Stochastic volatility for Lévy processes

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

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)345-382
Number of pages38
JournalMathematical Finance
Volume13
Issue number3
DOIs
StatePublished - Jul 2003

Keywords

  • Leverage
  • Lévy marginal
  • Martingale marginal
  • OU equation
  • Static arbitrage
  • Stochastic exponential
  • Variance gamma

ASJC Scopus subject areas

  • Accounting
  • Finance
  • Social Sciences (miscellaneous)
  • Economics and Econometrics
  • Applied Mathematics

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