Introduction to Corporate Finance

8: Options

This is not, however, the only option pricing model used by capital market analysts and traders.

An option pricing model that is perhaps more famous, but more formidable in its mathematics, is that

created by Fischer Black, Myron Scholes and Robert Merton. This model is driven by a recognition that

an option that fully hedges a capital market exposure will thereby reduce the portfolio risk to that of the

market risk-free rate; and this insight provides us with the appropriate discount rate to use in what is

really a continuous time variant of the present value formula. We examine the Black–Scholes model, as

it is conventionally known, in more detail in section 8-4b.

8-4a THe BINOMIal MOdel

The binomial option pricing model recognises that investors can combine options (either calls or puts) with

shares of the underlying asset to construct a portfolio with a risk-free payoff. Any asset with a risk-free

payoff is relatively easy to value – just discount its future cash flows at the risk-free rate. But if we can

value a portfolio containing options and shares, then we can also calculate the value of the options by

subtracting the value of the shares from the value of the portfolio.

Let’s work through an example that shows how to price an option using the binomial method. The

example proceeds in three distinct steps. First, we must find a portfolio of shares and options that

generates a risk-free payoff in the future. Second, given that the portfolio offers a risk-free cash payment,

we can calculate the present value of that portfolio by discounting its cash flow at the risk-free rate.

Third, given the portfolio’s present value, we can determine how much of the portfolio’s value comes

from the shares and how much comes from the option. By subtracting the value of the underlying shares

from the value of the portfolio, we obtain the option’s market price.

Step 1: Create a risk-free Portfolio

Suppose that shares of Financial Engineers Ltd currently sell for $55. We want to determine the price of

a call option on Financial Engineers shares with an exercise price of $55 and an expiration date in one

year. Assume the risk-free rate is 4%.

The binomial model begins with an assumption about the volatility of the underlying shares. Specifically,

the model assumes that by the time the option expires, the share will have increased or decreased to a

particular dollar value. In this problem, we assume that one year from now, Financial Engineers’ share price

will have risen to $70 or fallen to $40. Figure 8.7 provides a simple diagram of this assumption.^12

12 How can we possibly know that the price of Financial Engineers’ shares will be either $70 or $40? Of course, we cannot know that. Almost

any price is possible one year in the future. Soon, we will illustrate that this assumption, which seems completely ridiculous now, isn’t really

necessary in a more complex version of the binomial model. But let’s understand the simple version first.

LO8.4

binomial option

pricing model

A model that uses the

principle of ‘no arbitrage’ to

calculate call and put values

FIGure 8.7 BINOMIAL OPTION PRICING

The figure shows that in one year,

Financial Engineers’ shares will

each be worth $70 or $40. If there

is a call option on these shares,

with a strike price of $55, then

they will be worth $15 or $0 when

that option expires in one year.

Share

price

today

$55

$70

$40

$15

$0

Share

price

in one year

Call

option

payoff