Stochastic Volatility for L¶evy Processes

Three processes re°ecting persistence of volatility are formulated by evaluating three L¶evy processes at a time change given by the integral of a square root process. A positive stock price process is then obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating the processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. Our empirical results on index options and single name options suggest advantages to employing higher dimensional L¶evy systems for index options and lower dimensional structures for single names. In general, mean corrected exponentiation performs better than employing the stochastic exponential. Martingale laws for the mean corrected exponential are also studied and two new concepts termed L¶evy and martingale marginals are introduced. ¤We would like to thank George Panayotov for assistance with the computations reported in this paper. Dilip Madan would like to thank Ajay Khanna for important discussions and perspectives on the problems studied here. Errors are our own responsibility.