Generalized pricing formulas for stochastic volatility jump diffusion models applied to the exponential Vasicek model

TL;DR

This paper develops a method to incorporate jump processes into path integral pricing formulas for stochastic volatility models, specifically extending the exponential Vasicek model to include jump diffusion, validated by Monte Carlo simulations.

Contribution

It introduces a novel approach to adapt path integral formulas for stochastic volatility models with jump diffusion, enhancing their realism and applicability.

Findings

Extended the exponential Vasicek model to include jump diffusion.

Validated the extended model with Monte Carlo simulations.

Provided a framework for non-Gaussian fluctuations in option pricing.

Abstract

Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We present a method to adapt formulas for both the path-integral propagators and the option prices themselves, so that jump processes are taken into account in conjunction with the usual drift and diffusion terms. In particular, we focus on stochastic volatility models, such as the exponential Vasicek model, and extend the pricing formulas and propagator of this model to incorporate jump diffusion with a given jump size distribution. This model is of importance to include non-Gaussian fluctuations beyond the Black-Scholes model, and moreover yields a lognormal distribution of the volatilities, in agreement with results from superstatistical analysis. The results obtained in the present formalism are checked with Monte Carlo simulations.