Stochastic Volatility for Lévy Processes

TLDR

The study aims to formulate and investigate martingale marginals by constructing martingales in altered filtrations that preserve the one‑dimensional marginal distributions of the process at each future date. We develop three Lévy‑process volatility models driven by a mean‑reverting square‑root time change, extend the time change to non‑Gaussian Ornstein‑Uhlenbeck processes with stock correlation, exponentiate and mean‑correct to obtain positive stock prices, and derive option prices from their characteristic functions via fast Fourier transform. Mean‑corrected exponentiation yields better performance than stochastic exponentiation, yet the mean‑corrected exponential model fails to be a martingale in its original filtration.

Abstract

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.