Asset Pricing with Stochastic Volatility

Abstract

We consider a market consisting of a stock S_t and a bond B_t governed by the following equations:

$$d{S_t} = a(t,{S_t}){S_t}dt + {\sigma _t}{S_t}d{W_t} $$

(13.1)

and

$${\text{d}}{{\text{B}}_{\text{t}}}{\text{ = }}{{\text{r}}_{\text{t}}}{{\text{B}}_{\text{t}}}{\text{dt,}} {{\text{B}}_{\text{0}}}{\text{ = 1}}$$

(13.2)

where W_t is a Brownian motion, S₀ is a given random variable independent of W, r_t is a bounded, non-negative, progressively measurable interest rate process.