# American-Bermudan-Asian option fixed strike using finite differences

I'm trying to price the same American-Bermudan-Asian option described in Longstaff Schwartz (2001). Specifically, using finite difference methods with an explicit scheme to solve

\begin{aligned} \frac{\partial V}{\partial t} + \frac 12\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} + \frac{S - A}{\tau + t}\frac{\partial V}{\partial A} &= rV, &(t,S,A)\in \overline\Omega,\\ V(T, S, A) &= (A - K)^+, & (S,A)\in\overline\Omega_S\times\overline\Omega_A,\\ V(t, 0, A) &= e^{-r(T-t)}\left(\frac{\tau + t}{\tau + T}A - K\right)^+, &(t, A)\in\overline\Omega_T\times\overline\Omega_A\\ \frac{\partial V}{\partial S}(t, \overline S, A) &= \frac{T-t}{\tau + T}e^{-r(T-t)}, &(t, A)\in\overline\Omega_T\times\overline\Omega_A\\ V(t, S, \overline A) &= e^{-r(T-t)}\left(\frac t\tau K + \frac{T-t}{\tau + T}S\right), &(t, S)\in\overline\Omega_T\times\overline\Omega_S\\ \end{aligned}

where $$\overline\Omega = \overline\Omega_T\times \overline\Omega_S\times\overline\Omega_A \triangleq [0,T[\times]0, \overline S[\times]0, \overline A[$$ with $$\overline S\gg S_0$$ and $$\overline A = \left(1 + \frac T\tau\right)K$$. These boundary conditions have been constructed following Kemna Vorst (1990). In particular, $$\overline A$$ has been chosen such that the exercise is sure for any $$t$$ once $$A$$ reaches that value (recall the running sum is not decreasing).

In particular, for a lookback period of $$\tau = 1/4$$, a maturity of $$T=2$$, and a strike price of $$K=100$$, the exercise at the beginning is sure only for $$\overline A = 900$$!

Since often, one wants to solve this problem close to the ATM (i.e. $$A_0 = 100$$), this grid seems to be too big. In particular, it leads to numerically unstabilities when pricing OTM (eg. $$(S,A) = (80,90)$$).

Is there any way/trick to simplify this boundary problem besides increasing the number of steps in the $$A$$ direction?