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A general class of truncated Lévy processes is introduced, and possible ways of fitting parameters of the constructed family of truncated Lévy processes to data are discussed. For a market of a riskless bond and a stock whose log-price follows a truncated Lévy process, TLP-analogs of the Black-Scholes equation, the Black-Scholes formula, the Dynkin derivative and the Leland's model are obtained, a locally risk-minimizing portfolio is constructed, and an optimal exercise price for a perpetual American put is computed.

We consider perpetual American options, assuming that under a chosen equivalent martingale measure the stock returns follow a Lévy process. For put and call options, their analogues for more general payoffs, and a wide class of Lévy processes that contains Brownian motion, normal inverse Gaussian processes, hyperbolic processes, truncated Lévy processes, and their mixtures, we obtain formulas for the optimal exercise price and the fair price of the option in terms of the factors in the Wiener-Hopf factorization formula, i.e., in terms of the resolvents of the supremum and infimum processes, and derive explicit formulas for these factors. For calls, puts, and some other options, the results are valid for any Lévy process.We use Dynkin's formula and the Wiener-Hopf factorization to find the explicit formula for the price of the option for any candidate for the exercise boundary, and by using this explicit representation, we select the optimal solution.We show that in some cases the principle of the smooth fit fails and suggest a generalization of this principle. AMS subject classifications. 60G40, 90A09, 93E20PII. S0363012900373987 1. Introduction. Consider the market of a riskless bond and a stock whose returns follow a Lévy process. If the Lévy process is neither a Brownian motion nor a Poisson process, then the market is incomplete. According to the modern martingale approach to option pricing [16], arbitrage-free prices can be obtained as expectations under any equivalent martingale measure (EMM), which is absolutely continuous w.r.t. the historic measure.Let the riskless rate r > 0 and the dividend rate λ ≥ 0 be fixed, let S = {S t } t≥0 , S t = exp X t , be the price process of the stock, and let Q be an EMM chosen by the market. Let {X t } be a Lévy process under Q, and (Ω, F, Q) the corresponding probability space. (For general definitions of the theory of Lévy processes, see, e.g., [32], [5], and [33].)Let g(X t ) be the payoff function for a perpetual American option on the stock (e.g., for a put, g(x) = K − e x , and for a call, g(x) = e x − K, where K is the strike price; for the formulation of our results, it is more convenient to use g(X t ) rather than max{g(X t ), 0}). Set q = r + λ, and denote by V * (x), where x = ln S, the rational price of the perpetual American option. It is given by

We derive explicit formulas for barrier options of European type and touch-and-out options assuming that under a chosen equivalent martingale measure the stock returns follow a Lévy process from a wide class, which contains Brownian motions (BM), normal inverse Gaussian processes (NIG), hyperbolic processes (HP), normal tilted stable Lévy processes (NTS Lévy), processes of the KoBoL family and any finite mixture of independent BM, NIG, HP, NTS Lévy and KoBoL processes. In contrast to the Gaussian case, for a barrier option, a rebate must be specified not only at the barrier but for all values of the stock on the other side of the barrier. We consider options with a constant or exponentially decaying rebate and options which pay a fixed rebate when the first barrier has been crossed but the second one has not. We obtain pricing formulas by solving boundary problems for the generalized Black-Scholes equation. We use the representation of the q-order harmonic measure of a set relative to a point in terms of the q-potential measure, the Wiener-Hopf factorization method and elements of the theory of pseudodifferential operators.

We construct fast and accurate methods for (a) approximate Laplace inversion, (b) approximate calculation of the Wiener-Hopf factors for wide classes of Lévy processes with exponentially decaying Lévy densities, and (c) approximate pricing of lookback options. In all cases, we use appropriate conformal change-of-variable techniques, which allow us to apply the simplified trapezoid rule with a small number of terms (the changes of variables in the outer and inner integrals and in the formulas for the Wiener-Hopf factors must be compatible in a certain sense). The efficiency of the method stems from the properties of functions analytic in a strip (these properties were explicitly used in finance by Feng and Linetsky 2008). The same technique is applicable to the calculation of the pdfs of supremum and infimum processes, and to the calculation of the prices and sensitivities of options with lookback and barrier features.