DMCA
Stochastic Volatility for Lévy Processes (2001)
Cached
Download Links
| Citations: | 209 - 12 self |
BibTeX
@MISC{Carr01stochasticvolatility,
author = {Peter Carr and Dilip B. Madan and Marc Yor},
title = {Stochastic Volatility for Lévy Processes},
year = {2001}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
Three processes re°ecting persistence of volatility are initially formulated by evaluating three L¶evy 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 OU (Ornstein-Uhlenbeck) equations driven by one sided discontinuous L¶evy 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 ¯ltration in which it is originally de¯ned. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered ¯ltrations consistent with the one dimensional marginal distributions of the level of the process at each future date. 1
Keyphrases
stochastic volatility vy process martingale marginals exponentiation performs discontinuous evy log price dimensional marginal distribution non-gaussian model future date fast fourier transform mean reverting time change positive stock price process option price characteristic function mean reverting square root process important property stochastic exponential altered ltrations time change exponential model
