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.

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 $f$).

• Thank you for you answer. I actually have updated my question and wrote the payoff I was thinking about. It is not what you are referring to. Sorry for not making my original question clear enough. Commented Nov 24, 2015 at 22:19
• Is it American or European? Commented Nov 25, 2015 at 20:05
• It is "European" option in this case Commented Nov 25, 2015 at 21:15

You introduce a discretized auxiliary variable which represents $S_t$ to solve $N$ PDEs on $[t, t+\tau]$ using finite differences which will give you the present value of the option at time $t$ conditional on $S_t$. Then you solve one PDE using finite differences on $[0, t]$ to obtain the the present value at time $0$.

This is the same methodology than that used for pricing path dependent options using finite differences. The general idea is to transform a non markovian problem into a markovian problem of higher dimension by adding auxiliary variables that capture the past.

• Make sense! Do you have any good reference on this topic/approach? Commented Nov 25, 2015 at 21:20
• I don't have a specific reference in mind but it is widely used for a variety of path dependent options: asian options, lookback options, etc. The advantage over Monte Carlo is that you retain the accuracy of finite differences, especially for greeks and american type exercice. Commented Nov 26, 2015 at 15:51
• It will be more helpful if you can add more specific details. Commented Nov 26, 2015 at 20:44