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Dynkin game of convertible bonds and their optimal strategy. (English) Zbl 1305.91060

Summary: This paper studies the valuation and optimal strategy of convertible bonds as a Dynkin game by using the reflected backward stochastic differential equation method and the variational inequality method. We first reduce such a Dynkin game to an optimal stopping time problem with state constraint, and then in a Markovian setting, we investigate the optimal strategy by analyzing the properties of the corresponding free boundary, including its position, asymptotics, monotonicity and regularity. We identify situations when call precedes conversion, and vice versa. Moreover, we show that the irregular payoff results in the possibly non-monotonic conversion boundary. Surprisingly, the price of the convertible bond is not necessarily monotonic in time: it may even increase when time approaches maturity.

MSC:

91A80 Applications of game theory
91A15 Stochastic games, stochastic differential games
91A60 Probabilistic games; gambling
60G40 Stopping times; optimal stopping problems; gambling theory
91G20 Derivative securities (option pricing, hedging, etc.)
91G80 Financial applications of other theories
60H30 Applications of stochastic analysis (to PDEs, etc.)
49N90 Applications of optimal control and differential games
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References:

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