Abstract. A finite-difference method for integro-differential equations arising from Lévy driven asset processes in finance is discussed. The equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula. The discretization is shown to be second-order accurate independently of the degree of the singularity in the Lévy measure. The singularity is dealt with by means of an integration by parts technique. An application of the fast Fourier transform gives the overall amount of work O(M N log N ), rendering the method fast.Key words. Partial integro-differential equations, collocation method, option pricing.AMS subject classifications. 45K05, 45D05, 65R20, 91B281. Introduction. In a seminal paper from 1973, Fischer Black and Myron Scholes [8] derived a partial differential equation for option prices, when asset prices behave according to the Geometric Brownian Motion. The pricing formulas obtained in this paper represented a major breakthrough in understanding financial derivatives, to such an extent that financial institutions and traders immediately adopted the new methodology.Later empirical studies revealed that the normality of the log-returns, as assumed by Black and Scholes, could not capture features like heavy tails and asymmetries observed in market-data log-returns densities [13]. The Black-Scholes model assumes in addition constant parameters, which contradicts the existence of the socalled volatility smile: A numerical inversion of the Black and Scholes formula based on data from different strikes and fixed maturity resembles a skew or a smile. This inconsistency is said to be one of the causes for famous market crashes.To explain these empirical observations, a number of alternate models have appeared in the financial literature: Stochastic volatility [24,26]; deterministic local volatility [18,21]; jump-diffusion [27,31]; infinite activity Lévy models [7,16,22]. Each of these models has its advantages and disadvantages. Jump-diffusion and infinite activity models are attractive since they can capture the jump patterns exhibited by some stocks and they are more realistic when pricing options close to maturity [19]. Processes with infinite activity, without a diffusion component represent a family that describes the high activity of the prices, while at the same time they reflect the empirical features, desirable in a good model. The pricing equations are, however, numerically more challenging, and the market turns out to be incomplete in the sense that a hedging strategy leading to instantaneous risk free portfolio does not exist in general [17,32].Due to the close link between the martingale approach and the PDE approach [23,28] the field of computational finance has gained a tremendous impulse from well-established numerical techniques for PDEs. For an overview we refer to the introductory book [34]. The development of analogous reliable techniques under Lévy markets is a subject of present research [5,6,20] for jump-diffusions and [3, 4, 17...