Lévy processes and option pricing by recursive quadrature

In this paper we present a quadrature method for pricing discretely monitored path-dependent options when the dynamics of the underlying assets are described by L´evy processes. The literature mainly deals with the continuous monitoring case and assumes that the log-returns are normally distributed. The contribution of the present paper consists in providing a flexible computational method that allows to deal with the discrete monitoring rule that is common for exotic options and general enough to deal with non-Gaussian models. Since it is well known that the convergence of the discrete monitoring price to its continuous monitoring counterpart is very slow (Broadie et al. (1997, 1999)), fast and accurate pricing methods for the discrete monitoring case become particularly important. Also, effective alternatives to the Black-Scholes world, such as L´evy processes, are becoming increasingly popular, Cont and Tankov ([8]). The hybrid computational method proposed combines numerical inversion of the characteristic function and Gaussian quadrature extending the approach proposed by Tse et al. (2001) and Sullivan (2000) to the case of exponential L´evy processes. The higher accuracy of Gaussian quadrature provides accurate results up to a very large number of monitoring dates, much larger than the number of monitoring dates usually considered in the existing literature. We present an error analysis for the numerical method proposed and we provide detailed numerical examples for plain vanilla options, discrete barrier options (with a single or a double barrier), and Bermudian options for a variety of L´evy processes, such as the Gaussian, CGMY, Kou Double-Exponential, Merton Jump-Diffusion.