Modeling the Short Rate as a Levy Process and Option Pricing with the FFT

Abstract

This paper describes a practical algorithm for modeling interest rate derivatives with the short rate following a Levy process using the fast Fourier transform algorithm (FFT). It can be used with any Levy process for which we have a closed form formula for the characteristic function, this includes a large variety of ‘fat-tailed’ jump diffusion processes. This model allows for the computation of forward rates and can be used to price American and Bermudan exercise options. Pricing algorithms are provided for option bonds and swaptions. The model is effectively equivalent to a tree approach except that diffusion is done by FFT instead of by branching. Pricing algorithms using trees or finite difference method can be easily adapted. Under the Normal distribution it can replicate the single factor Hull-White and Black-Karasinski models. The model supports mean reversion of interest rates. Monte Carlo simulations can be efficiently performed as well.