Consider a stochastic volatility model for stock price $[$here the volatility is denoted by $]$:

$d{S}_t=r{S}_{t}dt+{}_t{S}_{t}d{W}_t$

$d{}_t=a(-{}_t)dt+d{B}_t.$

Here $r,a,,$ are all positive constants, and $(W,B)$ is a two$-$dimensional Brownian motion with covariance matrix

$\left[\begin{array}{cc}1& \\ & 1\end{array}\right].$

In this model the volatility $$ is itself a $($mean$-$reverting$)$ stochastic process. We are interested in estimating the price of a vertical spread with maturity $T.$ That is$,$

$[$:$(S_T-K{)}^{+}=(exp(Y_T)-K{)}^{+}\}{Y}_t$hat$(Y{)}_{t_{i+1}}=$hat$(Y{)}_{t_i}+(r-\frac{1}{2}$hat$({)}_{t_i}^2)(t_{i+1}-t_i)+$hat$({)}_{t_i}\sqrt[\phantom{2}]{t_{i+1}-t_i}{Z}_{i+1}$

hat$(Y{)}_{t_0}=Y_0$