Algorithms of Finite Difference for Pricing American Options under Fractional Diffusion Models

Jun Xi,Yanqing Chen,Jianwen Cao

doi:10.1155/2014/364868

Abstract

It is well known that linear complementarity problem (LCP) involving partial integro differential equation (PIDE) arises from pricing American options under Lévy Models. In the case of infinite activity process, the integral part of the PIDE has a singularity, which is generally approximated by a small Brownian component plus a compound Poisson process, in the neighborhood of origin. The PIDE can be reformulated as a fractional partial differential equation (FPDE) under fractional diffusion models, including FMLS (finite moment log stable), CGMY (Carr-Madan-Geman-Yor), and KoBol (Koponen-Boyarchenko-Levendorskii). In this paper, we first present a stable iterative algorithm, which is based on the fractional difference approach and penalty method, to avoid the singularity problem and obtain numerical approximations of first-order accuracy. Then, on the basis of the first-order accurate algorithm, spatial extrapolation is employed to obtain second-order accurate numerical estimates. Numerical tests are performed to demonstrate the effectiveness of the algorithm and the extrapolation method. We believe that this can be used as necessary tools by the engineers in research.

Highlights

The holder of American options has the right to exercise at any date prior to maturity and at the expiry date, while the holder of European options can exercise his right only at the expiry date
Cartea and del-Castillo-Negrete [26] presented the fractional partial differential equation (FPDE) under some infinite activity Levy models (FMLS, KoBol, and CGMY) as a generalization to partial integro differential equation (PIDE) and gave a finite difference method using the numerical valuation of fractional derivatives
Our study focuses on the discretization of fractional derivatives, spatial extrapolation, and design of stable iterative algorithms for pricing American options in the scheme of FPDE

Summary

Introduction

The holder of American options has the right to exercise at any date prior to maturity and at the expiry date, while the holder of European options can exercise his right only at the expiry date. Cartea and del-Castillo-Negrete [26] presented the fractional partial differential equation (FPDE) under some infinite activity Levy models (FMLS, KoBol, and CGMY) as a generalization to PIDE and gave a finite difference method using the numerical valuation of fractional derivatives. The numerical solutions of three fractional partial differential equations have been compared by Marom and Momoniat [27] Their methods are of first-order accuracy and are only for evaluation of European options. Our study focuses on the discretization of fractional derivatives, spatial extrapolation, and design of stable iterative algorithms for pricing American options in the scheme of FPDE. We first present a stable iterative algorithm, which is based on the fractional difference approach and penalty method, to avoid the singularity problem and obtain numerical approximations of first-order accuracy.

Mathematical Model for American Options

Space and Time Discretization

The Penalty Method

Numerical Results

Conclusions