Computing Exponential Moments of the Discrete Maximum of a Levy Process and Lookback Options

27 Pages Posted: 29 Aug 2008

See all articles by Liming Feng

Liming Feng

University of Illinois at Urbana-Champaign - Department of Industrial and Enterprise Systems Engineering

Vadim Linetsky

Northwestern University - Department of Industrial Engineering and Management Sciences

Date Written: May 9, 2008

Abstract

We present a fast and accurate method to compute exponential moments of the discretely observed maximum of a Levy process. The method involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the (Esscher transformed) Levy process. It can be discretized with exponentially decaying errors of the form O(exp(-aMb)) for some a, b > 0, where M is the number of discrete points used to compute the Hilbert transform. The discrete approximation can be efficiently implemented using the Toeplitz matrix-vector multiplication algorithm based on the fast Fourier transform, with total computational cost of O(NM log(M)), where N is the number of observations of the maximum. The method is applied to the valuation of European style discretely monitored floating strike, fixed strike, forward start and partial lookback options (both newly written and seasoned) in exponential Levy models.

Keywords: Levy processes, discrete maximum, exponential moments, Esscher transform, discrete lookback options, Fourier transform, Hilbert transform, Sinc expansion

JEL Classification: G13

Suggested Citation

Feng, Liming and Linetsky, Vadim, Computing Exponential Moments of the Discrete Maximum of a Levy Process and Lookback Options (May 9, 2008). Finance and Stochastics, Forthcoming, Available at SSRN: https://ssrn.com/abstract=1260934

Liming Feng (Contact Author)

University of Illinois at Urbana-Champaign - Department of Industrial and Enterprise Systems Engineering ( email )

104 S. Mathews Avenue
Urbana, IL 61801
United States

Vadim Linetsky

Northwestern University - Department of Industrial Engineering and Management Sciences ( email )

Evanston, IL 60208-3119
United States

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