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

Chapter 7

Pricing Contingent Claims: Option

Pricing Theory and Evidence

1. We can use the Black-Scholes formula (equation 7.36 pricing European calls.

= − −rfT

1 2 C SN(d ) Xe N(d )

where

f

1

ln(S/X) r T

d (1/ 2 T

T

= + + )σ

σ

d2 = d1 - σ T

Substituting the correct values into d1, we have

d1 = + + ln(28/40) .06(.5)

.5( .5)( .5)

.5 .5

d1 =

.356675 .03

.25

.5

− + + = -.40335

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-.06(.5) (.1829)

C = 9.6152 - 40(.970446)(.1829)

= $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 parameters, according to equation 7.8

C0 - P0 = f 0

f

(1 r )S X

(1 r )

+ −

+

Solving for P0, the present value of a put, and converting the formula to continuous rather than

discrete compounding, we have

= − + −rfT

0 0 0 P C S X e

78 Copeland/Shastri/Weston * Financial Theory and Corporate Policy, Fourth Edition

First, calculate the value of the corresponding call option, according to the Black-Scholes formula

(equation 7.36)

C = SN(d1) - e−rf T XN(d2)

where:

d1 = f ln(S/X) r T

(1/ 2) T

T

+ + σ

σ

Solving for d1, we have

d1 =

ln(20 / 20) (.08)(.5)

(1/ 2)(.6)( .5)

(.6)( .5)

+ +

= .3064

and d2 is equal to

d2 = d1 - .6 .5

= .3064 - .4243

= -.1179

From the Table of Normal Areas, we have

N(d1) = .5 + .12172

= .62172

N(d2) = .5 - .04776

= .45224

Substituting these values into the Black-Scholes formula yields

C = 20(.62172) - e-(.08) (.5)(20)(.45224)

= 12.434 - 8.690

= 3.744

Solving for the value of a put,

P = 3.744 - 20 + (20)e-(.08)(.5)

= -16.256 + 19.216

= $2.96

Chapter 7 Pricing Contingent Claims: Option Pricing Theory and Evidence 79

3. (a) Figure S7.1a shows the payoffs from selling one call (-C), selling one put (-P), and from the

combination (-C-P).

Figure S7.1a Payoffs from selling one call and one put

The portfolio (-C-P) is the opposite of a straddle. It earns a positive rate of return if the stock

price does not change much from its original value. If the instantaneous variance of the stock

increases, the value of the call increases, since

∂C

∂σ

> 0. Suppose the value of the call increases by

some amount a > 0,

C1 = C0 + a

Then by put-call parity, the value of the put also increases by a.

P1 = C1 - S + Xe-rt

= C0 + a - S + Xe-rt

= P0 + a

If you sold one call and one put for prices of C0 and P0, and the options' true values were C0 + a

and P0 + a, this represents an opportunity loss to you of -2a. Given the inside information, the

portfolio strategy would be to buy both the put and the call (at P0 and C0) for a gain of 2a.

(b) Figure S7.1b shows the payoffs from buying one call (+C), selling one put (-P) and from the

combination (C - P).

Figure S7.1b Payoffs from buying one call and selling one put

80 Copeland/Shastri/Weston * Financial Theory and Corporate Policy, Fourth Edition

The return from this portfolio (C - P) remains unchanged by an increase in the instantaneous

variance, since, by put-call parity, the increase in the