Financial Theory and Corporate Policy (4th Edition) Solution Chapter 7

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Chapter 7
Pricing Contingent Claims: Option Pricing Theory and Evidence
1. We can use the Black-Scholes formula (equation 7.36 pricing European calls. C = SN(d1 ) − Xe− rf T N(d 2 ) where d1 = ln(S/ X) + rf T + (1/ 2 )σ T σ T

d2 = d1 – σ T Substituting the correct values into d1, we have d1 = d1 = ln(28/40) + .06(.5) + .5( .5)( .5) .5 .5 −.356675 + .03 + .25 = –.40335 .5

d2 = –.40335 – ( .5)( .5) = –.90335 Using the Table for Normal Areas, we have N(d1) = .5 – .1566 = .3434 N(d2) = .5 – .3171 = .1829 Substituting these values back into the Black-Scholes formula, we have C = 28(.3434) – 40e = $2.52 2. We know from put-call parity that the value of a European put can be determined using the value of a European call with the same
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Substituting these values into the formula yields C = 44.375(.5714) – 45(.9704)(.4920) = 25.3559 – 21.4847 = $3.87 5. Compare the payoffs at maturity of two portfolios. The first is a European put option with exercise price X1, and the second is a European put option written on the same stock, with the same time to maturity, but with exercise price X2 < X1. The payoffs are given in Table S7.1. Because portfolio A has a value either greater than or equal to the value of portfolio B in every possible state of nature, the put with a higher exercise price is more valuable. P(S, T, X1) > P(S, T, X2) Table S7.1 Portfolio a) P(S, T, X1) b) P(S, T, X2) Comparative Value of A and B S < X2 X1 – S X2 – S VA > VB if X2 < X1. X1 ≤ S 0 0 VA = VB

State Contingent Payoffs of Put Portfolios X2 ≤ S < X1 X1 – S 0 VA > VB

Chapter 7

Pricing Contingent Claims: Option Pricing Theory and Evidence

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6. The total value of the firm, $5MM, is V=S+B We can use the OPM to determine the market value of the stock (expressed as a call option on the firm), and subtract this from $5MM to find B, the market value of the debt. For Rf = .05, we have S = VN(d1) – D e − rf T N(d2) where S = market value of the stock V = market value of the firm D = face value of the debt d1 = ln(5MM / 4MM) + (.05)(10) + (1/ 2 ( .5)( 10)

( .5 )( 10 )

Note that T, the time to maturity, is the time until the firm’s debt matures, ten

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