The Meixner Process: Theory and Applications in Finance

The Meixner process is a special type of Levy process which origi-nates from the theory of orthogonal polynomials. It is related to the Meixner-Pollaczek polynomials by a martingale relation. We discuss sev-eral properties of the Meixner process. We apply the Meixner process to nancial data. First, we shows that the Normal distribution is a very poor model to t log-returns of nancial assets like stocks or indices. In order to achieve a better t we replace the Normal distribution by the more sophisticated Meixner distribution, taking into account, skewness and excess kurtosis. We show that the underlying Meixner distribution allows a much better t to the data by performing a number of statistical tests. Secondly, we introduce stock price models based on the Meixner process in order to price nancial derivatives. A rst signicant improvement can be achieved with respect to the famous Black-Scholes model (BS-model) by replacing its Brownian motion by the more exible Meixner process. However, there still is a discrepancy between market and theoretical prices. The main feature which these Levy models are missing is the fact that volatility or more generally the environment is changing stochastically over time. By making business time stochastic, an idea which was developed in [9], one can incorporate these stochastic volatility eects. The resulting option prices can be calibrated almost perfectly to empirical prices. 2