Transforming Arithmetic Asian Option PDE to the Parabolic Equation with Constant Coefficients

Zieneb Ali Elshegmani,Saiful Hafiza Jaaman,Rokiah Rozita Ahmad,Roza Hazli Zakaria

doi:10.1155/2011/401547

Abstract

Arithmetic Asian options are difficult to price and hedge, since at present, there is no closed‐form analytical solution to price them. Transforming the PDE of the arithmetic the Asian option to a heat equation with constant coefficients is found to be difficult or impossible. Also, the numerical solution of the arithmetic Asian option PDE is not very accurate since the Asian option has low volatility level. In this paper, we analyze the value of the arithmetic Asian option with a new approach using means of partial differential equations (PDEs), and we transform the PDE to a parabolic equation with constant coefficients. It has been shown previously that the PDE of the arithmetic Asian option cannot be transformed to a heat equation with constant coefficients. We, however, approach the problem and obtain the analytical solution of the arithmetic Asian option PDE.

Highlights

Asian options are path-dependent options whose payoff depends on the average value of the underlying assets during a specific set of dates across the life of the option
Zhang 4 presents a theory of continuously sampled Asian option pricing; he solves the PDE with perturbation approach, and he shows that the PDE of the arithmetic the Asian option cannot be transformed to the heat equation with constant coefficients
Dewynne and Shaw 9 provide a simplified means of pricing arithmetic Asian options by PDE approach they derive an analytical formula for the Laplace transform in time of the Asian option, and they obtain asymptotic solutions for the Black-Scholes PDE for Asian options for low-volatility limit which is the big problem on using Laplace transform

Summary

Introduction

Asian options are path-dependent options whose payoff depends on the average value of the underlying assets during a specific set of dates across the life of the option. The two basic forms of averages in Asian options arithmetic and geometric both can be structured as calls or puts. A geometric average Asian option is easy to price because a closed-form solution is available 1 the most difficult task

Results

Conclusion