Introduction to Corporate Finance

ParT 2: ValuaTION, rISk aNd reTurN

The call option we want to price has a strike of $55. Therefore, if the underlying share reaches $70

in one year, the call option will be worth $15. However, if Financial Engineers’ price falls to $40, the call

option will be worthless.

Here is the crux of the first step. We want to find some combination of Financial Engineers’ shares

and the call option which yields the same payoff whether the share goes up or down over the next year.

In other words, we want to create a risk-free combination of shares and calls. To begin, suppose we

purchase one share and h call options. At the moment, we do not know the value of h, but we can solve

for it. Because our portfolio objective is to generate the same cash payment one year from now, whether

our share rises or falls, we can write down the portfolio’s payoffs in each possible scenario and then

choose h so that the payoffs are equal.

Cash flows one year from today

If the share price goes up to $70 If the share price drops to $40

One share is worth $70 $40

h options are worth $15h $0h

Total portfolio is worth $70 + $15h $40 + $0h

A portfolio that contains one share and h call options will have the same cash value in one year if we

choose the value of h that solves this equation:

70 + 15h = 40 +0h

h = –2

The value of h represents the number of call options in our risk-free portfolio. Because h equals –2,

we must sell two call options and combine that position with our single share to create a risk-free portfolio.

Why do we sell options to achieve this objective? Remember that the value of a call option rises as the

share price rises. If we own a share and a call option (or several call options) on that share, the assets in

our portfolio will move together, rising and falling at the same time. To create a portfolio that behaves

like a risk-free bond, we need the movements in the shares and the call option to offset each other. If

the share price movements exactly cancel out fluctuations in the call, then the portfolio’s payout will not

move at all, just like a risk-free bond. Therefore, if we buy a share, we must sell call options to create

offsetting movements between the assets in our portfolio.

What happens to our portfolio if we buy one share and sell two calls? You can see the answer in two

ways. First, just plug the value –2 back into the equation that we used to solve for h. You get:

40 = 40

This expression says that the portfolio payoff will be $40 whether the share price increases or

decreases. Another way to see this is to lay out the payoffs of each asset in the portfolio in a table like this.

Cash flows one year from today

If the share price goes up to $70 If the share price drops to $40

One share is worth $70 $40

Two short options are worth –30 0

Total portfolio is worth 40 40

The first line of the table is self-explanatory. The second line indicates that if we sell two call options

and the share price equals $70 next year, then we will owe the holder of the calls $15 per option, or $30