Stochastic Volatility for Levy Processes
AbstractThree 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.
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Bibliographic InfoPaper provided by Paris Dauphine University in its series Economics Papers from University Paris Dauphine with number 123456789/1392.
Date of creation: Jul 2003
Date of revision:
Publication status: Published in Mathematical Finance, 2003, Vol. 13, no. 3. pp. 345-382.Length: 37 pages
Risque de marché; Gestion du risque; Volatilité (finances); Risk management; Volatility (finance); Stochastic processes; Processus stochastiques; Finances; Modèles mathématiques;
Find related papers by JEL classification:
- C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
- G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
- G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies; Insider Trading
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