BKM_Sol_Ch_16

# BKM_Sol_Ch_16 - Chapter 16 - Option Valuation Chapter 16...

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Chapter 16 - Option Valuation Chapter 16 Option Valuation 1. Put values also increase as the volatility of the underlying stock increases. We see this from the parity relationship as follows: C = P + S 0 – PV(X) – PV(Dividends) Given a value of S and a risk-free interest rate, if C increases because of an increase in volatility, so must P in order to keep the parity equation in balance. Numerical example : Suppose you have a put with exercise price 100, and that the stock price can take on one of three values: 90, 100, 110. The payoff to the put for each stock price is: Stock price 90 100 110 Put value 10 0 0 Now suppose the stock price can take on one of three alternate values also centered around 100, but with less volatility: 95, 100, 105. The payoff to the put for each stock price is: Stock price 95 100 105 Put value 5 0 0 The payoff to the put in the low volatility example has one-half the expected value of the payoff in the high volatility example. 2. a. Put A must be written on the lower-priced stock. 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 lower price. This would explain its higher value. c. Call B. 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. This would explain its higher price. e. Not enough information. The call with the lower exercise price sells for more than the call with the higher exercise price. The values given are consistent with either stock having higher volatility. 16-1

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Chapter 16 - Option Valuation 3. Note that, as the option becomes progressively more in the money, its hedge ratio increases to a maximum of 1.0: X Hedge ratio 120 0/30=0.000 110 10/30=0.333 100 20/30=0.666 90 30/30=1.00 4. S d 1 N(d 1 ) 45 -0.0268 0.4893 50 0.5 0.6915 55 0.9766 0.8356 5. a. When S = 130, then P = 0. When S = 80, then P = 30. The hedge ratio is: [(P u – P d )/(uS 0 – dS 0 ) = [(0 – 30)/(130 – 80)] = –3/5 b. Riskless portfolio S =80 S = 130 3 shares 240 390 5 puts 150 0 Total 390 390 Present value = \$390/1.10 = \$354.545 c. Portfolio cost = 3S + 5P = \$300 + 5P = \$354.545 Therefore 5P = \$54.545 P = \$54.545/5 = \$10.91 16-2
Chapter 16 - Option Valuation 6. The hedge ratio for the call is: [(C u – C d )/(uS 0 – dS 0 )] = (20 – 0)/(130 – 80) = 2/5 Riskless portfolio S =80 S = 130 2 shares 160 260 Short 5 calls 0 -100 Total 160 160 –5C + 200 = \$160/1.10 = \$145.455 C = \$10.91 Put-call parity relationship: P = C – S 0 + PV(X) \$10.91 = \$10.91 + (\$110/1.10) – \$100 = \$10.91 7. d 1 = 0.3182 N(d 1 ) = 0.6248 d 2 = –0.0354 N(d 2 ) = 0.4859 Xe –rT = \$47.56 C = S 0 N(d 1 ) - Xe –rT N(d 2 ) = \$8.13 8. P = \$5.69 This value is from our Black-Scholes spreadsheet, but note that we could have derived the value from put-call parity: P = C – S 0 + PV(X) = \$8.13 – \$50 + \$47.56 = \$5.69 9. A straddle is a call and a put. The Black-Scholes value is: C + P = S 0 e T N(d 1 ) - Xe –rT

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## This note was uploaded on 08/25/2009 for the course FNCE 4330 taught by Professor Jianyang during the Spring '09 term at University of Colorado Denver.

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BKM_Sol_Ch_16 - Chapter 16 - Option Valuation Chapter 16...

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