Perpetual American Options Under Lévy Processes

Abstract

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

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