Optimal stopping problems for maxima and minima in models with asymmetric information

Abstract

We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American withdrawable standard and lookback put and call options in an extension of the Black-Merton-Scholes model with asymmetric information. It is assumed that the contracts are withdrawn by their writers at the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original necessarily two-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and normal-reflection conditions. We prove that the optimal exercise boundaries are the maximal and minimal solutions of some first-order nonlinear ordinary differential equations.

Keywords:

• Optimal stopping problem
• Brownian motion
• last hitting time
• asymmetric information
• first passage time
• running maximum and minimum processes
• stochastic boundary
• free-boundary problem
• instantaneous stopping and smooth fit
• normal reflection
• a change-of-variable formula with local time on surfaces
• perpetual American standard and lookback options

Acknowledgements

The authors are grateful to the Editor and two anonymous Referees for their valuable suggestions which helped to improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).