Unformatted text preview: ; therefore, the time premium is, in general, larger for a call option.
(6 / 12) ) = .0594 6 / 12 ) = –.3154 443 16. The stock price can either increase 15 percent, or decrease 15 percent. The stock price at expiration will either be: Stock price increase = $54(1 + .15) = $62.10 Stock price decrease = $54(1 – .15) = $45.90 The payoff in either state will be the maximum stock price minus the exercise price, or zero, which is: Payoff if stock price increases = Max[$62.10 – 50, 0] = $12.10 Payoff if stock price decreases = Max[$45.90 – 50, 0] = $0 To get a 15 percent return, we can use the following expression to determine the riskneutral probability of a rise in the price of the stock: Riskfree rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) .08 = (ProbabilityRise)(.15) + (1 – ProbabilityRise)(–.15) ProbabilityRise = .7667 And the probability of a stock price decrease is: ProbabilityFall = 1 – .7667 = .2333 So, the risk neutral value of a call option will be: Call value = [(.7667 × $12.10) + (.2333 × $0)] / (1 + .08) Call value = $8.59 17. The stock price increase, decrease, and option payoffs will remain unchanged since the stock price change is the same. The new risk neutral probability of a stock price increase is: Riskfree rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) .05 = (ProbabilityRise)(.15) + (1 – ProbabilityRise)(–.15) ProbabilityRise = .6667 And the probability of a stock price decrease is: ProbabilityFall = = 1 – .6667 = .3333 So, the risk neutral value of a call option will be: Call value = [(.6667 × $12.10) + (.3333 × $0)] / (1 + .05) Call value = $7.68 444 Intermediate 18. If the exercise price is equal to zero, the call price will equal the stock price, which is $75. 19. If the standard deviation is zero, d1 and d2 go to +8, so N(d1) and N(d2) go to 1. This is the no risk call option formula, which is: C = S – Ee–rt C = $86 – $80e–.05(6/12) = $7.98 20. If the standard deviation is infinite, d1 goes to positive infinity so N(d1) goes to 1, and d2 goes to negative infinity so N(d2) goes to 0. In this case, the call price is equal to the stock price, which is $35. 21. We can use the BlackScholes model to value the equity of a firm. Using the asset value of $15,800 as the stock price, and the face value of debt of $15,000 as the exercise price, the value of the firm’s equity is: d1 = [ln($15,800/$15,000) + (.05 + .382/2) × 1] / (.38 × d2 = .4583 – (.38 × N(d1) = .6766 N(d2) = .5312 Putting these values into the BlackScholes model, we find the equity value is: Equity = $15,800(.6766) – ($15,000e–.05(1))(.5312) = $3,111.31 The value of the debt is the firm value minus the value of the equity, so: D = $15,800 – 3,111.31 = $12,688.69 22. a. We can use the BlackScholes model to value the equity of a firm. Using the asset value of $17,000 as the stock price, and the face value of debt of $15,000 as the exercise price, the value of the firm if it accepts project A is: d1 = [ln($17,000/$15,000) + (.05 + .552/2) × 1] / (.55 × d2 = .5935 – (.55 × N(d1) = .7236 N(d2) = .5173 Putting these values into the BlackScholes model, we find the equity value is: EA = $17,000(.7236) – ($15,000e–.05(1))(.5173) = $4,919.05
1 ) = .0435 1 ) = .5935 1 ) = .0783 1 ) = .4583 445 The value of the debt is the firm value minus the value of the equity, so: DA = $17,000 – 4,919.05 = $12,080.95 And the value of the firm if it accepts Project B is: d1 = [ln($17,400/$15,000) + (.05 + .342/2) × 1] / (.34 × d2 = .7536 – (.34 × N(d1) = .7745 N(d2) = .6604 Putting these values into the BlackScholes model, we find the equity value is: EB = $17,400(.7745) – ($15,000e–.05(1))(.6604) = $4,052.41 The value of the debt is the firm value minus the value of the equity, so: DB = $17,400 – 4,052.41 = $13,347.59 b. Although the NPV of project B is higher, the equity value with project A is higher. While NPV represents the increase in the value of the assets of the firm, in this case, the increase in the value of the firm’s assets resulting from project B is mostly allocated to the debtholders, resulting in a smaller increase in the value of the equity. Stockholders would, therefore, prefer project A even though it has a lower NPV. c. Yes. If the same group of investors have equal stakes in the firm as bondholders and stockholders, then total firm value matters and project B should be chosen, since it increases the value of the firm to $17,400 instead of $17,000. d. Stockholders may have an incentive to take on riskier, less profitable projects if the firm is leveraged; the higher the firm’s debt load, all else the same, the greater is this incentive. 23. We can use the BlackScholes model to value the equity of a firm. Using the asset value of $27,200 as the stock price, and the face value of debt of $25,000 as the exercise price, the value of the firm’s equity is: d1 = [ln($27,200/$25,000) + (.05 + .532/2) × 1] / (.53 × d2 = .5185 – (.53 × N(d1) = .6979 N(d2) = .4954 Putting these v...
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This note was uploaded on 07/10/2010 for the course FIN 6301 taught by Professor Eshmalwi during the Spring '10 term at University of TexasTyler.
 Spring '10
 eshmalwi
 Finance, Corporate Finance

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