Pricing Discretely Monitored Barrier Options and Defaultable Bonds in Lévy Process Models: A Fast Hilbert Transform Approach

Abstract

This paper presents a novel method to price discretely monitored single-and doublebarrier 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.

Chapter 8 Discrete Barrier and Lookback Options

“…under Black-Scholes model or Merton (1976) normal jump diffusion model). For example, as it is pointed out in Feng and Linetsky (2005) it may take 0.01 seconds for Broadie-Yamamoto to achieve accuracy of 10 −12 under the Black-Scholes model, while it may take 0.04 seconds for Feng-Linetsky method to achieve accuracy of 10 −8 . The beauty of Feng-Linetsky method is that it works for general Lévy processes with very reasonable computational time.…”

“…the fast Gaussian transform method developed in Broadie and Yamamoto (2003) and the Hilbert transform method in Feng and Linetsky (2005). This is basically due to the fact that the joint probability of the first passage time and the terminal value of a discrete random walk can be written as m-dimensional probability distribution (hence a m-dimensional integral or convolution.)…”

“…As shown in Feng and Linetsky (2005), a Hilbert transform can be computed efficiently by using approximation theory in Hardy spaces which leads to a simple trapezoidal-like quadrature sum.…”

“…Now the Fourier transform in the backward induction will involve Fourier transform of product of the indicator function and another function. The key observation in Feng and Linetsky (2005) is that Fourier transform of the product of the indicator function and a function can be written in terms of Hilbert transform. More precisely,…”

“…One is the fast Gaussian transform in Broadie and Yamamoto (2003) in which 352 S.G. Kou the convolution is computed very fast under the Gaussian assumption. The second method is the Hilbert transform method in Feng and Linetsky (2005), in which they recognize an interesting linking between Fourier transform of indicator functions and Hilbert transform. Feng and Linetsky method is more general, as it works as long as the asset returns follow a Lévy process.…”