I'm a bit stuck with the pricing of an option where the underlying stock is dependent with an additional process.
Setting: Assume that we have a probability space where under $Q$ the dynamics of the stock and an additional process are given by
$dS(t)=S(t)(rdt+\sigma dW_1 (t))$
$d\lambda(t)=c\lambda(t)dt+\xi dW_2 (t)$
where $dW_2 (t)=\rho dW_1 (t)+ \sqrt{1-\rho^2}dZ(t)$ with $W_1 (t)$ and $Z(t)$ are independent brownian motions.
The question is now how to determine the following conditional risk-neutral valuation:
$E^Q [e^{-\int_0^T\lambda(v)dv} max(S(T),K)|e^{-\int_0^T\lambda(v)dv} =x]$
In the last expression, we can rewrite it as
$ x E^Q [max(S(T),K)|e^{-\int_0^T\lambda(v)dv} =x]$
but then I'm stuck how we can deal with the dependence between $S$ and $\lambda$.
Thanks a lot in advance for your help!
N.B: $\lambda$ is not the interest rate but just a stochastic discount factor.