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

## TL;DR

This paper introduces a high-order compact finite difference scheme for option pricing in stochastic volatility jump models, offering improved accuracy and efficiency over standard methods, validated through numerical experiments.

## Contribution

A novel fourth-order accurate in space and second-order in time finite difference scheme for complex option pricing models, outperforming standard discretizations and finite element methods.

## Key findings

- Outperforms standard second-order schemes in accuracy and efficiency.
- Requires only one LU-factorization for the entire computation.
- Uses less memory and computational effort compared to finite element methods.

## Abstract

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in time. Numerical experiments for the European option pricing problem are presented. We validate the stability of the scheme numerically and compare its performance to standard finite difference and finite element methods. The new scheme outperforms a standard discretisation based on a second-order central finite difference approximation in all our experiments. At the same time, it is very efficient, requiring only one initial $LU$-factorisation of a sparse matrix to perform the option price valuation. Compared to finite element approaches, it is very parsimonious in terms of memory requirements and computational effort, since it achieves high-order convergence without requiring additional unknowns, unlike finite element methods with higher polynomial order basis functions. The new high-order compact scheme can also be useful to upgrade existing implementations based on standard finite differences in a straightforward manner to obtain a highly efficient option pricing code.