We thank George Panayotov for assistance with the computations reported in this paper. Dilip Madan thanks Ajay Khanna for important discussions and perspectives on the problems studied here. Errors are our own responsibility.
Stochastic Volatility for Lévy Processes
Article first published online: 29 MAY 2003
Volume 13, Issue 3, pages 345–382, July 2003
How to Cite
Carr, P., Geman, H., Madan, . D. B. and Yor, M. (2003), Stochastic Volatility for Lévy Processes. Mathematical Finance, 13: 345–382. doi: 10.1111/1467-9965.00020
Manuscript received October 2001; final revision received June 2002.
- Issue published online: 29 MAY 2003
- Article first published online: 29 MAY 2003
- Accepted 2001 October 12. Received 2001 October 11; in original form 2001 March 21
- variance gamma;
- static arbitrage;
- stochastic exponential;
- OU equation;
- Lévy marginal;
- martingale marginal
Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.