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# Chap021 - Chapter 21 Option Valuation CHAPTER 21 OPTION...

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Chapter 21 - Option Valuation CHAPTER 21: OPTION VALUATION PROBLEM SETS 1. The value of a put option also increases with the volatility of the stock. We see this from the put-call parity theorem as follows: P = C – S 0 + PV(X) + PV(Dividends) Given a value for S and a risk-free interest rate, then, if C increases because of an increase in volatility, P must also increase in order to maintain the equality of the parity relationship. 2. A \$1 increase in a call option’s exercise price would lead to a decrease in the option’s value of less than \$1. The change in the call price would equal \$1 only if: (i) there were a 100% probability that the call would be exercised, and (ii) the interest rate were zero. 3. Holding firm-specific risk constant, higher beta implies higher total stock volatility. Therefore, the value of the put option increases as beta increases. 4. Holding beta constant, the stock with a lot of firm-specific risk has higher total volatility. The option on the stock with higher firm-specific risk is worth more. 5. A call option with a high exercise price has a lower hedge ratio. This call option is less in the money. Both d 1 and N(d 1 ) are lower when X is higher. 6. a.Put A must be written on the stock with the lower price. Otherwise, given the lower volatility of Stock A, Put A would sell for less than Put B. b. Put B must be written on the stock with the lower price. This would explain its higher price. c.Call B must have the lower time to expiration. Despite the higher price of Stock B, Call B is cheaper than Call A. This can be explained by a lower time to expiration. d. Call B must be written on the stock with higher volatility. This would explain its higher price. e.Call A must be written on the stock with higher volatility. This would explain its higher price. 21-1

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Chapter 21 - Option Valuation 7. Exercise Price Hedge Ratio 120 0/30 = 0.000 110 10/30 = 0.333 100 20/30 = 0.667 90 30/30 = 1.000 As the option becomes more in the money, the hedge ratio increases to a maximum of 1.0. 8. S d 1 N(d 1 ) 45 -0.0268 0.4893 50 0.5000 0.6915 55 0.9766 0.8356 9. a.uS 0 = 130 P u = 0 dS 0 = 80 P d = 30 The hedge ratio is: 5 3 80 130 30 0 dS uS P P H 0 0 d u - = - - = - - = b. Riskless Portfolio S = 80 S = 130 Buy 3 shares 240 390 Buy 5 puts 150 0 Total 390 390 Present value = \$390/1.10 = \$354.545 c.The portfolio cost is: 3S + 5P = 300 + 5P The value of the portfolio is: \$354.545 Therefore: P = \$54.545/5 = \$10.91 21-2
Chapter 21 - Option Valuation 10. The hedge ratio for the call is: 5 2 80 130 0 20 dS uS C C H 0 0 d u = - - = - - = Riskless Portfolio S = 80 S = 130 Buy 2 shares 160 260 Write 5 calls 0 -100 Total 160 160 Present value = \$160/1.10 = \$145.455 The portfolio cost is: 2S – 5C = \$200 – 5C The value of the portfolio is: \$145.455 Therefore: C = \$54.545/5 = \$10.91 Does P = C + PV(X) – S? 10.91 = 10.91 + 110/1.10 – 100 = 10.91 11. d 1 = 0.3182 N(d 1 ) = 0.6248 d 2 = –0.0354 N(d 2 ) = 0.4859 Xe - r T = 47.56 C = \$8.13 12. P = \$5.69 This value is derived from our Black-Scholes spreadsheet, but note that we could have derived the value from put-call parity: P = C + PV(X) – S 0 = \$8.13 + \$47.56 - \$50 = \$5.69 13. a.C falls to \$5.5541 b. C falls to \$4.7911 c.C falls to \$6.0778 d. C rises to \$11.5066 e. C rises to \$8.7187 21-3

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Chapter 21 - Option Valuation 14. According to the Black-Scholes model, the call option should be priced at: [\$55 × N(d 1 )] – [50 × N(d 2 )] = (\$55 × 0.6) – (\$50 × 0.5) = \$8 Since the option actually sells for more than \$8, implied volatility is greater than 0.30.
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Chap021 - Chapter 21 Option Valuation CHAPTER 21 OPTION...

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