Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options

Abstract

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 * Corresponding author Email addresses: gianluca.fusai@unipmn.it, gianluca.fusai.1@city.ac.uk (Gianluca Fusai), g.germano@ucl.ac.uk, g.germano@lse.ac.uk (Guido Germano), daniele.marazzina@polimi.it (Daniele Marazzina) Preprint submitted to ElsevierNovember 27, 2015the error decays exponentially with the number of grid points.

Pricing methods for α-quantile and perpetual early exercise options based on Spitzer identities

“…In equtions (1) and (2) ξ is introduced as the Fourier variable conjugated to x and thus is a real number. However, later ξ will be extended to the complex plane for use with Wiener-Hopf methods as in the barrier option pricing work of Fusai et al (2016). Let S(t) be the price of an underlying asset at time t ≥ 0 and x(t) = log(S(t)/S 0 ) be its log-price, which we model with a Lévy process.…”

“…They have a fixed expiry; the payoff at expiry depends on the path of the underlying up to that date. Two such examples are lookback options, priced by Fusai et al (2016), and quantile options. Fixed-strike lookback options have a payoff similar to European options except that, instead of being a function of the underlying asset price at expiry, it uses its maximum or minimum over the monitoring period.…”

“…This approach was extended to general Lévy processes by Green (2009), who developed direct methods based on his formulation of the Spitzer identities for calculating the distribution of the supremum and infimum of processes. Later developments include the numerical refinements carried out by Haslip and Kaishev (2014) and the implementation of an option pricing method for lookback options with Lévy processes in Fusai et al (2016), where the error converges exponentially with the size of the log-price grid and the CPU time is independent of the number of monitoring dates. In this paper we address the gap in the literature for a general method for pricing α-quantile options with general Lévy processes and a date-independent computational time.…”

On the calibration of the 3/2 model

“…The FFT method has proven useful for a variety of interesting applications in quantitative finance, see e.g. (Albanese et al, 2004;Chen et al, 2014;Fusai et al, 2016). However, a major drawback of the method in our setting is that a very large number of integration nodes are required to obtain a satisfactory level of accuracy, since the integration nodes need to be equally spaced and that prohibits the use of genuinely efficient integration schemes.…”

Semi-Analytical method for the pricing of barrier options in case of time-dependent parameters (with Matlab® codes)

“…The hit of the barrier can be either discretely or continuously monitored. In the first case the benchmark methods are FFT based methods such as [6][7][8]. Otherwise, for continuously monitored"barrier options", pricing is traditionally based on Monte Carlo methods or on domain methods (such as Finite Element Methods and Finite Difference methods) but Monte Carlo methods are affected by high computational costs and inaccuracy due to their slow convergence and domain methods have some troubles particularly in unbounded domains.…”