21 10 chapter 21 option valuation 32 the two possible

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Here, the value of the call is greater than the value in the lower-volatility scenario. 21-10

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Chapter 21 - Option Valuation 32. The two possible stock prices and the corresponding put values are: uS 0 = 120 P u = 0 dS 0 = 80 P d = 20 The hedge ratio is: 2 1 80 120 20 0 dS uS P P H 0 0 d u - = - - = - - = Form a riskless portfolio by buying one share of stock and buying two puts. The cost of the portfolio is: S + 2P = 100 + 2P The payoff for the riskless portfolio equals \$120: Riskless Portfolio S = 80 S = 120 Buy 1 share 80 120 Buy 2 puts 40 0 Total 120 120 Therefore, find the value of the put by solving: \$100 + 2P = \$120/1.10 P = \$4.545 According to put-call parity: P + S = C + PV(X) Our estimates of option value satisfy this relationship: \$4.545 + \$100 = \$13.636 + \$100/1.10 = \$104.545 33. If we assume that the only possible exercise date is just prior to the ex-dividend date, then the relevant parameters for the Black-Scholes formula are: S 0 = 60 r = 0.5% per month X = 55 σ = 7% T = 2 months In this case: C = \$6.04 If instead, one commits to foregoing early exercise, then we reduce the stock price by the present value of the dividends. Therefore, we use the following parameters: S 0 = 60 – 2e (0.005 × 2) = 58.02 r = 0.5% per month X = 55 σ = 7% T = 3 months In this case, C = \$5.05 The pseudo-American option value is the higher of these two values: \$6.04 21-11
Chapter 21 - Option Valuation 34. True. The call option has an elasticity greater than 1.0. Therefore, the call’s percentage rate of return is greater than that of the underlying stock. Hence the GM call responds more than proportionately when the GM stock price changes in response to broad market movements. Therefore, the beta of the GM call is greater than the beta of GM stock. 35. True. The elasticity of a call option is higher the more out of the money is the option. (Even though the delta of the call is lower, the value of the call is also lower. The proportional response of the call price to the stock price increases. You can confirm this with numerical examples.) Therefore, the rate of return of the call with the higher exercise price responds more sensitively to changes in the market index, and therefore it has the higher beta. 36. As the stock price increases, conversion becomes increasingly more assured. The hedge ratio approaches 1.0. The price of the convertible bond will move one-for-one with changes in the price of the underlying stock. 37. Salomon believes that the market assessment of volatility is too high. Therefore, Salomon should sell options because the analysis suggests the options are overpriced with respect to true volatility. The delta of the call is 0.6, while that of the put is 0.6 – 1 = –0.4. Therefore, Salomon should sell puts and calls in the ratio of 0.6 to 0.4. For example, if Salomon sells 2 calls and 3 puts, the position will be delta neutral: Delta = (2 × 0.6) + [3 × (–0.4)] = 0 38. If the stock market index increases 1%, the 1 million shares of stock on which the options are written would be expected to increase by: 0.75% × \$5 × 1 million = \$37,500 The options would increase by: delta × \$37,500 = 0.8 × \$37,500 = \$30,000 In order to hedge your market exposure, you must sell \$3,000,000 of the market index portfolio so that a 1% change in the index would result in a \$30,000 change in the value of the portfolio.

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