Solution of Derivative

000 6000 3000 option price exercise at perpetual

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Unformatted text preview: erefore, rho is positive (remember, rho for a call option is positive, because a call entails paying the fixed strike price to receive the stock and a higher interest rate reduces the present value of the strike). For stock prices smaller than $ 100, the put dominates and we know that the rho of a put is negative. 94 Chapter 12 The Black-Scholes Formula Tables for 12.18 Inputs Stock Price Exercise Price Volatility Risk-free interest rate Dividend Yield 50 60 40.000% 6.000% 3.000% Inputs Stock Price Exercise Price Volatility Risk-free interest rate Dividend Yield 50 60 40.000% 6.000% 4.000% Inputs Stock Price Exercise Price Volatility Risk-free interest rate Dividend Yield 50 60 40.000% 7.000% 3.000% Inputs Stock Price Exercise Price Volatility Risk-free interest rate Dividend Yield 50 60 50.000% 6.000% 3.000% Option Price Exercise at: Perpetual Options Call Put 26.35183 23.07471 317.3092 22.6908 Option Price Exercise at: Perpetual Options Call Put 22.75128 23.82482 248.2475 21.75248 Option Price Exercise at: Perpetual Options Call Put 27.10008 21.2744 334.9193 25.08067 Option Price Exercise at: Perpetual Options Call Put 29.83555 27.62938 412.5475 17.45254 Question 12.18 a) The price of the perpetual call option is $ 26.35. It should be exercised when the stock price reaches the barrier of $ 317.31. b) The price of the perpetual call option is now $ 22.75. It should be exercised when the stock price reaches the barrier of $ 248.25. The higher dividend yield makes it more costly to forego the dividends and wait for an increase in the stock price before exercising the option. Therefore, the option is worth less and it is optimal to exercise after a smaller increase in the underlying stock price. c) The price of the perpetual call option is now $ 27.10. It should be exercised when the stock price reaches the barrier of $ 334.92. The higher interest rate increases the value of the call option and makes it attractive to wait a bit longer before you exercise the option, as you can 95 Part 3 Options continue to earn interest on the strike before you exercise. Therefore, the option is worth more and it is only optimal to exercise after a larger increase in the underlying stock price. d) The price of the perpetual call option is $ 29.84. It should be exercised when the stock price reaches the barrier of $ 412.55. Options love volatility. The chances of an even larger increase in the stock price are high with a large standard deviation (and your risk is capped at the downside). Therefore, the option is worth more and you wait longer until you forego the future potential and exercise. Question 12.20 a) b) C(100, 90, 0.3, 0.08, 1, 0.05) = 17.6988 P(90, 100, 0.3, 0.05, 1, 0.08) = 17.6988 c) The prices are equal. This is a result of the mathematical equivalence of the pricing formulas. To see this, we need some algebra. We start from equation (12.3) of the text, the formula for the European put option: ln S + r − δ + 0.5σ 2 T ln S + r − δ − 0.5σ 2 T K K P (• ) = K × exp − rT × N − − S × exp − δ T × N − σT σT ( ) ( ) ( ) ( ) Now we replace: K = S , r = δ , δ = r, S = K Then: K ln + (δ − r − 0.5σ 2 )T S = S × exp (− δT ) × N − σT K S Since ln = − ln S K S ln − (δ − r − 0.5σ 2 )T K = S × exp (− δT ) × N σT K ln + (δ − r + 0.5σ 2 )T − K × exp (− rT ) × N − S σT S ln − (δ − r + 0.5σ 2 )T − K × exp (− rT ) × N K σT = S × exp (− δT ) × N (d 1 ) − K × exp (− rT ) × N (d 2 ) = C (• ) 96 Chapter 13 Market-Making and Delta-Hedging Question 13.2. Using the Black Scholes formula we can solve for the put premium and the put’s delta: P = 1.9905 and = −0.4176. If we write this option, we will have a position that moves with the stock price. This implies our delta hedge will require shorting 41.76 shares (receiving $41.76 (40) = $1670.4). As before, we must look at the three components of the profit. There will now be interest earned since we are receiving both the option premium 199.05 as well as the 1670.40 on the short sale. This $2369.45 will earn (rounding to the nearest penny) 2369.45 e.08/365 − 1 = .52 in interest. If the stock falls to 39 we make 41.76 on our short sale and if the stock price rises to 40.5 we lose 20.88 on our short sale. If the stock prices falls to 39 or rises to 40.5 the price of the put option we wrote will be (using T = 90/365) P (39) = 2.4331 or P (40.5) = 1.7808. This implies our option position will lose 243.31 − 199.05 = 44.26 if the stock falls by $1 and make 199.05 − 178.08 = 20.97 if the stock rises by $0.50. Combining these results, our profit will be 41.76 − 44.26 + .52 = −1.98 (1) −20.88 + 20.97 + .52 = .61. (2) if the stock price falls to $39 and Notice that, as in the case of the call option, the large change implies a loss and the small change involves a profit. Question 13.4. The 45-strike put has a premium of 5.0824 and a delta of −.7185 and the 40-strike put has a premium of 1.9905...
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