The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 453 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 453

Let’s now generalize the price process of the underlying stock. We

represent the underlying stock price process as a generic Itô process.

Recall from Chapter 8 that a generic univariate Itô process can be repre-

sented through the differential stochastic equation:

dSt = μ(St, t)dt + σ(St, t)dBt ; S 0 = x

where x is the initial condition, B is a standard Brownian motion, and

μ(St,t) and (St,t) are given functions R × [0,∞) → R. The geometric

Brownian motion is a particular example of an Itô process.

Let’s now define the bond price process. We retain the risk-free

nature of the bond but let the interest rate be stochastic. Recall that in a

discrete-state, discrete-time setting, a bond was defined as a process

that, at each time step, exhibits the same return for each state though

the return can be different in different time steps. Consequently, in con-

tinuous-time we define a bond price process as the following integral:

t

∫rS(u, u)ud

V^0

t = V 0 e

where r is a given function that represents the stochastic rate. In fact,

the rate r depends on the time t and on the stock price process St. Appli-

cation of Itô’s lemma shows that the bond price process satisfies the fol-

lowing equation:

dVt = VtrS( t, t)dt

We can now use the same reasoning that led to the Black-Scholes

formula. Suppose that there are both an option pricing function Yt =

C(St,t) and a replicating self-financing trading strategy

Y^12

t = θt Vt + θt St

We can now write a stochastic differential equation for the process Yt in

two ways:

1. Applying Itô’s lemma to Yt = C(St,t)
   1 2


2. Directly to Yt = θt Vt + θt St

The first approach yields