Dynkin Game of Convertible Bonds and Their Optimal Strategy

TL;DR

This paper models the valuation and optimal strategies of convertible bonds as a Dynkin game, analyzing the free boundary and revealing non-intuitive behaviors such as non-monotonic bond prices over time.

Contribution

It introduces a novel approach combining reflected backward stochastic differential equations and variational inequalities to analyze convertible bonds as Dynkin games, including boundary properties and irregular payoffs.

Findings

Call precedes conversion in certain conditions

Conversion boundary can be non-monotonic due to irregular payoffs

Convertible bond price may increase as maturity approaches

Abstract

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.