Prices and Sensitivities of Barrier and First-Touch Digital Options in Levy-Driven Models

Abstract

We present a fast and accurate FFT-based method of computing the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a Levy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying Levy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short.

Keywords: Option pricing, greeks, barrier options, first-touch digitals, Levy processes, KoBoL processes, CGMY model, Normal Inverse Gaussian processes, Variance Gamma processes, Fast Fourier transform, Carr's randomization, Wiener-Hopf factorization

JEL Classification: C63, G13