Small Dimension Pde for Discrete Asian Options

Abstract

This paper presents an efficient method for pricing discrete Asian options. Its contribution to the existing literature consists in targeting at smile and non proportional dividend effects. Using an homogeneity property, we show how to reduce an n +1 dimensional problem to a 2 or 3 dimensional one. We derive a PDE for the Asian option and solve it with the standard Crank Nicholson method. The dimension reduction imposes us to interpolate and extrapolate our conditional price at each fixing date. Within a deterministic volatility structure consistent with the smile, the homogeneity property is roughly conserved, thanks to a vega correction term. This allows us to stay in a two dimensional framework as in the Black Scholes case. We examine different numerical specifications of our finite difference (interpolation method, grid boundaries, time and space steps) as well as the extension to the case of non proportional discrete dividends, using a jump condition. We benchmark our results with Quasi Monte-Carlo simulation and a multi-dimensional PDE.

JEL Classification: G12, G13