Principles of Corporate Finance_ 12th Edition

554 Part Six Options

bre44380_ch21_547-572.indd 554 10/05/15 12:53 PM

◗ FIGURE 21.2

Present and possible future

prices of Google stock assum-

ing that in each three-month

period the price will either rise

by 17.09% or fall by 14.60%.

Figures in parentheses show

the corresponding values of

a six-month call option with

an exercise price of $530. The

interest rate is .5% a quarter.

$530.00

Now (?)

$726.65

($196.65)

$386.57

($0)

$530.00

($0)

Month 6

Month 3

$620.59

(?)

$452.64

(?)

Example: The Two-Step Binomial Method

Dividing the period into shorter intervals doesn’t alter the basic approach for valuing a call

option. We can still find at each point a levered investment in the stock that gives exactly the

same payoffs as the option. The value of the option must therefore be equal to the value of this

replicating portfolio. Alternatively, we can pretend that investors are risk-neutral and expect

to earn the interest rate on all their investments. We then calculate at each point the expected

future value of the option and discount it at the risk-free interest rate. Both methods give the

same answer.

If we use the replicating-portfolio method, we must recalculate the investment in the stock

at each point, using the formula for the option delta:

Option delta =

spread of possible option prices

__________________________

spread of possible stock prices

Recalculating the option delta is not difficult, but it can become a bit of a chore. It is simpler

in this case to use the risk-neutral method, and that is what we will do.

Figure 21.2 is taken from Figure 21.1(b) and shows the possible prices of Google stock,

assuming that in each three-month period the price will either rise by 17.09% or fall by

14.60%.^7 We show in parentheses the possible values at maturity of a six-month call option

with an exercise price of $530. For example, if Google’s stock price turns out to be $386.57 in

month 6, the call option will be worthless; at the other extreme, if the stock value is $726.65,

the call will be worth $726.65 – $530 = $196.65. We haven’t worked out yet what the option

will be worth before maturity, so we will just put question marks there for now.

We continue to assume an interest rate of 1% for 6 months, which is equivalent to about

.5% a quarter. We now ask: If investors demand a return of .5% a quarter, what is the prob-

ability (p) at each stage that the stock price will rise? The answer is given by our simple

formula:

p =

interest rate − downside change

____________________________

upside change − downside change

=

.005 − (−.1460)

_______________

.1709 − (−.1460)

= .4764

We can check that if there is a 47.64% chance of a rise of 17.09% and a 52.36% chance of a

fall of 14.60%, then the expected return must be equal to the .5% risk-free rate:

(.4764 × 17.09) + (.5236 × −14.60) = .5

(^7) We explain shortly why we picked these figures.
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