Optimal stopping games in models with various information flows

Abstract

We study zero-sum optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under various information flows. In this type of contracts, the writers have the right to withdraw the bonds, before the holders convert them into assets. We derive closed-form expressions for the associated value function and optimal exercise boundaries in the model with an accessible dividend rate policy which is described by a continuous-time Markov chain with two states. We further consider the optimal stopping game in the model with inaccessible dividend rate policy and prove that the optimal exercise times are the first times at which the asset price process hits monotone boundaries depending on the running state of the filtering dividend rate estimate. We finally present the value of the optimal stopping game for the model in which the dividend rate policy is accessible to the writers but remains inaccessible to the holders of the bonds.

Keywords:

• Optimal stopping game
• full and partial information
• continuous-time Markov chain
• filtering estimate (Wonham filter)
• perpetual convertible bond
• stochastic dividend rate
• free-boundary problem
• change-of-variable formula with local time on surfaces

MATHEMATICS SUBJECT CLASSIFICATION 2010:

• Primary 60G40
• 91G20
• 34K10. Secondary 60J60
• 60J27
• 62M20

Acknowledgments

The authors are indebted to the Editor for providing the kind opportunity to revise the paper and the Associate Editor and an anonymous Referee for their suggestions which helped to essentially improve the presentation of the paper. The first author is also grateful to Goran Peskir for a fruitful discussion at the Isaac Newton Institute in Cambridge and to Mihail Zervos for a fruitful discussion at the London School of Economics and Political Science.