Exchange Option Pricing Under Variance Gamma-Like Models

In this article, we focus on the pricing of exchange options when the risk-neutral dynamic of log-prices follows either the well-known variance gamma or the recent variance gamma++ process introduced in Gardini et al. (2022. “The Variance Gamma++ Process and Applications to Energy Markets.” Applied Stochastic Models in Business and Industry 38 (2): 391–418. https://doi.org/10.1002/asmb.v38.2.). 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 tractabilities. 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.