Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

•We deal with the finite time distribution of the discrete extrema for a Lévy process.•We propose a procedure based on the combined use of Hilbert- and z-transforms.•Our results solve in an efficient and accurate way a long-standing problem.•Our results are relevant in many fields: Operational research, finance, insurance, engineering, physics.•As a concrete application, we contribute to the option pricing literature.

The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Lévy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Lévy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points.