I have two coupled SDEs \begin{align*} dS_t=rS_tdt+V_tdW_t^{(1)},\\ dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\ \end{align*} where $W_t^{(1)}$ and $W_t^{(2)}$ are independent Brownian motions, initial input data are $S_0$ and $V_0$, $a(\cdot)$ and $b(\cdot)$ are sufficiently well-behaved, and I use an Euler-Maruyama discretisation with $N$ timesteps. How exactly should one calculate the derivative of a payoff function $\mathbb{E}f(S)$ with respect to $S_0$ in this case?

In particular, I am confused as to how to apply the chain rule in this case due to the dependence of $S_t$ on $V_t$? For example, say with the likelihood ratio method, would my integral form of the probability density be formed by $2N$ integrals ($N$ with respect to the $S$ and $N$ with respect to $V$)? (Similarly for the pathwise sensitivity approach, I am confused as to how construct the chain rule.)

Any help is greatly appreciated!


Are W1 and W2 independent? I would assume there is some correlation structure? Cholesky decomposition would help in generating the path. It's very similar to heston model where Vt is volatility.


  • $\begingroup$ Yes they are completely independent, they are coupled via V_t. $\endgroup$
    – user107224
    Apr 17 at 16:34

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