The Multivariate Variance Gamma Model: Basket Option Pricing and Calibration

Abstract

A basket option is an option whose underlying is a portfolio of individual stock prices. Due to the unknown dependence structure between stocks, basket option pricing relies in general on approximations or numerical methods like Monte Carlo simulation. We propose a methodology for pricing basket options in a multivariate Variance Gamma model. The stock prices composing the basket are then modeled by time changed geometric Brownian motions with a common Gamma subordinator. Using the additivity property of comonotonic stop-loss premiums together with Gauss-Laguerre polynomials, we derive a closed-form expression for the basket option price as a linear combination of Black & Scholes prices. This technique manages to approximate the real basket option price in an accurate way. Furthermore, our new basket option pricing formula enables us to calibrate the multivariate VG model in a fast way provided option quotes on the components and the basket itself are available. As an illustration, we show that the multivariate VG model can closely match the observed Dow Jones index options.