Introduction to Corporate Finance

8: Options

present value of the exercise price, adjusted for the probability that when the option expires, the share

price will exceed the strike price (the probability that the option expires in the money).

example

Shares of Cloverdale Food Processors currently sell for $40. A European call option on Cloverdale shares has an

expiration date six months in the future and a strike price of $38. The estimate of the annual standard deviation

of Cloverdale shares is 45%, and the risk-free rate is 6%. What is the call worth? It is worth $6.58, as shown

below.

d 1

40 2

38

006

045

2

1

2

045

1

2

=









++









ln.

.

.

=

+

=

(. )(.)

.

.

0 0513 0 0806

0 3182

0 4146

d 21 dt0 4146 045

1

2

=−σ =−..=0 0964.

N(0.4146) = 0.6608 N(0.0964) = 0.5384

C = 40(0.608) – 38(2.718–(.06)(0.5)) (0.5384) = $6.58

By experimenting with Equations 8.2 and 8.3, we can study the effect of changes in each of the

key input variables on the price of a call option. For example, suppose we recalculate the value of the

Cloverdale call option, described earlier, by adjusting just one of the required inputs each time to see

the resulting effect on the option’s price. After just a few experiments, we could reach the following

conclusions:

■ The call value increases as the price of Cloverdale shares (S) increases.

■ The call value increases as the time to expiration (t) increases.

■ The call value increases as the standard deviation of Cloverdale shares (σ) increases.

■ The call value increases as the strike price (X) decreases.

■ The call value increases as the risk-free interest rate (r) increases.

We have already discussed the first four relationships above. Call values generally increase with

increases in the underlying share price, the time to expiration or the volatility of the underlying

shares, and calls are more valuable when the strike price is lower. The finding that call values increase

when the interest rate increases is new. Here is an intuitive explanation for that relationship. The

call option grants the holder the right to buy something and to pay for it at a later date. The right to

defer payment is more valuable when the interest rate is high, so call values increase when interest

rates do.

Though the Black–Scholes model and the binomial model look very different at first glance, they

share the same underlying logical principles. Both models calculate option values based on the notion

that combinations of options and underlying shares can mimic the payoffs of risk-free bonds. Both

models require essentially the same inputs (S, X, r, t and some assumption about volatility) to calculate

option values. And both models produce the same predictions about how changes in the input variables

affect option prices.

See the concept explained

step by step on the

CourseMate website.

SMarT

CONCEPTS

How does a share’s expected

return influence the price of a

call option?

thinking cap

question