The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 447 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 447

OPTION PRICING

We will now apply the concepts of arbitrage pricing to option pricing in a

continuous-state, continuous-time setting. Suppose that a market consists

of three assets: a risk-free asset (which allows risk-free borrowing and lend-

ing at the risk-free rate of interest), a stock, and a European option. We will

show that the price processes of a stock and of a risk-free asset fix the price

process of an option on that stock.

Suppose the risk-free rate is a constant r. Recall from Chapter 4 that

the value Vt of a risk-free asset with constant rate r evolves according to

the deterministic differential equation of continually compounding

interest rates:

dVt = rVtdt

The above is a differential equation with separable variables. After sep-

arating the variables, the equation can be written as

dVt

--------- = rdt

Vt

which admits the solution Vt = V 0 ert where V 0 is the initial value of

the bank account. This formula can also be interpreted as the price pro-

cess of a risk-free bond with deterministic rate r.

Stock Price Processes

Let’s now examine the price process of the stock. Consider the process y

= αt + σBt where Bt is a standard Brownian motion. From the definition

of Itô integrals, it can be seen that this process, which is called an arith-

metic Brownian motion, is the solution of the following diffusion equa-

tion:

dyt = αdt + σdBt

where α is a constant called the drift of the diffusion and σ is a constant

called the volatility of the diffusion. (αt + σB

Consider now the process S t)

t = S 0 e , t ≥ 0. Applying Itô’s

lemma it is easy to see that this process, which is called a geometric

Brownian motion, is an Itô process that satisfies the following stochastic

differential equation:

dSt = μStdt + σStdBt ; S 0 = x