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PRICING DISCRETELY MONITORED BARRIER OPTIONS AND DEFAULTABLE BONDS IN LÉVY PROCESS MODELS: A FAST HILBERT TRANSFORM APPROACH
Author(s) -
Feng Liming,
Linetsky Vadim
Publication year - 2008
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/j.1467-9965.2008.00338.x
Subject(s) - sinc function , mathematics , exponential function , double exponential function , toeplitz matrix , hilbert transform , matrix multiplication , fourier transform , fast fourier transform , matrix (chemical analysis) , algorithm , mathematical analysis , pure mathematics , physics , statistics , spectral density , materials science , quantum mechanics , composite material , quantum
This paper presents a novel method to price discretely monitored single‐ and double‐barrier options in Lévy process‐based models. The method involves a sequential evaluation of Hilbert transforms of the product of the Fourier transform of the value function at the previous barrier monitoring date and the characteristic function of the (Esscher transformed) Lévy process. A discrete approximation with exponentially decaying errors is developed based on the Whittaker cardinal series (Sinc expansion) in Hardy spaces of functions analytic in a strip. An efficient computational algorithm is developed based on the fast Hilbert transform that, in turn, relies on the FFT‐based Toeplitz matrix–vector multiplication. Our method also provides a natural framework for credit risk applications, where the firm value follows an exponential Lévy process and default occurs at the first time the firm value is below the default barrier on one of a discrete set of monitoring dates.