Introduction to Corporate Finance

ParT 2: ValuaTION, rISk aNd reTurN

example

We begin this problem with the same set of

assumptions from the last problem. Financial

Engineers shares each sell for $55, and may increase

to $70 or decrease to $40 in one year. The risk-free

rate equals 4%. We want to use the binomial model

to calculate the value of a one-year put option

with a strike price of $55. We begin by finding the

composition of a perfectly hedged portfolio. As

before, let’s write down the payoffs of a portfolio that

contains one share of share and h put options.

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 $0h $15h

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

Notice that the put option pays $15 when the

share price drops, and pays nothing when the share

price rises. Set the payoffs in each scenario equal to

each other and solve for h:

70 + 0h = 40+15h

h = 2

To create a perfectly hedged portfolio, we must

buy one share and two put options. Observe that in

this problem we are buying options, not selling them.

Put values increase when share values decrease, so

it is possible to form a risk-free portfolio containing

long positions in both shares and puts, because they

move in opposite directions. By plugging the value of

h = 2 back into the equation, we see that an investor

who buys one share and two put options essentially

creates a synthetic bond with a face value of $70:

70 + 0(2) = 40 + 15(2)

70 = 70

Given a risk-free rate of 4%, the present value

today of $70 is $67.31. It would cost $67.31 to buy a

one-year, risk-free bond paying $70, so it must also

cost $67.31 to buy the synthetic version of that bond,

consisting of one share and two puts. Given that the

current share price is $55, and letting P denote the

price of the put, we find that the put option is worth

$6.16 (rounding to the nearest cent):

Cost of one share + 2 puts = $67.31 = $55 + 2P

$12.31 = 2P

$6.16 = P

What factors influence the prices

of call and put options?

thinking cap
question

Take a moment to look over the last two examples of pricing options that use the binomial approach.

Make a list of the data needed to price these options:

■ the current price of the underlying shares.

■ the amount of time remaining before the option expires.

■ the strike price of the option.

■ the risk-free rate.

■ the possible values that the underlying shares could take in the future.

On this list, the only unknown is the fifth item. You can easily find the other four necessary values by

looking at current market data.

At this point, we want to pause and ask one of our all-time favourite exam questions. Look back at

Figure 8.7. What assumption are we making about the probability of an up-and-down move in Financial

Engineers’ shares? Most people see that the figure shows two possible outcomes and guess that the

probabilities must be 50–50. That is not true. At no point in our discussion of the binomial model did we

make any assumption about the probabilities of up-and-down movements in the shares. We don’t have

to know what those probabilities are to value the option, which is convenient, because estimating them

could be very difficult.