3330 PS10 sol - Cornell University Fall 2010 Economics...

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Cornell University Fall 2010 Economics 3330: Problem Set 10 Solutions 1. Consider the following parameters: current stock price $50, standard deviation 50%, exercise price $50, interest rate 4%. a. Use the Black-Scholes formula to find the value of a call option with an exercise price of $50 and expiration of 1 year. d 1 = 0.33 N(d 1 ) = .63 d 2 = –0.17 N(d 2 ) = .43 Xe –rT = $52.05 C = S 0 N(d1) - Xe –rT N(d2) = $10.68 b. Use the Black-Scholes formula to find the value of a put option with an exercise price of $50 and expiration of 6 months. P= 8.73 c. Verify put-call parity. P = C – S 0 + PV(X) = $10.68 – $50 + $48.07 = $8.73 2. Find how the value of the call option from problem one changes when each of the following changes in parameters is made individually . In other words, for each of the following, you are only changing one parameter relative to problem one . a. Time to expiration 3 months. C falls to 5.20 b. Standard deviation 25%. C falls to 5.92 c. Exercise price $55. C falls to 8.79 d. Interest rate 10%. C rises to 9.96 3. Option deltas a.What is the delta of a call option? How is it approximated in the Black-Scholes model? b.Fill in the following table for a call option with exercise stock price 50, standard deviation 50%, interest rate 4%, and expiration in one year. Sketch the resulting graph. 10 20 30 40 45 50 55 60 70 80 90 100 140 Price 0 .24 1.73 5.23 7.74 10.68 14.01 17.65 25.68 34.40 43.57 53.03 92.19 Delta 0 .07 .24 .45 .55 .63 .52 .69 .84 .90 .93 .96 .99 c. Now do the same for a put option, including sketching the resulting graph. 4. A hedge fund has a net asset value of $100 per share and a high water mark of $110. The standard deviation of the fund’s annual returns is 40% and the risk-free rate is 2%. The incentive fee is 20%. All answers should be given in terms of one year. a. According to Black-Scholes, what is the value of the incentive fee? First, compute the Black-Scholes value of a call option with the following parameters: S 0 = 100 X = 110 R = 0.02 = 0.4 T = 1 year Therefore: C = $12.84. The value of the annual incentive fee is: 0.20 × C = 0.20 × $12.84 = $2.57
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b. What would the incentive fee be worth if the fund had no high water mark?
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3330 PS10 sol - Cornell University Fall 2010 Economics...

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