The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 448 Wednesday, February 4, 2004 1:08 PM

448 The Mathematics of Financial Modeling and Investment Management

where x is an initial value, μ = α + ¹
₂σ^2 and Bt is a standard Brownian

motion. We assume that the stock price process follows a geometric

Brownian motion and that there is no payoff-rate process.

Now consider a European call option which gives the owner the right

but not the obligation to buy the underlying stock at the exercise price K

at the expiry date T. Call Yt the price of the option at time t. The price of

the option as a function of the stock price is known at the final expiry

date. If the option is rationally exercised, the final value of the option is

YT = max(ST – K, 0 )

In fact, the option can be rationally exercised only if the price of the

stock exceeds K. In that case, the owner of the option can buy the

underlying stock at the price K, sell it immediately at the current price St

and make a profit equal to (ST – K). If the stock price is below K, the

option is clearly worthless. After T, the option ceases to exist.

How can we compute the option price at every other date? We can

arrive at the solution in two different but equivalent ways: (1) through

hedging arguments and (2) the equivalent martingale measures. In the

following sections we will introduce hedging arguments and equivalent

martingale measures.

Hedging

To hedge means to protect against an adverse movement. The seller of an

option is subject to a liability as, from his point of view, the option has a

negative payoff in some states. In our context, hedging this option means

to form a self-financing trading strategy formed with the stock plus the

risk-free asset in appropriate proportions such that the option plus this

hedging portfolio is risk free. Hedging the option implies that the hedging

portfolio perfectly replicates the option payoff in every possible state.

A European call option has only one payoff at the expiry date. It

therefore suffices that the hedging portfolio replicates the option payoff

at that date. Suppose that there is a self-financing trading strategy

(θt^1 , θ^2 t ) in the bond and the stock such that

1 2

θt VT + θt ST = YT

To avoid arbitrage, the price of the option at any moment must be equal

to the value of the hedging self-financing trading strategy. In fact, sup-

pose that at any time t < T the self-financing strategy (θt

1 2

, θt ) has a

value lower than the option: