Stochastic Volatility for Levy Processes

35 Pages Posted: 4 Jun 2002

See all articles by Peter Carr

Peter Carr

New York University Finance and Risk Engineering

Hélyette Geman

University of London - Economics, Mathematics and Statistics

Dilip B. Madan

University of Maryland - Robert H. Smith School of Business

Marc Yor

Universite Paris

Date Written: April 2001

Abstract

Three processes reflecting persistence of volatility are formulated by evaluating three Levy processes at a time change given by the integral of a square root process. A positive stock price process is then obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating the processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. Our empirical results on index options and single name options suggest advantages to employing higher dimensional Levy systems for index options and lower dimensional structures for single names. In general, mean corrected exponentiation performs better than employing the stochastic exponential. Martingale laws for the mean corrected exponential are also studied and two new concepts termed Levy and martingale marginals are introduced.

Suggested Citation

Carr, Peter P. and Geman, Helyette and Madan, Dilip B. and Yor, Marc, Stochastic Volatility for Levy Processes (April 2001). EFA 2002 Berlin Meetings Presented Paper. Available at SSRN: https://ssrn.com/abstract=314979 or http://dx.doi.org/10.2139/ssrn.314979

Peter P. Carr (Contact Author)

New York University Finance and Risk Engineering ( email )

6 MetroTech Center
Brooklyn, NY 11201
United States
9176217733 (Phone)

HOME PAGE: http://engineering.nyu.edu/people/peter-paul-carr

Helyette Geman

University of London - Economics, Mathematics and Statistics ( email )

Malet Street
London, WC1E 7HX
United Kingdom

Dilip B. Madan

University of Maryland - Robert H. Smith School of Business ( email )

College Park, MD 20742-1815
United States
301-405-2127 (Phone)
301-314-9157 (Fax)

Marc Yor

Universite Paris ( email )

223 Rue Saint-Honore
Paris, 75775
France

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