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

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?

• Did not look deeply into it but Least Squares Monte Carlo (e.g. Longstaff-Schwartz alogrithm) seems like a good candidate. Also $I_t$ is $\mathcal{F}_t$ measurable if $S_t$ is adapted to the latter filtration so I don't understand your statement concerning $I_j$ etc. – Quantuple Apr 5 '17 at 7:30
• For the usual Asian option using PDE approach, the differential $d\ I_t$ can be represented by $I_t$ and $S_t.$ But in this case, we can't, it is the reason for the failure of PDE approach I think. – A.Oreo Apr 5 '17 at 8:07
• You can at the very least get estimates of the upper and lower bounds - for the lower bound price it without the american exercise, and for the upper bound you can price where early exercise is determined by looking at the future of each path (i.e. the value of an american option has to be worth more to someone who can tell the future, so you can say this is an upper bound on your price...). If you're lucky they'll be close together! – will Apr 5 '17 at 8:08

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.
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.