ACT4000, MIDTERM #1 ADVANCED ACTUARIAL TOPICS FEBRUARY 9, 2009 HAL W. PEDERSEN You have 70 minutes to complete this exam. When the invigilator instructs you to stop writing you must do so immediately. If you do not abide by this instruction you will be penalised. All invigilators have full authority to disqualify your paper if, in their judgement, you are found to have violated the code of academic honesty. Each question is worth 10 points. Provide sufficient reasoning to back up your answer but do not write more than necessary. This exam consists of 8 questions. Answer each question on a separate page of the exam book. Write your name and student number on each exam book that you use to answer the questions. Good luck! Question 1. You are considering investments in a single-period binomial market. (As you know, this means that we are currently at time 0 and at time 1 the world will be in one of two states which we will call the upstate and the downstate.) There are two assets available for trade. The first asset currently sells for 10 and at time 1 will be worth 12 in the upstate and 8 in the downstate. The second asset currently sells for 10 and at time 1 will be worth 15 in the upstate and 2 in the downstate. (i) (3 points) What is the current price of an asset that at time 1 pays 1 in the upstate and 0 in the downstate? (ii) (2 points) What is the current price of an asset that at time 1 pays 0 in the upstate and 1 in the downstate? (iii) (5 points) If you do not want any risk, is it possible for you to deposit 100 at time 0 and receive a certain payoff at time 1? Explain how this can be done or why it cannot be done. If this can be done, what is the effective interest-rate you will earn over the period? Question 2. An equity securities market model follows a multi-period binomial model. At each node of the binomial tree the current stock price S will branch to us in the upstate and ds in the downstate. You are given that the initial stock price is 10, u =1.35, d =0.85 and the interest-rate is 5% effective per period. The stock does not pay dividends. (i) (4 points) Compute the price of a European put option on the stock which expires in 4 periods and has a strike price of 9.0. (ii) (6 points) Compute the price of an American put option on the stock which expires in 4 periods and has a strike price of 9.0 and describe the optimal exercise policy for this American put option. 1
2 ACT4000 MIDTERM #1 Question 3. The insurance company you work for has recently began issuing a stock index GIC. The essence of the contract is that the investor places an amount of principal in an account for two years and the investor is guaranteed some minimum effective return over the two-year period. The investor s returns are based on the returns on the TSE 35 index. As pricing actuary, you are told that the product is to guarantee a 0% return and you are to set the maximum total return the investor will receive over the two years so that the insurance company will break even. The continuously compounded risk-free interest rate is 4% and the stock index is currently at 50 and will go to either 25 or 75 at the end of the two years. (i) (5 points) Compute the maximum total return the investor will receive over the two years so that the insurance company will break even. If this break even return does not exist then explain why. (ii) (2 points) What are the embedded options in this contract? (iii) (3 points) Write a general algebraic expression for the cost to the insurance company per $1 invested of providing a maximum return of R, foreachr>0? Question 4. Assume the Black-Scholes option pricing model. Consider a standard European call option on a stock. The strike price of the option is equal to the current price of the stock (i.e. the option is at-the-money). The option has one year to maturity and the stock does not pay dividends. Is the option s delta greater than 0.5, less than 0.5, or equal to 0.5. Justify your answer. Question 5 (Text Question 12.7, page 407). You are given the following data. S = $100 K = $95 σ = 30% r =0.08 δ =0.03 T =0.75 Compute the Black-Scholes price of a call.
ACT4000 MIDTERM #1 3 Question 6 (Text Question 13.2, page 439). You are given the following data assuming a Black-Scholes model. S = $40 σ = 30% r =0.08 δ =0 Suppose you sell a 40-strike put with 91 days to expiration. (i) (5 points) What is delta? (ii) (5 points) If the option is on 100 shares, what investment is required for a deltahedged portfolio? Question 7 (Text Question 10.1, page 338). You are given the following data. S = $100 K = 105 r = 8% (continuously compounded) T =0.5 δ =0 You are given u =1.3 andd =0.8. For a single-period binomial model compute the premium, and B for a European call option. Question 8. For a two-period binomial model, you are given the following data. Each period is one year. The current price for a non-dividend paying stock is $20. u =1.2840, where u is one plus the rate of capital gain on the stock per period if the stock price goes up. d =0.8607, where d is one plus the rate of capital loss on the stock per period if the stock price goes down. The continuously compounded risk-free interest rate is 5%. Calculate the price of an American call option on the stock with a strike price of $22.
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u 1.3500 p* 0.40000 American 0.68467 European 0.60470 d 0.8500 Put Price Put Price cont. int_rate 0.0488 n 4 delta 0.0000 K 9.00 American 3.20038 European 3.20038 h 1.0000 Call Price Call Price Stock Prices 10 r=ln(1.05) 201.06556 9 (i) Answer = 0.60470 148.93745 126.59683 8 (ii) Answer = 0.68467 110.32404 93.77543 79.70912 7 Exercise at period 3 if stock has fallen three times. 81.72151 69.46328 59.04379 50.18722 6 (Exercise when intrinsic value is greater than value of future cash flows if not exercised.) 60.53445 51.45428 43.73614 37.17572 31.59936 5 44.84033 38.11428 32.39714 27.53757 23.40693 19.89589 4 33.21506 28.23280 23.99788 20.39820 17.33847 14.73770 12.52704 3 24.60375 20.91319 17.77621 15.10978 12.84331 10.91681 9.27929 7.88740 2 18.22500 15.49125 13.16756 11.19243 9.51356 8.08653 6.87355 5.84252 4.96614 1 13.50000 11.47500 9.75375 8.29069 7.04708 5.99002 5.09152 4.32779 3.67862 3.12683 0 S_0 10.00 8.50000 7.22500 6.14125 5.22006 4.43705 3.77150 3.20577 2.72491 2.31617 1.96874 0 1 2 3 4 5 6 7 8 9 10 Put Intrinsic Value European Put Payoff 10 0.00000 10 0.00000 9 0.00000 0.00000 9 0.00000 0.00000 8 0.00000 0.00000 0.00000 8 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.70931 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.70931 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 0.50000 1.77500 2.85875 3.77994 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 0.00000 0.00000 0.00000 3.77994 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0 0 American Put Prices European Put Prices 10 0.00000 10 0.00000 9 0.00000 0.00000 9 0.00000 0.00000 8 0.00000 0.00000 0.00000 8 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.13235 0.23161 0.40532 0.70931 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.13235 0.23161 0.40532 0.70931 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 Price 0.68467 1.10994 1.78798 2.85875 3.77994 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 Price 0.60470 0.96999 1.54308 2.43018 3.77994 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Call Intrinsic Value European Call Payoff 10 0.00000 10 0.00000 9 0.00000 0.00000 9 0.00000 0.00000 8 0.00000 0.00000 0.00000 8 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 24.21506 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 24.21506 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 15.60375 11.91319 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 11.91319 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 9.22500 6.49125 4.16756 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 4.16756 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 4.50000 2.47500 0.75375 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 American Call Prices European Call Prices 10 0.00000 10 0.00000 9 0.00000 0.00000 9 0.00000 0.00000 8 0.00000 0.00000 0.00000 8 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 24.21506 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 24.21506 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 16.03232 11.91319 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 16.03232 11.91319 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 10.06173 6.91982 4.16756 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 10.06173 6.91982 4.16756 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 5.85781 3.54335 1.58764 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 5.85781 3.54335 1.58764 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 Price 3.20038 1.69546 0.60482 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 Price 3.20038 1.69546 0.60482 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
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Ddt6;; f'j(rl,) J,. t-( s/1<-)!'-j L A,) = A 7'". \ I 7 z: Without dividends, the standard Black and Scholes (1973) pricing formula for the European call opt.ion is given by c( t) S(t)N(dd - e-r(t-t) J( N(d2), where In ( ) + (r + 4a2)(T - t) a.jy=t T - t d1 - avt - t. ' and The option's "delta" is given by g~~~~ = N(d1). With the option struck at-the-money, S(t) = K, and thus, In e~») = 0 [remember that In(l) = 0]. All other terms in d1 are positive. Therefore, d1 > 0, and N(dd > 0.5 (remember that N(O) = 0.5 and N( ) is an increasing function of its argument). Thus, an at-the-money option on a non-dividend-paying stock always has a delta slightly greater than one-half.
Black-Scholes Option Pricing Model S_0 Stock Price 100.00000 Summary K Strike Price 95.00000 Call Price 14.38631 sigma Volatility 0.30000 Call Delta 0.66626 r Interest Rate 0.08000 Call Gamma 0.01343 T Time to Expiration (Years) 0.75000 Put Price 3.85394 delta Dividend Yield 0.03000 Put Delta -0.31149 Put Gamma 0.01343 0.07125 0.12254 d_1 0.47167 d_2 0.21186 C(S,K,sigma,r,T,delta) = S_0*exp(-delta*T)*N(d_1)-K*exp(-r*T)*N(d_2) N(d_1) 0.68142 N(d_2) 0.58389 97.77512 P(S,K,sigma,r,T,delta) = K*exp(-r*T)*N(-d_2)-S_0*exp(-delta*T)*N(-d_1) 89.46763 Answer: Call Price 14.38631 Call Price = 14.38631 Put Price 3.85394
Black-Scholes Option Pricing Model S_0 Stock Price 40.00000 Summary K Strike Price 40.00000 Call Price 2.78040 sigma Volatility 0.30000 Call Delta 0.58240 r Interest Rate 0.08000 Call Gamma 0.06516 T Time to Expiration (Years) 0.24932 Put Price 1.99049 delta Dividend Yield 0.00000 Put Delta -0.41760 Put Gamma 0.06516 0.03116 0.03116 d_1 0.20805 d_2 0.05825 C(S,K,sigma,r,T,delta) = S_0*exp(-delta*T)*N(d_1)-K*exp(-r*T)*N(d_2) N(d_1) 0.58240 N(d_2) 0.52323 40.00000 P(S,K,sigma,r,T,delta) = K*exp(-r*T)*N(-d_2)-S_0*exp(-delta*T)*N(-d_1) 39.21010 Answer: Call Price 2.78040 (i) Put Delta = -0.41760 Put Price 1.99049 (ii) Short 100*Delta = 41.76 shares & deposit short sale proceeds plus 199.05 put option premiums in the bank.
One Period Interest Rate 0.0408 Cash Flow Matrix 1.0408 130.00 1.0408 80.00 Traded Asset -- Bank Account 1.0408 1 1.0408 Cash Flow Matrix Inverse -1.5372631 2.4980525 0.0200000-0.0200000 Time 0 1 Replicating Portfolio (Bank Account & Stock) -38.4315776 Traded Asset -- Stock 130.00 0.5000000 100.00 80.00 Price of Cash Flows 11.5684224 Time 0 1 Answer: Cash Flows to Price 25.00 Delta = 0.5, B=-38.4316, Price=11.5684 0.00 Time 0 1 Note: One Period Interest Rate = Exp(0.08*0.5)-1
u 1.2840 p* 0.45020 American 2.50300 European 1.96488 d 0.8607 Put Price Put Price int_rate 0.0500 n 2 delta 0.0000 K 22.00 American 2.05845 European 2.05845 h 1.0000 Call Price Call Price Stock Prices Put Intrinsic Value 10 Answer: 243.60165 9 American Call Price = 2.05845 189.72091 163.29279 8 147.75772 127.17507 109.45958 7 115.07611 99.04601 85.24890 73.37373 6 89.62314 77.13863 66.39322 57.14465 49.18440 5 69.79995 60.07682 51.70812 44.50518 38.30561 32.96963 4 54.36133 46.78880 40.27112 34.66135 29.83303 25.67729 22.10044 3 42.33749 36.43987 31.36380 26.99482 23.23444 19.99789 17.21218 14.81452 2 32.97312 28.37996 24.42664 21.02401 18.09536 15.57468 13.40512 11.53779 9.93058 1 25.68000 22.10278 19.02386 16.37384 14.09296 12.12981 10.44013 8.98582 7.73409 6.65673 0 S_0 20.00 17.21400 14.81609 12.75221 10.97583 9.44689 8.13094 6.99830 6.02344 5.18437 4.46219 0 1 2 3 4 5 6 7 8 9 10 European Put Payoff 10 0.00000 10 0.00000 9 0.00000 0.00000 9 0.00000 0.00000 8 0.00000 0.00000 0.00000 8 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 4.78600 7.18391 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 0.00000 7.18391 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 American Put Prices European Put Prices 10 0.00000 10 0.00000 9 0.00000 0.00000 9 0.00000 0.00000 8 0.00000 0.00000 0.00000 8 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 Price 2.50300 4.78600 7.18391 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 Price 1.96488 3.75706 7.18391 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Call Intrinsic Value European Call Payoff 10 10 0.00000 0.00000 9 0.00000 0.00000 9 0.00000 0.00000 8 0.00000 0.00000 0.00000 8 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 10.97312 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 10.97312 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 3.68000 0.10278 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.10278 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 American Call Prices European Call Prices 10 0.00000 10 0.00000 9 0.00000 0.00000 9 0.00000 0.00000 8 0.00000 0.00000 0.00000 8 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 6 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 10.97312 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 10.97312 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 4.75295 0.10278 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 4.75295 0.10278 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 Price 2.05845 0.04401 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 Price 2.05845 0.04401 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Solution to (4) Answer: (C) First, we construct the two-period binomial tree for the stock price. Year 0 Year 1 Year 2 32.9731 25.680 20 22.1028 17.214 14.8161 The calculations for the stock prices at various nodes are as follows: S u = 20 1.2840 = 25.680 S d = 20 0.8607 = 17.214 S uu = 25.68 1.2840 = 32.9731 S ud = S du = 17.214 1.2840 = 22.1028 S dd = 17.214 0.8607 = 14.8161 The risk-neutral probability for the stock price to go up is rh 0.05 e d e 0.8607 p* = = = 0.4502. u d 1.2840 0.8607 Thus, the risk-neutral probability for the stock price to go down is 0.5498. If the option is exercised at time 2, the value of the call would be C uu = (32.9731 22) + = 10.9731 C ud = (22.1028 22) + = 0.1028 C dd = (14.8161 22) + = 0 If the option is European, then C u = e 0.05 [0.4502C uu + 0.5498C ud ] = 4.7530 and C d = e 0.05 [0.4502C ud + 0.5498C dd ] = 0.0440. But since the option is American, we should compare C u and C d with the value of the option if it is exercised at time 1, which is 3.68 and 0, respectively. Since 3.68 < 4.7530 and 0 < 0.0440, it is not optimal to exercise the option at time 1 whether the stock is in the up or down state. Thus the value of the option at time 1 is either 4.7530 or 0.0440. Finally, the value of the call is C = e 0.05 [0.4502(4.7530) + 0.5498(0.0440)] = 2.0585.
Remark: Since the stock pays no dividends, the price of an American call is the same as that of a European call. See pages 294-295 of McDonald (2006). The European option price can be calculated using the binomial probability formula. See formula (11.17) on page 358 and formula (19.1) on page 618 of McDonald (2006). The option price is e r(2h) 2 [ p * 2 2 2 Cuu + p *(1 p*) Cud + (1 p*) 2 Cdd ] 2 1 0 = e 0.1 [(0.4502) 2 10.9731 + 2 0.4502 0.5498 0.1028 + 0] = 2.0507