Abstract

Consider a small investor who holds a stock that is subject to default risk and seeks to identify the optimal time to sell the asset in the sense of minimizing the prophet's drawdown, which is the ratio of the ultimate maximum of the stock price at the time of default and the value of the stock price at the moment of sale. Assuming that default occurs at a constant rate and that at the moment of default there is a random recovery value, we solve this stochastic optimisation problem explicitly in the case the log-price of the stock prior to default is modelled by a general spectrally negative Levy process. Our results reveal a decomposition of the critical drift levels of the log-stock (at which the optimal strategy changes) into gap-risk, default-risk and volatility-risk components. Moreover, we provide an algorithm for the computation of the optimal exercise policy in terms of the Levy measure, volatility and drift parameters of the Levy process and apply this algorithm to a number of widely used models in the literature.

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