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Unformatted text preview: CHAPTER 21: OPTION VALUATION 1. The value of a put option also increases with the volatility of the stock. We see this from the putcall parity theorem as follows: P = C S + PV(X) + PV(Dividends) Given a value for S and a riskfree 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 be written on the stock with 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 the higher option premium. 3. Exercise Price Hedge Ratio 115 85/150 = 0.567 100 100/150 = 0.667 75 125/150 = 0.833 50 150/150 = 1.000 25 150/150 = 1.000 10 150/150 = 1.000 As the option becomes more in the money, the hedge ratio increases to a maximum of 1.0. 4. S d 1 N(d 1 ) 450.0268 0.4893 50 0.5000 0.6915 55 0.9766 0.8356 211 5. a.uS = 130 P u = 0 dS = 80 P d = 30 The hedge ratio is: 5 3 80 130 30 dS uS P P H d u = = = b. Riskless Portfolio S = 80 S = 130 Buy 3 shares 240 390 Buy 5 puts 150 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 20 dS uS C C H d u = = = Riskless Portfolio S = 80 S = 130 Buy 2 shares 160 260 Write 5 calls100 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 = 10.91 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 = $8.13 212 8. P = $5.69 This value is derived from our BlackScholes spreadsheet, but note that we could have derived the value from putcall parity: P = C + PV(X) S = $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 BlackScholes 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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 Spring '08
 Vizanko

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