Post by Sapphire Capital on Aug 6, 2008 22:08:04 GMT 4
Pricing Discretely Monitored Barrier Options and Defaultable Bonds in Lévy Process Models: A Fast Hilbert Transform Approach
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
Mathematical Finance, Vol. 18, Issue 3, pp. 337-384, July 2008
Abstract:
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 matrixvector 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.
papers.ssrn.com/sol3/papers.cfm?abstract_id=1144911
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
Mathematical Finance, Vol. 18, Issue 3, pp. 337-384, July 2008
Abstract:
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 matrixvector 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.
papers.ssrn.com/sol3/papers.cfm?abstract_id=1144911