Multivariate Subordination using Generalised Gamma Convolutions with Applications to V.G. Processes and Option Pricing

TL;DR

This paper develops a broad class of multivariate Variance-Gamma models for option pricing by subordinating multivariate Brownian motion to generalized Gamma convolution subordinators, enabling more flexible dependency structures and infinite variation processes.

Contribution

It introduces a wide-ranging multivariate V.G. model framework based on Thorin class subordinators, extending previous models and providing explicit formulas for applications in option pricing.

Findings

Models are invariant under Esscher transforms.

Explicit expressions for measures and densities are derived.

Applications demonstrated on multi-asset European and American options.

Abstract

We unify and extend a number of approaches related to constructing multivariate Variance-Gamma (V.G.) models for option pricing. An overarching model is derived by subordinating multivariate Brownian motion to a subordinator from the Thorin (1977) class of generalised Gamma convolution subordinators. A class of models due to Grigelionis (2007), which contains the well-known Madan-Seneta V.G. model, is of this type, but our multivariate generalization is considerably wider, allowing in particular for processes with infinite variation and a variety of dependencies between the underlying processes. Multivariate classes developed by P\'erez-Abreu and Stelzer (2012) and Semeraro (2008) and Guillaume (2013) are also submodels. The new models are shown to be invariant under Esscher transforms, and quite explicit expressions for canonical measures (and transition densities in some cases) are obtained, which permit applications such as option pricing using PIDEs or tree based methodologies. We illustrate with best-of and worst-of European and American options on two assets.