Perpetual Options for Lévy Processes in the Bachelier Model

Author: E. Mordecki.


Posted: 30/10/2000, Status: Submitted.

Abstract: Solution to the optimal stopping problem

V(x)=sup{E[exp(-dT)g(x+X(T))]; T is a stopping time}

is given, where X is a Lévy process, d, a constant greather or equal to 0 is a discount rate, and the reward function g takes the form gc(x)=max(x-K , 0) or gp(x)=max(K-x , 0). Results, interpreted as option prices of perpetual options in Bachelier's model are expressed in terms of the distribution of the overall supremum in case g=gc and overall infimum in case g=gp of the process X killed at rate d. Closed form solutions are obtained under mixed exponentially distributed positive jumps with arbitrary negative jumps for gc, and under arbitrary positive jumps and mixed exponentially distributed negative jumps for gp. In case g=gc a prophet inequality comparing prices of perpetual look-back call options and perpetual call options is obtained.


Keywords: Optimal stopping, Lévy process, mixture of exponential distributions, Perpetual American options, prophet inequalities, Classification: MSC 60G40, 60J30, 62L15, 90A09

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