Splitting and Matrix Exponential approach for jump-diffusion models with Inverse Normal Gaussian, Hyperbolic and Meixner jumps

TL;DR

This paper extends a matrix exponential and splitting method to solve PIDEs for jump-diffusion models with Inverse Normal Gaussian, Hyperbolic, and Meixner jumps, ensuring stability and positivity.

Contribution

It introduces a second-order operator splitting approach combined with matrix exponential techniques for new jump models, improving stability and accuracy.

Findings

The schemes are unconditionally stable in time.

Solutions preserve positivity.

Numerical experiments validate the approach.

Abstract

This paper is a further extension of the method proposed in Itkin, 2014 as applied to another set of jump-diffusion models: Inverse Normal Gaussian, Hyperbolic and Meixner. To solve the corresponding PIDEs we accomplish few steps. First, a second-order operator splitting on financial processes (diffusion and jumps) is applied to these PIDEs. To solve the diffusion equation, we use standard finite-difference methods. For the jump part, we transform the jump integral into a pseudo-differential operator and construct its second order approximation on a grid which supersets the grid that we used for the diffusion part. The proposed schemes are unconditionally stable in time and preserve positivity of the solution which is computed either via a matrix exponential, or via P'ade approximation of the matrix exponent. Various numerical experiments are provided to justify these results.