On the Pricing of Perpetual American Compound Options

Abstract

We present explicit solutions to the perpetual American compound option pricing problems in the Black-Merton-Scholes model. The method of proof is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. The latter are solved through their associated one-sided free-boundary problems and the subsequent martingale verication. We also obtain a closed form solution to the perpetual American chooser option pricing problem, by means of the analysis of the equivalent two-sided free-boundary problem.

Keywords: Perpetual American compound options, the Black-Merton-Scholes model, geometric Brownian motion, multi-step optimal stopping problem, first hitting time, free-boundary problem, local time-space formula

JEL Classification: G13