Some pricing tools for the Variance Gamma model

TL;DR

This paper derives fast-converging closed-form pricing formulas for various options under the Variance Gamma model using Mellin transform and complex analysis, improving numerical efficiency especially for short-term options.

Contribution

It introduces new series-based pricing formulas for the Variance Gamma model, enhancing computational speed and accuracy over existing numerical methods.

Findings

Series formulas converge rapidly for short-term options

Comparison shows improved efficiency over Fourier and Monte Carlo methods

Extension to asymmetric Variance Gamma processes is provided

Abstract

We establish several closed pricing formula for various path-independent payoffs, under an exponential L\'evy model driven by the Variance Gamma process. These formulas take the form of quickly convergent series and are obtained via tools from Mellin transform theory as well as from multidimensional complex analysis. Particular focus is made on the symmetric process, but extension to the asymmetric process is also provided. Speed of convergence and comparison with numerical methods (Fourier transform, quadrature approximations, Monte Carlo simulations) are also discussed; notable feature is the accelerated convergence of the series for short term options, which constitutes an interesting improvement of numerical Fourier inversion techniques.