High-Order ADI Scheme for Option Pricing in Stochastic Volatility Models

Abstract

We propose a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. Our approach combines different high-order spatial discretisations with Hundsdorfer and Verwer's ADI time-stepping method, to obtain an efficient method which is fourth-order accurate in space and second-order accurate in time. Numerical experiments for the European put option pricing problem using Heston's stochastic volatility model confirm the high-order convergence.

High-order compact finite difference scheme for option pricing in stochastic volatility jump models

“…A straightforward extension of this paper is the introduction of the so-called SVCJ model which allows for jumps in both returns and volatility. As a second extension, one can combine the method presented in this paper with high-order alternating direction implicit methods [18] and with sparse grids methods [21,16]. We leave these extensions for future research.…”

“…[25] where different efficient methods for solving the American option pricing problem for the Heston model are proposed. Other approaches include finite element-finite volume [37], multigrid [8], sparse wavelet [24], FFT-based [30], spectral [36], hybrid tree-finite difference [5] methods and operator splitting techniques [23,15,18,21,16].…”

A Simulation of the Heston Model with Stochastic Volatility Using the Finite Difference Method

“…However, the method requires a large number of time steps. Many authors instead use the implicit method or alternative implicit method (Douglas & Rachford, 1956;Peaceman & Rachford, 1955;Craig & Sneyd, 1988;Hundsdorfer, 2002;Hout, 2007;During & Miles, 2017) which do not limit the number of time steps but require at each time step the solution of large sets of equations. Other authors follow the ideas of the splitting method as a combination of operator splitting and iterative methods, and transform a 2dimenstional problem into quasi 1-dimensional ones (Safaei et al, 2018;Li & Huang, 2020).…”