Introduction to Corporate Finance

8: Options

When you first encounter it, the Black–Scholes option pricing equation looks rather intimidating.

As a matter of fact, Black and Scholes’s paper was originally rejected by the editor at the prestigious

academic journal in which they eventually published their prize-winning formula. The editor felt it was

too technical and not of interest to a wide audience. Although it is true that the derivation of the formula

requires a rather high level of mathematics, the intuition behind the equation is fairly straightforward. In

fact, the logic of the Black–Scholes model mirrors that of the binomial model.

Black and Scholes began their research by asking a question. Suppose investors can buy and sell

shares, options on those shares, and risk-free bonds. Does a combination of options and shares exist that

provides a risk-free payoff? This should sound familiar, because it is exactly how you begin when you

price an option using the binomial model. However, Black and Scholes’ approach to valuing an option

differs from the binomial method in several important ways.

First, recall that the binomial model assumes that over a given time period, the share price will move

up or down by a known amount. In Figure 8.8, we showed that by shortening the length of the period

during which the share price moves, we increase the number of different prices that the shares may

reach by the option’s expiration date. The Black–Scholes model takes this approach to its logical extreme.

It presumes that share prices can move at every instant. If we were to illustrate this assumption by

drawing a binomial tree like the ones in Figure 8.8, the tree would have an infinite number of branches,

and on the option’s expiration date the share price could take on almost any value.

Second, Black and Scholes did not assume that they knew precisely what the up-and-down

movements in shares would be at every instant. They recognised that these movements were essentially

random, and therefore unpredictable. Instead, they assumed that the volatility, or standard deviation, of

a share’s movements was known.

With these assumptions in place, Black and Scholes calculated the price of a European call option

(on a non-dividend-paying share) with the following equations:

Eq. 8.2 C = SN(d 1 ) – Xe−rt N(d 2 )

Eq. 8.3 =









 σ







σ

=−σ

d

S

X

rt

t

dd t

In ++

2

1

2

21

Let’s dissect this carefully. We have seen most of the terms in the equation before:

S = current market price of underlying share

X = strike price of option

t = amount of time (in years) before option expires

r = annual risk-free interest rate

σ = annual standard deviation of underlying share’s returns

e = 2.718 (approximately)

N(X) = the probability of drawing a value less than or equal to X from the standard normal distribution.

Does this list of variables look familiar? It should, because it is nearly identical to the list of inputs required

to use the binomial model. The stock or share price (S), the strike price (X), the time until expiration (t)

and the risk-free rate (r) are all variables that the binomial model uses to price options. The new item the

Black–Scholes model requires is the standard deviation, σ, of the underlying asset’s returns.

standard normal

distribution

A normal distribution with a

mean of zero and a standard

deviation of 1