Stochastic Volatility for Lévy Processes

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…