ = p + 125 56. e - 0 03 0 1865^.^ ´^. p = 2 1512.
month put option on the index with a strike price of 300 if it is (a) European and (b) American? (a) The price is 14.39 as indicated by the tree in Figure S15.1. (b) The price is 14.97 as indicated by the tree in Figure S15. Figure S15.1 Tree for valuing the European option in Problem 15. Figure S15.2 Tree for valuing the American option in Problem 15. Problem 15.25. At each node: Upper value = Underlying Asset Price Lower value = Option Price Values in red are a result of early exercise. Strike price = 300 Discount factor per step = 0. Time step, dt = 0.1667 years, 60.83 days Growth factor per step, a = 1. Probability of up move, p = 0. Up step size, u = 1.0851 383. Down step size, d = 0.9216 0
0 325.5227 325. 5.042274 0 300 300 14.3917 10. 276.4784 276. 25.37969 23.
Node Time: 0.0000 0.1667 0.3333 0. At each node: Upper value = Underlying Asset Price Lower value = Option Price Values in red are a result of early exercise. Strike price = 300 Discount factor per step = 0. Time step, dt = 0.1667 years, 60.83 days Growth factor per step, a = 1. Probability of up move, p = 0. Up step size, u = 1.0851 383. Down step size, d = 0.9216 0
0 325.5227 325. 5.042274 0 300 300 14.97105 10. 276.4784 276. 26.631 23.
Node Time: 0.0000 0.1667 0.3333 0.
c. Show that your answer to (a) does not depend on interest rates provided that the interest rate differential between the two currencies, r – rf , remains the same. (a) A put with a strike price of 1.25 is worth $0.019. By trial and error DerivaGem can be used to show that the strike price of a call that leads to a call having a price of $0.019 is 1.3477. This is the higher strike price to create a zero cost contract. (b) The company should sell a put with strike price 1.25 and buy a call with strike price 1.3477. This ensures that the exchange rate it pays for the euros is between 1.2500 and 1.3477. (c) If the interest rates change so that the spread between the dollar and euro interest rates remains the same, forward prices remain the same. From equations (15.10) and (15.11). changes to r have the same proportional effect on both c and p. If the relationship c = p holds for one value of r , it holds for all values of r. as a result the answer to (a) is unchanged when the spread between the two rates is held the same. Problem 15. In Business Snapshot 15.1 what is the cost of a guarantee that the return on the fund will not be negative over the next 10 years? In this case the guarantee is valued as a put option with S 0 = 1000, K = 1000, r = 5%, q = 1%, s = 15%, and T =10. The value of the guarantee is given by equation (15.5) as 38.46 or 3.8% of the value of the portfolio. Problem 15. The one-year forward price of the Mexican peso is $0.0750 per MXN. The U.S. risk-free rate is 1.25%. The exchange rate volatility is 13%. What is the value of one-year European call and put options with a strike price of $0.0800. Using equations (15.10) and (15.11) the values of the call and put are 0.0020 and 0.0069, respectively Note that we do not need the Mexican risk-free rate when we use forward prices for the valuation.
