Essentially high-order compact schemes with application to stochastic volatility models on non-uniform grids

TL;DR

This paper develops high-order compact finite difference schemes for stochastic volatility PDEs, demonstrating improved accuracy over standard methods on non-uniform grids with applications to option pricing.

Contribution

Introduction of fourth-order accurate compact schemes for stochastic volatility PDEs on non-uniform grids, with comprehensive numerical validation.

Findings

Achieved fourth-order spatial accuracy for zero and non-zero correlation.

Outperformed standard second-order finite difference schemes.

Validated effectiveness on option pricing models.

Abstract

We present high-order compact schemes for a linear second-order parabolic partial differential equation (PDE) with mixed second-order derivative terms in two spatial dimensions. The schemes are applied to option pricing PDE for a family of stochastic volatility models. We use a non-uniform grid with more grid-points around the strike price. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical convergence study we achieve fourth-order accuracy also for non-zero correlation. A combination of Crank-Nicolson and BDF-4 discretisation is applied in time. Numerical examples confirm that a standard, second-order finite difference scheme is significantly outperformed.