The Valuation of Convertible Bonds With Credit Risk

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Convertible bonds are typically issued by firms which have both relatively high growth and quite high risk. Convertibles can be difficult to value, given their hybrid nature of containing elements of both debt and equity. Further complications arise due to the frequent presence of additional options such as callability and puttability, and contractual complexities such as trigger prices and "soft call" provisions, in which the ability of the issuing firm to exercise its option to call is dependent upon the history of its stock price. This paper explores the valuation of convertible bonds subject to credit risk using an approach based on the numerical solution of a system of coupled linear complementarity problems. We argue that many of the existing modes, such as that of Tsiveriotis and Fernandes (1998), are unsatisfactory in that they do not explicitly specify what happens in the event of a default by the issuing firm. In fact, many of the differences between existing models appear to arise from varying implicit assumptions about this. In existing models it is assumed that upon a default either nothing happens to the firm's stock price or else it instantly jumps to zero. Neither of these alternatives seems to be entirely appealing: while it is a significant event, implying that there will be some market reaction to the news of a default, a sudden and complete collapse is rare. Consequently, we propose a model where the firm's stock price falls by some specified percentage between 0% and 100% (which includes the limiting cases implicit in existing models). We also present a detailed description of our numerical algorithm, which uses a partially implicit method to decouple the system of linear complementarity problems at each timestep. Numerical examples illustrating the convergence properties of the algorithm are provided.
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