Robust and Nearly Exact Option Pricing with Bilateral Gamma Processes

Abstract

Bilateral gamma processes generalize the variance gamma process and allow one to capture, more precisely, the differences between upward and downward moves of financial returns, notably in terms of jump speed, frequency, and size. Like in most other pure jump models, option pricing under bilateral gamma processes relies heavily on numerical evaluation of Fourier integrals. In this article, we combine the Mellin transform and residue calculus to establish closed-form pricing formulas for several vanilla and exotic European options. These formulas take the form of series whose terms are straightforward to evaluate in practice and achieve an arbitrary degree of precision, without requiring sophisticated numerical tools; moreover, the convergence of the series is particularly accelerated for short maturity options, which are the most challenging to price for competing Fourier methods. Accuracy of the formulas is assessed thanks to several comparisons with state-of-the-art Fourier methods, with reference prices provided for future research.