Fast and Accurate Pricing of Barrier Options Under Levy Processes

Abstract

We suggest two new fast and accurate methods, Fast Wiener-Hopf method (FWH-method) and Iterative Wiener-Hopf method (IWH-method), for pricing barrier options for a wide class of Lévy processes. Both methods use the Wiener-Hopf factorization and Fast Fourier Transform algorithm. Using an accurate albeit relatively slow finite-difference algorithm developed in (FDS-method), we demonstrate the accuracy and fast convergence of the two methods for processes of finite variation. We explain that the convergence of the methods must be better for processes of infinite variation, and, as a certain supporting evidence, demonstrate with numerical examples that the results obtained by two methods are in extremely good agreement. Finally, we use FDS, FWH and IWH-methods to demonstrate that Cont and Volchkova method (CV-method), which is based on the approximation of small jumps by an additional diffusion, may lead to sizable relative errors, especially near the barrier and strike. The reason is that CV-method presumes that the option price is of class C 2 up to the barrier, whereas for processes without Gaussian component, it is typically not of class C 1 at the barrier. (2006) (FDS-method), on démontre la précision et la convergence rapide des deux méthodes pour des processusà variation finie. On explique que la convergence des méthodes doitêtre meilleure pour des processusà variation infinie et on montre sur des exemples numériques la cohérence des résultats obtenus par les deux méthodes. Finalement on utilise les méthodes FDS, FWH and IWHmethods pour montrer que la méthode de Cont and Volchkova (CV-method), qui est basée sur l'approximation des petits sauts par une diffusion additionnelle peut entrainer des erreurs relativement importantes notamment près de la barrière et du strike. Cela s'explique par le fait que cette méthode présuppose que le prix de l'option est de classe C 2 jusqu'à la barrière, alors que pour des processus sans composante gaussienne, il n'est en général pas de classe C 1à la barrière.

Prices of Barrier and First-Touch Digital Options in Lévy-Driven Models, Near Barrier

“…The authors are grateful to the participants of the Financial and Insurance Mathematics seminar at ETH Zürich (November 13, 2008) for the suggestion to calculate the asymptotics of the price of barrier options near the barrier, and to the participants of Mathematical Finance seminars at University of Edinburgh (October 4, 2009) and University of Chicago (November 6, 2009) and participants of the 6th Congress of the Bachelier Finance Society, Toronto (June [22][23][24][25][26][27] 2010) for useful discussions about the paper. The authors are especially grateful to the anonymous referee for valuable suggestions.…”

“…Other methods of pricing barrier options that use models with a tangible diffusion component to approximate models with zero diffusion component (such as the method of Cont and Voltchkova [18]) suffer from the same problem (see [25,27] for an analysis of the errors that result from applying these methods). This issue is important because empirical studies of financial markets (see, e.g., [16]) show that, typically, the dynamics of a stock has zero diffusion component.…”