Barrier Option Pricing Under Meromorphic Levy Processes Observed at Poisson Arrival Times

Abstract

Calculating the expected value of options is a prominent task in financial mathematics literature. Exotic options, such as barrier options, provide a further specialisation that often requires vastly differing techniques. The first explicit continuous-time option pricing formula, the Black-Scholes model, assumed that the underlying security could be modelled using a stochastic process called Brownian motion. While this was reasonably effective, empirical research has shown that financial markets have path properties and distributions that cannot be achieved by Brownian motion. Accordingly, over the past two decades, the literature has investigated the use of other L´evy processes in option pricing models. In these models, the expected value of barrier options may be represented through a system of integro-differential equations, which can be solved numerically using the Wiener-Hopf factors of the process. However, many L´evy processes do not have known expressions for the Wiener-Hopf factors, so previous algorithms, such as the Simple Wiener-Hopf (SWH) method [33], have required methods of approximating these functions. Additional calculations and errors in these approximations reduce the accuracy and time efficiency of these algorithms. In this thesis, we investigate the family of meromorphic L´evy processes, which have known Wiener-Hopf factors and are more suitable than Brownian motion. We derive an algorithm to price barrier option when the model is driven by a meromorphic process, calling this new algorithm the Meromorphic Wiener-Hopf (MWH) method. By comparing the results with Monte Carlo simulations and the SWH method, we demonstrate that the MWH method provides an accurate estimate of barrier option prices. While there is currently no significant difference in computational speed between the MWH and SWH methods, we suggest potential means to optimise the MWH method. Regardless, the MWH method is a beneficial addition to the suite of option pricing models because it may be applied in different conditions than other algorithms. In particular, the MWH method allows for processes of either finite or infinite path variation, while the SWH requires finite variation.