Application of Operator Splitting Methods in Finance

TL;DR

This paper reviews operator splitting methods for numerically solving PDEs in option pricing, demonstrating their efficiency and stability across various models including Black-Scholes, jump-diffusion, and stochastic volatility models.

Contribution

It introduces and compares different operator splitting schemes tailored for complex financial PDEs, including ADI and IMEX methods, with practical implementation insights.

Findings

Splitting schemes show stable and convergent results for multiple models.

Numerical experiments validate efficiency in European and American options.

Methods effectively handle nonlocal jump operators and multidimensional problems.

Abstract

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps.