Finite-Difference Pricing of American–Bermudan Asian Options
Summary
The document describes a finite-difference setup for pricing an American–Bermudan Asian option with a fixed strike. The option value depends on time, the underlying price, and the running average, so its pricing equation includes derivatives in both the underlying and average dimensions. The questioner uses an explicit scheme and boundary conditions informed by prior work, with the terminal payoff based on the average exceeding the strike.
A key numerical challenge is that the chosen upper boundary for the average must be high enough to guarantee exercise, which can make the computational grid large when the starting average is near the strike. The author reports numerical instability for an out-of-the-money case and asks whether the boundary problem can be simplified without adding more average-grid steps. The document provides the setup and motivation but no answer or validation, so it does not establish a stable scheme, a better boundary condition, or pricing accuracy. Its value is in highlighting the interaction between exercise assumptions, boundary placement, and grid resolution.
Key ideas
- The pricing state includes time, the underlying price, and the running average.
- The explicit finite-difference scheme must handle boundary conditions in both price dimensions.
- A high exercise-guaranteeing average boundary can enlarge the grid substantially.
- The question reports instability for an out-of-the-money case but gives no tested remedy.
Tags
Full text
# American-Bermudan-Asian option fixed strike using finite differences
# 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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.