Bilateral Gamma distributions and processes in financial mathematics

TL;DR

The paper introduces Bilateral Gamma processes as a four-parameter Lévy class formed by the difference of two Gamma subordinators, offering a flexible yet tractable model for financial fluctuations. It derives deep distributional properties, links to generalized hyperbolic, CGMY, and Variance Gamma families, and develops estimation, simulation, and measure-change tools. It applies these processes to exponential Lévy stock models and to Heath-Jarrow-Morton type term-structure models, producing closed-form option pricing and bond pricing formulas, and demonstrates calibration to real data (DAX 1996–1998). The results provide a practical and interpretable framework for fitting, pricing, and risk management in markets with jump-diffusion dynamics and finite-variation paths.

Abstract

We present a class of Lévy processes for modelling financial market fluctuations: Bilateral Gamma processes. Our starting point is to explore the properties of bilateral Gamma distributions, and then we turn to their associated Lévy processes. We treat exponential Lévy stock models with an underlying bilateral Gamma process as well as term structure models driven by bilateral Gamma processes and apply our results to a set of real financial data (DAX 1996-1998).

Figures (3)

• Figure 1: DAX, 1996-1998.
• Figure 2: Call Option prices $C_{\lambda}(5000,5000;t,t+100)$ for $\lambda \in (1,\infty)$.
• Figure 3: Empirical density and fitted bilateral Gamma density.

Theorems & Definitions (32)

• Lemma 2.1
• proof
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Proposition 6.1
• proof