How to price the American style Asian option with recent N day average

Question

How to price the American style Asian option with recent N day average, for example, we exercise at t day, then the payment is $$\Psi(t) = \dfrac{1}{N}\sum\limits^t_{i=t - N+1}S_i$$ Since the early exercise and path dependence, we can not use the Monte Carlo simulation and tree method. And since the average is not from the begin day to today, we are not allowed to use the addition variable: $$I_j = \sum\limits^j_{i=j - N+1}S_i$$ namely, we can't use the PDE method. This is because, we don't know $I_{j+1}$ just from $I_j$ and $S_{j+1}.$

I only know above three methods to price the option. Is there any reference or advanced method to price such option?

Answer 1

In convertible bond pricing there is something similar called a "soft call" with similar properties so you might want to search for literature on them. The main difference is that soft calls are an exercise condition rather than an exercise price.

One key point is that, if expiration $t$ is distant, very little error is introduced by ignoring the softness. This is because there is so much variance remaining that paths exceeding the exercise price tend to do so by a lot, and for a long while.

Otherwise we are in the realm of "tricks", among which I count Longstaff Schwartz (LS). In practice LS is not used much because it is so slow and tricky to metaparameterize.

One good trick is to pretend N is a much smaller number (say 2 or 3) at greater intervals (like 10 day periods). Then you can add a couple dimensions $I^n_j$ to your PDE solver (often on a reasonably small grid).

Another trick used by practitioners is to associate any given level of $S$ with a probability of exercise based on the brownian bridge. This is easy to accommodate in a PDE scheme.