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. – mth_mad Nov 24 '15 at 22:19
• Is it American or European? – parsiad Nov 25 '15 at 20:05
• It is "European" option in this case – mth_mad Nov 25 '15 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? – mth_mad Nov 25 '15 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. – Antoine Conze Nov 26 '15 at 15:51
• It will be more helpful if you can add more specific details. – Gordon Nov 26 '15 at 20:44