Problem Set 3 Econ 236 Question 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 points On May 15th, 2015, the closing price of Target stock was $78.53. A call option with strike price X = $70.00 and maturity date June 19th, 2015 costs $8.30. A put option with the same strike and maturity costs $0.31. Assume a continuously-compounded risk-free rate of 0.00096 (0.096%) per annum. Further, assume that the options are European and that the stock does not pay dividends. (a) (5 points) If put-call parity holds, what should the price of the put option be? Solution: According to put-call parity c + Ke−rT = p + S0 . Since there are exactly 35 days until expiration, T = 35/365 = 0.09589. Thus, p = c + Ke−rT − S0 = 8.3 + 70e−0.00096×0.09589 − 78.53 = −0.2364, or -$0.23. Since a put price can never be negative, the price should be zero. (b) (5 points) If put-call parity holds, what should the price of the call option be? Solution: Again, by the put-call parity relationship, c = p + S0 − Ke−rT 0.31 + 78.53 − 70e−0.00096×0.09589 = 8.846, or $8.85. Question 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 points Consider the following options portfolio. You write an April call option on IBM with exercise price $85. You write an April IBM put option with exercise price $80. The price of the call option is $0.95 and the price of the put is $2.45.

(a) (5 points) Graph the payoff of this portfolio at option expiration as a function of IBM’s stock price at that time.

(b) (5 points) What will be the profit/loss on this position if IBM is selling at $83 on the option maturity date. What if IBM is selling at $90? Solution: If IBM is selling at $83 on the maturity date, then neither the call option nor the put option are in the money so the profit is $0.95 + $2.45 = $3.40. If IBM is selling for $90 instead, then the call option is in the money. In this case, profit is $3.40 + ($85 − $90) = −$1.60. (c) (5 points) At what two stock prices will you just break even on your investment? Solution: We will break even when either option is in the money by an amount equal to $3.40. For the call option, this price is $85 + $3.40 = $88.40. For the put option, this price is $80 − $3.40 = $76.60. (d) (5 points) What kind of bet is this investor making; that is, what must this investor believe

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about IBM’s stock price to justify this position? Solution: The investor believes that IBM’s stock price will have low volatility (in both directions) from now until option maturity. Question 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 points A stock currently trades at a price of $100. The stock price can go up 10 percent or down 15 percent. The risk-free rate is 6.5 percent. (a) (7 points) Use a one-period binomial model to calculate the price of a call option with an exercise price of $90. Solution: Assume that the maturity date is one year from now. The value of the stock can either go up to 100(1.10) = $110 (in which case the option is worth 110−90 = $20) or go down to 100(0.85) = $85 (in which case the option is worthless). $110 fu = $20

p $100 f

1−

p $85 fd = $0

There are two ways to find the option price. For the first approach, we’ll build a riskless portfolio by going long on shares and by shorting 1 option. Let ∆ be the number of shares we want to hold in the portfolio to totally hedge against risk. By definition, a riskless portfolio will have the same value whether the stock goes up or down. Therefore, 110∆ − 20 = 85∆ ⇒ ∆ = 0.8 and the value of the portfolio at maturity is 110(0.8) − 20 = $68. The present value of the portfolio is thus 68e−0.065(1) = $63.72. But we also know that the present value of the portfolio is 100(0.8) − f where f is spot price of the option. This gives f = 100(0.8) − 63.72 = $16.28.

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The second way to price the option is via risk neutrality. Letting p be the probability of the stock going up, the value of today’s portfolio compounded at the risk-free rate must equal the expected future value of the portfolio: 110p + 85(1 − p) = 100e0.065(1) ⇒ p = 0.8686 Furthermore, the present value of the option must be equal to the discounted expected future value of the option: f = [20p + 0(1 − p)]e−0.065(1) = $16.28

(b) (4 points) Suppose the call price is currently $17.50. Show how to execute an arbitrage transaction that will earn more than the risk-free rate. Use 100 call options. Solution: Build the riskless portfolio as in the previous part by going long 80 shares and shorting 100 options. The current value of this portfolio is 100(80) − 17.50(100) = $6250. The future value of this portfolio (no matter whether the stock goes up or 6800 down) is $6800. This implies a 1-year interest rate of ln( 6250 ) = 0.08434 which exceeds the risk-free rate. (c) (4 points) Suppose the call price is currently $14. Show how to execute an arbitrage transaction that replicates a loan that will earn less than the risk-free rate. Use 100 call options. 6800 Solution: The riskless portfolio now implies a 1-year interest rate of ln( 100(80)−14(100) )= 0.02985 which is less than the risk-free rate. The arbitrage play here is to short the riskless portfolio (by shorting shares and going long on the options) and investing the proceeds at the risk-free rate.

Question 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 points Consider a two-period binomial model in which a stock currently trades at a price of $65. The stock price can go up 20 percent or down 17 percent each period. The risk-free rate is 5 percent. (a) (10 points) Calculate the price of a European put option expiring in two periods with exercise price of $60.

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Solution: Assume that each of the two periods lasts one year (∆t = 1). $93.6 fuu = $0

p $78 fu

p

1−

p

$65 f

$64.74 fud = $0

1−

p

p $53.95 fd

1−

p $44.78 fdd = $15.22

The first step is to calculate the option prices in the second period, fuu , fud , and fdd . The next step is to calculate the prices of the options in the first period (fu and fd ) using the prices in the second period. Since fuu = fud = 0, fu must also have zero value. To find fd , we must first find p using the fact that the the value of the stock invested at the risk-free rate must be equal to the expected future value of the stock: 53.95e0.05(1) = 64.74p + 44.78(1 − p) ⇒ p = 0.5980 Then, fd must be equal to the present value of the expected value of the options in the second period: fd = [0 · p + 15.22(1 − p)]e−0.05(1) = $5.82 This same argument can be used to find the price of the option at time zero, f : f = [0 · p + 5.82(1 − p)]e−0.05(1) = $2.23 (b) (5 points) Based on your answer in part (a), calculate the number of units of the underlying

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stock that would be needed at each point in the binomial tree in order to construct a riskfree hedge. Use 10,000 puts. Solution: We need to calculate ∆, ∆u , and ∆d : 0 − 5.82 fu − fd = = −0.2420 78 − 53.95 78 − 53.95 fuu − fud 0−0 ∆u = = =0 93.6 − 64.74 93.6 − 64.74 0 − 15.22 fud − fdd = = −0.7625 ∆d = 64.74 − 44.78 64.74 − 44.78 ∆=

Unlike for call options, ∆’s for put options are negative. This means we would need to go long on both the shares and the options or short on both the shares and the options in order to construct a riskless portfolio. At time zero, we would need to go long |10000∆| = 2420 shares and 10000 puts. If the stock goes up after one year, we would not be able to hedge risk without using a different option. If the stock goes down after one year, we would need to go long |10000∆d | = 7625 shares and 10000 puts. (c) (5 points) Is the price of an American put option different? If so, compute the price. Solution: If the option were American instead of European, then its price could potentially change if it were optimal to exercise the option early. For this problem, if the stock goes down after one year, then it is optimal to exercise the option early since 60 − 53.95 = 6.05 > fd . Therefore, the price of the American option is f = [0 · p + 6.05(1 − p)]e−0.05(1) = $2.31

(d) (10 points) Calculate the price of a European put option expiring in two periods with exercise price of $70. Solution: As in part (a), p = 0.5980 and fuu = 0. However, fud and fdd are different than in part (a).

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$93.6 fuu = $0

p $78 fu

p

1−

p

$65 f

$64.74 fud = $5.26

1−

p

p $53.95 fd

1−

p $44.78 fdd = $25.22

fu = [0 · p + 5.26(1 − p)]e−0.05(1) = $2.01 fd = [5.26p + 25.22(1 − p)]e−0.05(1) = $12.64 f = [2.01p + 12.64(1 − p)]e−0.05(1) = $5.98

(e) (5 points) Based on your answer in part (d), calculate the number of units of the underlying stock that would be needed at each point in the binomial tree in order to construct a riskfree hedge. Use 10,000 puts.

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Solution: 2.01 − 12.64 fu − fd = = −0.4420 78 − 53.95 78 − 53.95 fuu − fud 0 − 5.26 ∆u = = = −0.1823 93.6 − 64.74 93.6 − 64.74 fud − fdd 5.26 − 25.22 ∆d = = = −1 64.74 − 44.78 64.74 − 44.78 ∆=

(f) (5 points) Is the price of an American put option different? If so, compute the price. Solution: If the option were American, then fd changes from $12.64 to 70 − 53.95 = $16.05. Therefore, f = [2.01p + 16.05(1 − p)]e−0.05(1) = $7.28

Question 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 points Suppose you are a financial adviser and that you have a client who believes the common stock price of TRT materials (currently $58 per share) could move substantially in either direction in reaction to an expected court decision involving the company. The client currently owns no TRT shares, but asks you for advice about implementing a strangle strategy to capitalize on the possible stock price movement. Suppose that a 90-day call option with strike of $60 is priced at $5 and that a 90-day put option with strike of $55 is priced at $4. (a) (3 points) Would you recommend a long strangle or short strangle strategy to achieve the client’s objective? Solution: The client should implement a long strangle to take advantage of high volatility in the stock price. (b) For the recommended strategy, calculate the: i. (4 points) Maximum possible loss per share. Solution: The maximum loss per share is 5 + 4 = $9 and is incurred when the stock price is between $55 and $60.

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ii. (4 points) Maximum possible gain per share. Solution: The maximum possible gain per share if the stock moves downward is 55 − 9 = $46. The possible gain per share if the stock moves upward is unlimited since in theory the stock price can go towards infinity. iii. (4 points) Break even stock price(s). Solution: The break even prices are 55 − 9 = $46 and 60 + 9 = $69.

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