Option pricing in affine generalized Merton models

TL;DR

This paper extends the Merton jump diffusion model by incorporating affine processes, including the log-Heston model, for improved option pricing, and proposes an approximation method for characteristic functions with numerical validation.

Contribution

It introduces affine generalizations of the Merton model with a novel approximation method for characteristic functions of complex components.

Findings

The characteristic function of the log-Heston component is known explicitly.

An approximation procedure for the second component's characteristic function is proposed.

Numerical examples demonstrate the effectiveness of the method.

Abstract

In this article we consider affine generalizations of the Merton jump diffusion model [Merton, J. Fin. Econ., 1976] and the respective pricing of European options. On the one hand, the Brownian motion part in the Merton model may be generalized to a log-Heston model, and on the other hand, the jump part may be generalized to an affine process with possibly state dependent jumps. While the characteristic function of the log-Heston component is known in closed form, the characteristic function of the second component may be unknown explicitly. For the latter component we propose an approximation procedure based on the method introduced in [Belomestny et al., J. Func. Anal., 2009]. We conclude with some numerical examples.