Optimal Stopping and Perpetual Options for Lévy processes.

Author: E. Mordecki.

Posted: 30/5/2000. Final version 17/12/2001. Finance and Stochastics. Volume VI (2002) 4, 473-493

Abstract: Consider a model of a financial market with a stock driven by a Lévy process and constant interest rate. A closed formula for prices of perpetual American call options in terms of the overall supremum of the Lévy process, and a corresponding closed formula for perpetual American put options involving the infimum of the after-mentioned process are obtained. As a direct application of the previous results, a Black-Scholes type formula is given. Also as a consequence, simple explicit formulas for prices of call options are obtained for a Lévy process with positive mixed-exponential and arbitrary negative jumps. In the case of put options, similar simple formulas are obtained under the condition of negative mixed-exponential and arbitrary positive jumps. Risk-neutral valuation is discussed and a simple jump-diffusion model is chosen to illustrate the results.

Keywords: Optimal stopping, Lévy process, mixtures of exponential distributions, American options, Derivative pricing. Classification: JEL index G12, MSC 60G40.

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