Exchange option pricing under variance gamma-like models

TL;DR

This paper develops new models for pricing exchange options using variance gamma and variance gamma++ processes, providing explicit formulas and calibration methods applied to energy markets.

Contribution

It introduces a Margrabe's formula for variance gamma and an integral-free formula for variance gamma++, along with multidimensional extensions and practical calibration techniques.

Findings

Derived explicit pricing formulas for exchange options.

Successfully calibrated models to energy market data.

Validated models with Fourier and Monte Carlo methods.

Abstract

In this article we focus on the pricing of exchange options when the dynamic of logprices follows either the well-known variance gamma or the recent variance gamma++ process introduced in Gardini et al [19]. In particular, for the former model we can derive a Margrabe's type formula whereas, for the latter one we can write an "integral free" formula. Furthermore, we show how to construct a general multidimensional versions of the variance gamma++ processes preserving both the mathematical and numerical tractability. Finally we apply the derived models to German and French energy power markets: we calibrate their parameters using real market data and we accordingly evaluate exchange options with the derived closed formulas, Fourier based methods and Monte Carlo techniques.