Pricing Forward Start Option with PDE

I am looking for references (books and papers) or suggestions on how to price forward starting calls using a PDE approach typically in the Heston model (In the BS world, the computation is trivial), with forward payoff

$$\left(\frac{S_{t+\tau}}{S_t}-K\right)^{+},$$ where $t$ and $\tau$ are positive numbers.

I feel like the only way to use a PDE approach would be to identify the fundamental solution of the PDE in order to be able to apply the tower property on the expectation of the payoff.

All I have read up to know focus computing the characteristic function, and the martingale approach.

Answer

Here's an approach that's easy to code (but FAR from the fastest). Let $f(T,S,v,K)$ denote the price of a European call in the Heston model with time-to-expire $T$, initial price $S$, initial volatility $v$, strike $K$. First, use the tower property to transform the pricing problem:

$$\begin{align*} V_{0} & =\mathbb{E}\left[e^{-r\left(t+\tau\right)}\left(S_{t+\tau}/S_{t}-K\right)^{+}\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}\mathbb{E}\left[e^{-r\tau}\left(S_{t+\tau}-KS_{t}\right)^{+}\mid\mathcal{F}_{t}\right]\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}f(\tau,S_{t},v_{t},KS_{t})\right]. \end{align*}$$ Now perform a Monte-Carlo simulation to approximate the above (the advantage here is that you can use existing procedures to compute $f$).