I want to price an American call option based on an underlying that follows a jump-diffusion process with an inhomogeneous jump frequency function.

My mathematical skills are not sufficient to derive the respective PDE for the option price (due to complexity, there we cannot find a closed-form solution anyway).

I want to apply the LSM by Longstaff and Schwartz (2001) to find the optimal exercise strategy and calculate the value of an american call option.

To do this, it still necessary to derive the respective PDE? If yes, why?


1 Answer 1


You do not need the PDE to implement the LSM algorithm.

The $T$ maturity American call price on time $t$ is $$v_t = \max_{\tau} E_t\left[e^{-\int_t^\tau r(u) du} (S_\tau - K)^+\right]$$ where the max is over all the stopping times $t \leq \tau \leq T$.

After discretizing time along a discrete time line $t_k$, this leads to the recursion $$ v_{t_k} = \max\left\{(S_{t_k} - K)^+, E_{t_k}\left[v_{t_{k+1}} e^{-\int_{t_k}^{t_{k+1}} r(u) du}\right] \right\} $$ where $E_{t_k}\left[v_{t_{k+1}} e^{-\int_{t_k}^{t_{k+1}} r(u) du}\right]$ is the option continuation value (the value if you choose not to exercise on $t_k$).

The Longstaff and Schwartz LSM algorithm is a "trick" to compute the continuation value at any point in the Monte Carlo simulation without having to resort to a new Monte Carlo simulation that would originate at that point.

As you can see you do not need a PDE for $v$ to implement the algorithm. The only thing you require is the SDE for $S_t$ for generating the Monte Carlo paths, as well as appropriate projection functions for the LS algorithm itself (there are many good references on the latter, for instance Monte Carlo Methods in Finance - Peter Jäckel).

  • $\begingroup$ @MrPefister If this answers your question please accept it by clicking the check mark. $\endgroup$
    – Bob Jansen
    Commented Dec 23, 2017 at 11:02

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