A Lévy process for the GNIG probability law with 2nd order stochastic volatility and applications to option pricing

Abstract

Here we derive the Lévy characteristic triplet for the GNIG probability law. This characterizes the corresponding Lévy process. In addition we derive equivalent martingale measures with which to price simple put and call options. This is done under two different equivalent martingale measures. We also present a multivariate Lévy process where the marginal probability distribution follows a GNIG Lévy process. The main contribution is, however, a stochastic process which is characterized by autocorrelation in moments equal and higher than two, here a multivariate specification is provided as well. The main tool for achieving this is to add an integrated Feller square root process to the dynamics of the second moment in a time-deformed Browninan motion. Applications to option pricing are also considered, and a brief discussion is held on the topic of estimation of the suggested process.