Fast and accurate pricing of barrier options under Lévy 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.

A Radial Basis Function Scheme for Option Pricing in Exponential Lévy Models

“…The contribution of the small (or smallest) jumps is sometimes approximated by an effective diffusion terms, cf. (Cont and Voltchkova, 2005;Wang et al, 2007), although this procedure was criticised in (Levendorskiȋ, 2004) and (Kudryavtsev and Levendorskiȋ, 2009), where it was argued that it can lead to sizeable numerical errors. Most of the cited FD schemes use implicit time-stepping for the local part and explicit time-stepping for the nonlocal term, with the exception of (Wang et al, 2007) who use a fully impicit scheme.…”

On the Monitoring Error of the Supremum of a Normal Jump Diffusion Process

“…They naturally incorporate extreme price movements due to jumps that are difficult to capture by Brownian motion which is underlying the celebrated Black-Scholes-Merton model [3], [20]. The extremum of such a process is of interest when derivatives with barrier and lookback features are evaluated [10], [11], [15], [16], [17], [18], [19], [22].…”

Pricing discrete barrier options and credit default swaps under Lévy processes

“…Continuously monitored barrier options with one or two barriers under Lévy models can be priced very efficiently using the BBL method, introduced in Levendorskiȋ (2002, 2009a) and , or a closely related approach developed by Kudryavtsev and Levendorskiȋ (2009). For the most difficult case of pure jump processes, the BBL method is extremely accurate even near the barrier, unlike other methods, such as jump-diffusion approximations, cf.…”