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SolCh21 - Chapter 21 Option Valuation CHAPTER 21 OPTION...

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Chapter 21 - Option Valuation 21-1 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.
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Chapter 21 - Option Valuation 21-2 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
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Chapter 21 - Option Valuation 21-3 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
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Chapter 21 - Option Valuation 21-4 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
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