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Chapter 21

# Chapter 21 - CHAPTER 21 OPTION VALUATION 1 The value of a...

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CHAPTER 21: OPTION VALUATION 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.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 maturity. Despite the higher price of Stock B, Call B is cheaper than Call A. This can be explained by a lower time to maturity. 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. 3. Exercise Price Hedge Ratio 115 5/30 = 0.167 100 20/30 = 0.667 75 30/30 = 1.000 50 30/30 = 1.000 25 30/30 = 1.000 10 30/30 = 1.000 As the option becomes more in the money, the hedge ratio increases to a maximum of 1.0. 21-1

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4. S d 1 N(d 1 ) 45 -0.0268 0.4893 50 0.5000 0.6915 55 0.9766 0.8356 5. 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 6. 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 21-2
7. 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 8. 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 9. 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 10. 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. 11. A straddle is a call and a put. The Black-Scholes value would be: C + P = S 0 N(d 1 ) - Xe –rT N(d 2 ) + Xe –rT [1 - N(d 2 )] - S 0 [1 - N(d 1 )] = S 0 [2N(d 1 ) - 1] + Xe –rT [1 - 2N(d 2 )] On the Excel spreadsheet (Spreadsheet 21.1), the valuation formula would be: B5*(2*E4 - 1) + B6*EXP( - B4*B3)*(1 - 2*E5) 12. 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.

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