Motivated by the pricing of lookback options in exponential Lévy models, we study the difference between the continuous and discrete supremums of Lévy processes. In particular, we extend the results of Broadie, Glasserman and Kou (1999) to jump diffusion models. We also derive bounds for general exponential Lévy models.

Abstract. Motivated by the pricing of lookback options in exponential Lévy models, we study the difference between the continuous and discrete supremum of Lévy processes. In particular, we extend the results of Broadie et al. (1999) to jump diffusion models. We also derive bounds for general exponential Lévy models.Key words. Exponential Lévy model, Lookback option, Continuity correction, Spitzer identity AMS subject classifications. 60G51, 60J75, 65N15, 91G20 JEL classification. C02, G131. Introduction. The payoff of a lookback option typically depends on the maximum or the minimum of the underlying stock price. The maximum can be evaluated in continuous or discrete time depending on the contract. In the Black-Scholes setting, Broadie, Glasserman and Kou (1999 and 1997) derived a number of results relating discrete and continuous path-dependent options. In particular, they obtained continuity correction formulas for lookback, barrier and hindsight options. The purpose of this paper is to establish similar results for exponential Lévy models. We will focus on lookback or hindsight options, leaving the treatment of barrier options to another paper.Our results are based on the analysis of the difference between the discrete and continuous maximum of a Lévy process. In the case of a Lévy process with finite activity and a non zero Brownian part, we extend (see Theorem 4.2) the theorem of Asmussen, Glynn and Pitman (1995) which is the key to the continuity correction formulas for lookback options in Broadie, Glasserman and Kou (1999). This allows us to extend these formulas to jump-diffusion models. We also establish estimates for the L 1 -norm of the difference of the continuous and discrete maximum of a general Lévy process. These estimates are based on Spitzer's identity, which relates the expectation of the supremum of sums of iid random variables to a weighted sum of the expectations of the positive parts of the partial sums. In the case of Lévy processes with finite activity, we derive an expansion up to te order o(1/n), where n is the number of dates in the discrete supremum, see Theorem 3.5. In the case of infinite activity, we have precise upper bounds (see Theorem 3.9). We also derive an expansion in the case of Lévy processes with finite variation (see Theorem 3.12).The paper is organized as follows. In the next section, we recall some basic facts about real Lévy processes. In section 3, we state Spitzer's identity for Lévy Processes and use it to analyse the expectation of the difference of the continuous and discrete maximum of a general Lévy process. Section 4 is devoted to the extension of the theorem of Asmussen et al. The last two sections are devoted to financial applications. In Section 5, we derive continuity corrections for lookback options in

The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.

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The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.