Numerical Contour Integral Methods for Free-Boundary Partial Differential Equations Arising in American Volatility Options Pricing

Yong Chen,Jianjun Ma

doi:10.1155/2018/1838521

Abstract

The aim of this paper is to study the numerical contour integral methods (NCIMs) for solving free-boundary partial differential equations (PDEs) from American volatility options pricing. Firstly, the governing free-boundary PDEs are modified as a unified form of PDEs on the fixed space region; then performing Laplace-Carson transform (LCT) leads to ordinary differential equations (ODEs) which involve the unknown inverse functions of free boundaries. Secondly, the inverse free-boundary functions are approximated and optimized by solving of the free-boundary values of the perpetual American volatility options. Finally, the ODEs are solved by the finite difference methods (FDMs), and the results are restored via the numerical Laplace inversion. Numerical results confirm that the NCIMs outperform the FDMs for solving free-boundary PDEs in regard to the accuracy and computational time.

Highlights

The pricing problems of European-style options written on volatility are widely studied in the literature by deriving the explicit formulas [1,2,3,4,5,6], by using the simulation approaches [7, 8], by using the partial differential equations (PDEs) approaches [5, 9,10,11,12,13], and by the empirical analysis [14]
The aim of this paper is to study the numerical contour integral methods (NCIMs) for solving free-boundary partial differential equations (PDEs) from American volatility options pricing
This paper studies the numerical contour integral methods (NCIMs) for solving free-boundary partial differential equations (PDEs) from American volatility options written on the volatility whose price follows four well-known models: geometric Brownian motion process (GBMP), mean-reverting Gaussian process (MRGP), mean-reverting square-root process (MRSRP), and mean reverting log process (MRLP)

Summary

Introduction

The pricing problems of European-style options written on volatility are widely studied in the literature by deriving the explicit formulas [1,2,3,4,5,6], by using the simulation approaches [7, 8], by using the PDE approaches [5, 9,10,11,12,13], and by the empirical analysis [14]. This paper studies the numerical contour integral methods (NCIMs) for solving free-boundary partial differential equations (PDEs) from American volatility options written on the volatility whose price follows four well-known models: GBMP, MRGP, MRSRP, and MRLP. The original idea of NCIMs is developed by Zhou, Ma, and Sun [16] for solving free-boundary problems of space-fractional diffusion equations; Ma and Zhu [8] prove the convergence rates of such methods under the regime-switching European option pricing, and it can be described as follows. V this paper has no such procedure by means of solving of the free-boundary values of the perpetual American volatility options, which can optimize the approximated inverse freeboundary functions. The remaining parts of this paper are arranged as follows: In Section 2, the free-boundary PDEs governing the American volatility option price are presented and some accompanied properties are introduced.

Model Descriptions

Numerical Examples

Findings

Conclusions