Chapter 15 Option on Stock Indices and Currencies Promble 15.1. When the index goes down to 7, the value of the portfolio can be expected to be 1*(7/8)=$8.75 million.(this assumes that the dividend yield on the portfolio equals the dividend yield on the index.) Buying put option on 1,,/8=12,5 times the index with a strike of 48 therefore provides protection against a drop in the value of the portfolio below $8.75 million. If each contract is on 1 times the index, a total of 125 contracts would be required. Promble 15.2. stock index is analogous to a stock paying a continuous dividend yield, the dividend yield being the dividend yield on the index. currency is analogous to a stock paying a continuous dividend yield, the dividend yield being the foreign risk-free interest rate. Promble 15.3. Promble 15.4. Finger 15.1 Tree for Promble 15.4
Promble 15.5. If a US Company that knows it will receive one million pounds after three month. In order to limit the downside risk of the exchange rate, the company could look in this exchange rate by entering into a short range forward contract that is to buy a European put option with the strike price of K 1 and sell a European call option with a strike price K 2. So the company locks in this exchange rate between K 1 and K 2. If the knows it was due to pay rather than receive one million pounds in three months, it could sell a European put option with the strike price of K 1 and buy a European call option with a strike price K 2. Promble 15.6. Promble 15.7.
Promble 15.8. In this case, we assume that the current price of one unit of currency for currency B is S. So the current price of one unit of currency B for currency is 1 S. nd for the put option, r = r, r = r, for the call option to buy one unit of currency B for currency at strike price 1/K, f B r = r, r = r. Substituting it to (15.11) and (15.12) f B 1 1 B c= e Nd ( 1) e Nd ( 2) S K p = S e N( d ) Ke N( d ) B 2 1 where d 1 = 2 ln( / ) + ( B + σ / 2) K S r r T σ T 2 ln( / ) + ( B σ / 2) S K r r T = = d σ T 2 d = d 2 1 So K B Kc= e N( d2) e N( d1) S 1 ( B ( 2 ) = Ke N d Se N ( d1 )) S t the same time, valuing the call option use the currency, but the put option is the currency B. So the put option to sell currency for currency B at strike price K is the same as a call option to buy B with currency at strike price 1/k. Promble 15.9.
Promble 15.1. Promble 15.11. Promble 15.12.
Promble 15.13. From the put-call parity for currency option, rt f c + Ke = p + Se If the forward price equal the strike price, that is So That is c ( r r ) T K= F = Se ( r rf ) T f c+ Ke = c+ Se e = c+ Se = p. Promble 15.14. f Promble 15.15. Promble 15.16.
Promble 15.17. Promble 15.18. In this case, S = 15, K = 14, T =, c= 154, p= 34.25, r =.5. So the implied dividend yield is
.5.5 154 34.25 + 14 e q = 2ln =.199 15 or 1.99%. Promble 15.19. Promble 15.2. Promble 15.21. Promble 15.22.
This proves equation (15.1). This proves equation (15.2). Portfolio and C are both worth max( S T, K) at time T. They must, therefore, be worth the same today, and the put-call parity result in equation(15.3) follows. Promble 15.23. In this case, S = 125.56, K = 126, c= 2.25, q=.3, r =.53, use the DerivaGem software, the implied volatility for the call option is 1.96%. nd from the put-call parity 2.25 + 126 e = p+ 125.56 e.53 1/6.3 1/6 so p= + e e =.53 1/6.3 1/6 2.25 126 125.56 2.281 Using the DerivaGem software, the implied volatility for the put option is 1.96% that equal the call option.
Promble 15.24. with the corresponding European and merican option prices is shown in Figure15.2. The European price is $11.88, while the merican price is $13.56. Figure 15.2 Tree to evaluate European and merican put option in Problem 15.24 t each node,upper number is the stock price; next number is the European put price; final number is the merican put price Promble 15.25. In this case S =.85, K =.85, r =.5, r =.4, σ =.4, T =.75. The option can be valued using equation(15.11) d 1 f ln(.85 /.85) + (.5.4 +.16 / 2).75 = =.2338.4.75 nd d2 = d1.4.75 =.1992 N( d 1)=.5924, N( d 2)=.5789 The value of the call, c, is given by c= e e =.4.75.5.75.85(.5924-.5789).147
i.e., it is 1.47 cents. From put-call parity So that rt f p + S e = c + Ke p = + e e.5.75.4.75.147.85.85 =.85 The option to buy.85 USD with 1. CDN is the same as the an option to sell one Canadian dollar for.85 USD. In other words, it is a put option on the Canadian dollar and its price is.85 USD. Promble 15.26.
Promble 15.27. 2 rate might be expected to be r rbrather than r r + σ. Promble 15.28. In this case F = 1.3, r =.4, r =.5, σ =.15, K1 = 1.25 (a) So the put option can be valued using equation(15.14) d 1 f 2 ln(1.3 /1.25) +.15 / 2.25 = =.564.15.25 B nd d2 = d1.15.25 =.4854 N( d 1)=.7124, N( d 2)=.6863
N(- d1)=1-n( d 1)=1-.7124=.2876, N(- d2)=1-n( d 2)=1-.6863=.3137 The value of the put, p, is given by.4 p= e.25 (1.25.3137-1.3.2876) =.181 In order to creating a zero-cost contract, the price of the call option must be the same as the put option,.181. So the strike price of the call option is 1.355. (b) Because the company will have to pay 1 million Euros, it could sell a European put option with the strike price 1.25 and buy a European call option with the strike price 1.355. (c) The answer don t dependent on the euro risk-free rate, because we use the forward price and the equation (15.14) don t include the euro risk-free rate. (In fact, the forward price include the euro risk-free rate as the dividend yield.) (d) The answer dependent on the USD risk-free rate. s the equation (15.14), the USD risk-free rate used as the risk-free rate.