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Stochastic Volatility for Levy Processes

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  • Geman, Hélyette
  • Carr, Peter
  • Madan, Dilip B.
  • Yor, Marc
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    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.

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    Bibliographic Info

    Paper provided by Paris Dauphine University in its series Economics Papers from University Paris Dauphine with number 123456789/1392.

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    Date of creation: Jul 2003
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    Publication status: Published in Mathematical Finance, 2003, Vol. 13, no. 3. pp. 345-382.Length: 37 pages
    Handle: RePEc:dau:papers:123456789/1392

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    Related research

    Keywords: Risque de marché; Gestion du risque; Volatilité (finances); Risk management; Volatility (finance); Stochastic processes; Processus stochastiques; Finances; Modèles mathématiques;

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    1. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
    2. Jin-Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32.
    3. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-24, October.
    4. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    5. Frank Milne & Dilip Madan, 1991. "Option Pricing With V. G. Martingale Components," Working Papers 1159, Queen's University, Department of Economics.
    6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
    7. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
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