TAILIEUCHUNG - Perpetual convertible bonds with credit risk

A convertible bond is a security that the holder can convert into a specified number of underlying shares. We enrich the standard model by introducing some default risk of the issuer. Once default has occured payments stop immediately. In the context of a reduced form model with infinite time horizon driven by a Brownian motion, analytical formulae for the no-arbitrage price of this American contingent claim are obtained and characterized in terms of solutions of free boundary problems. It turns out that the default risk changes the structure of the optimal stopping strategy essentially. Especially, the continuation region may become a disconnected subset of the state space | Perpetual convertible bonds with credit risk Christoph Kuhn Kees van Schaik Abstract A convertible bond is a security that the holder can convert into a specified number of underlying shares. We enrich the standard model by introducing some default risk of the issuer. Once default has occured payments stop immediately. In the context of a reduced form model with infinite time horizon driven by a Brownian motion analytical formulae for the no-arbitrage price of this American contingent claim are obtained and characterized in terms of solutions of free boundary problems. It turns out that the default risk changes the structure of the optimal stopping strategy essentially. Especially the continuation region may become a disconnected subset of the state space. Keywords convertible bonds exchangeable bonds default risk optimal stopping problems free-boundary problems smooth fit. Mathematics Subject Classification 2000 60G40 60J50 60G44 91B28. 1 Introduction The market for convertible bonds has been growing rapidly during the last years and the corresponding optimal stopping problems have attracted much attention in the literature on mathematical finance. One has to distinguish between reduced form models where the stock price process of the issuing firm is exogenously given by some stochastic process and structural models where the starting point is the firm value which splits in the total equity value and the total debt value. Within a firm value model the pricing problem is treated in Sirbu Pikovsky and Shreve 15 and Sirbu and Shreve 16 . In contrast to earlier articles of Brennan and Schwartz 4 and Ingersoll 11 12 15 16 includes the case where an earlier conversion of the bond can be optimal that necessitates to address a nontrivial free-boundary problem. In the context of a reduced form model Bielecki Crepey Jeanblanc and Rutkowski 2 made quite recently a comprehensive analysis of interesting features of convertible bonds. Especially they model the interplay between

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