This Demonstration shows a path of the (extended) Meixner process with four parameters and a cross-sectional ("marginal") density function of the process at a chosen moment in time. The kurtosis and skewness of the density at the given time are also displayed. The Meixner process is a pure-jump Lévy process with semi-heavy tails, which has been used successfully for stock price modelling and valuing derivative instruments. The Demonstration makes use of Mathematica 8's ability to generate random variates when an explicit formula for the probability density function is given.

The Meixner process is a three-parameter pure jump Lévy process that was introduced in [1] and applied to finance in [2]. As with other similar processes, one can add a "drift" parameter, creating a four-parameter process particularly convenient for pricing derivative instruments. The process originated in the theory of orthogonal polynomials. It is a pure jump Lévy process (i.e. it has no continuous component) and was defined by explicitly giving its density function, which plays the central role in this Demonstration.

References

[1] W. Schoutens and J. L. Teugels, "Lévy Processes, Polynomials and Martingales," Communications in Statistics: Stochastic Models,14, 1998 pp. 335–349.

[2] W. Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, New York: John Wiley & Sons, 2003.