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.

摘要

我们在 Black-Merton-Scholes 模型的扩展中研究了与永久可转换债券相关的零和最优停止博弈，在各种信息流下具有随机股息。在这种类型的合同中，出票人有权在持有人将其转换为资产之前撤回债券。我们推导出模型中相关价值函数和最优行使边界的闭式表达式，该模型具有可访问的股息率政策，该政策由具有两个状态的连续时间马尔可夫链描述。我们进一步考虑了模型中具有不可访问的股息率策略的最优停止博弈，并证明了最优行使时间是资产价格过程第一次达到单调边界的时间，这取决于过滤股息率估计的运行状态。