Solution of Derivative

5 therefore to buy a long butterfly spread we buy the

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: aiting to exercise the option, because exercising gives us a share of AOL, which does not pay interest in form of a dividend. However, by early exercising we will forego the interest we could earn on Apple. Therefore, it is again never optimal to exercise the American put option early. Question 9.16. a) Short call perspective: Transaction Sell Call Buy put Buy share Borrow @ rB TOTAL t=0 +CB − PA −SA + Ke − rBT C B − P A − S A + Ke − rBT time = T ST ≤ K 0 K − ST ST -K 0 time=T ST > K K − ST 0 ST −K 0 In order to preclude arbitrage, we must have: C B − P A − S A + Ke − rBT ≤ 0 62 Chapter 9 Parity and Other Option Relationships b) Long call perspective: Transaction Buy call Sell put Sell share Lend @ rL TOTAL time = T ST ≤ K 0 t=0 −CA ST − K + PB B − ST +S − rLT +K − Ke A B B − rLT 0 − C + P + S − Ke time=T ST > K ST − K 0 − ST +K 0 In order to preclude arbitrage, we must have: − C A + P B + S B − Ke − rLT ≤ 0 Question 9.18. a) Since the strike prices are symmetric, lambda is equal to 0.5. Therefore, to buy a long butterfly spread, we buy the 95-strike call, sell two 100 strike calls and buy one 105 strike call. C (K1 ) − C (K 2 ) 13.2 − 8.7 = 0 .9 = 100 − 95 K 2 − K1 and C (K 2 ) − C (K 3 ) 8.7 − 5.4 = 0.66 = 105 − 100 K3 − K2 and C (K 2 ) − C (K 3 ) 9 − 5.20 = 0.76 = 105 − 100 K3 − K2 Convexity holds. b) C (K1 ) − C (K 2 ) 12.90 − 9 = 0.78 = 100 − 95 K 2 − K1 Convexity holds. c) A market maker can buy a butterfly spread at the prices we sell it for. Therefore, the above convexity conditions are the ones relevant for market makers. Convexity is not violated from a market maker’s perspective. d) Let us check the conditions for the retail seller of a butterfly spread (she is the one who buys the 100 strike option). Please note that this is the same position as that of a market maker who buys the butterfly spread. We have: C (K1 ) − C (K 2 ) 12.90 − 9.20 = 0.74 and = 100 − 95 K 2 − K1 Therefore, convexity is indeed violated. 63 C (K 2 ) − C (K 3 ) 9.2 − 5.20 = 0 .8 = 105 − 100 K3 − K2 Chapter 10 Binomial Option Pricing: I Question 10.2. a) Using the formulas of the textbook, we obtain the following results: ∆ = 0 .7 B = −53.8042 price = 16.1958 b) If we observe a price of $17, then the option price is too high relative to its theoretical value. We sell the option and synthetically create a call option for $19.196. In order to do so, we buy 0.7 units of the share and borrow $53.804. These transactions yield no risk and a profit of $0.804. c) If we observe a price of $15.50, then the option price is too low relative to its theoretical value. We buy the option and synthetically create a short position in an option. In order to do so, we sell 0.7 units of the share and lend $53.8042. These transactions yield no risk and a profit of $0.696. Question 10.4. The stock prices evolve according to the following picture: 169 call-payoff: 74 104 call-payoff: 9 64 call-payoff: 0 130 100 80 Since we have two binomial steps, and a time to expiration of one year, h is equal to 0.5. Therefore, we can calculate with the usual formulas for the respective nodes: t=0, S=100 t=1, S=80 t=1, S=130 64 Chapter 10 Binomial Option Pricing: 1 ∆ = 0.691 ∆ = 0.225 ∆ =1 B = −49.127 price = 19.994 B = −13.835 price = 4.165 B = −91.275 price = 38.725 Question 10.6. The stock prices evolve according to the following picture: 169 put-payoff: 0 104 put-payoff: 0 64 put-payoff: 31 130 100 80 Since we have two binomial steps, and a time to expiration of one year, h is equal to 0.5. Therefore, we can calculate with the usual formulas for the respective nodes: t=0, S=100 ∆ = −0.3088 B = 38.569 price = 7.6897 t=1, S=80 ∆ = −0.775 B = 77.4396 price = 15.4396 t=1, S=130 ∆=0 B=0 price = 0 Question 10.8. We must compare the results of the equivalent European put that we calculated in exercise 10.6. with the value of immediate exercise. In 10.6., we calculated: t=1, S=80 ∆ = −0.775 B = 77.4396 price = 15.4396 immediate exercise = max(95 - 80, 0 ) = 15 t=1, S=130 ∆=0 B=0 price = 0 immediate exercise = max(95 − 130,0) = 0 65 Part 3 Options Since the value of immediate exercise is smaller than or equal to the continuation value (of the European options) at both nodes of the tree, there is no benefit to exercising the options before expiration. Therefore, we use the European option values when calculating the t = 0 option price: t=0, S=100 ∆ = −0.3088 B = 38.569 price = 7.6897 immediate exercise = max (95 - 100, 0 ) = 0 Since the option price is again higher than the value of immediate exercise (which is zero), there is no benefit to exercising the option at t = 0. Since it is never optimal to exercise earlier, the early exercise option has no value. The value of the American put option is identical to the value of the European put option. Question 10.10. We can calculate for the different nodes of the tree: delta B Call premium value of early exercise node uu node ud = du 1 0.8966 -92.5001 -79.532 56.6441 15.0403 54.1442 10.478 node dd 0...
View Full Document

Ask a homework question - tutors are online