Time-adaptive high-order compact finite difference schemes for option pricing in a family of stochastic volatility models

TL;DR

The paper addresses efficient pricing of options under a two-factor stochastic volatility model by deriving a time adaptive, fourth order accurate scheme that combines a high order compact semi-discrete spatial discretisation with a fourth order multistep time integrator for the resulting PDE $V(S,v,t)$. It introduces rectangular-domain transformations to enable a uniform grid, and employs a predictor-corrector framework with a Crank-Nicolson startup and a BDF-4 corrector, using local error estimates to adapt the time step. The main contributions are the formulation of a fully time adaptive, fourth order in space and time scheme and demonstrating its efficiency through numerical tests on European puts for models such as Heston and its variants, with adaptive time stepping concentrating computational effort where needed. The approach offers accurate and efficient option pricing in two-factor SV models, improving practical applicability for pricing and risk management.

Abstract

We propose a time-adaptive, high-order compact finite difference scheme for option pricing in a family of stochastic volatility models. We employ a semi-discrete high-order compact finite difference method for the spatial discretisation, and combine this with an adaptive time discretisation, extending ideas from [LSRHF02] to fourth-order multistep methods in time.

Figures (1)

• Figure 1: Adaptation factor $\xi_n$, time grid points distribution, and error threshold $\hat{\epsilon}$ (dotted red), local error $|| \epsilon_n||_2$ for adaptive (solid green) and equidistant time stepping (dashed blue): GARCH (left), $a=b=3/4$ model (right).