On the Pricing of Perpetual American Compound Options
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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 verification. 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.
KeywordsPerpetual 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
Mathematics Subject Classification (2010)91B28 60G40 34K10
The authors are grateful to Mihail Zervos for many useful discussions. The authors thank the Editor and two anonymous Referees for their careful reading of the manuscript and helpful suggestions. The second author gratefully acknowledges the scholarship of the Alexander Onassis Public Benefit Foundation for his doctoral studies at the London School of Economics and Political Science.
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