In recent years, the Malliavin calculus has found many applications in stochastic control and finance. At the same time, L´evy processes became important in financial modelling. With this in mind, we saw a need for a course that deals with Malliavin calculus not only for Brownian motion but for L'evy processes in general, and presents some of the most important and newest applications for finance. The aim of this course is to try to fill this need. In this monograph, we present a general Malliavin account for L'evy processes, covering both the Brownian motion case and the pure jump martingale case via Poisson random measurements, as well as a combination of the two. We also offer many of the latest applications for finance.
Stochastic processes and filters, process classes, Markov processes, Martingales, finite and infinite variation processes, characteristic functions, stochastic integrals and stochastic differential equations (SDE), Equivalent martingale scale, pricing formulas for European options, Black-Scholes model flaws,
Levy processes: Definition and properties, Examples of Levy processes: Poisson process, Compound Poisson process, Gamma process, Inverse Gaussian process, Generalized Inverse Gaussian process, Variance-Gamma process, Normal Inverse Gaussian process, CGMY process, Meixner process, Generalized Hyperbolic process.
Asset pricing models driven by Levy processes: parameter estimation, Levy market model, pricing formulas for European options.
Levy models with stochastic volatility: BNS model, BNS model with Gamma stochastic volatility, simulation techniques for Levy models.