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An Asset Based Model of Defaultable Convertible Bonds with Endogenised Recovery

We describe a two factor valuation model for convertible bonds when the firm may default. The underlying state variables are the asset value of the firm and the short riskless interest rate. Default can occur exogenously, or endogenously at a time a cash payment is made by the bond. We endogenize the recovery value of a defaulted bond through assumptions concerning the character of the reorganization period following default. We use a tailored Lagrange-Galerkin discretization, coupled with a Lagrange multiplier method for free boundaries, to value convertibles in the model. Our framework enables us to specify numerically and financially consistent boundary conditions and inequality constraints. We investigate the affect of changing the default, recovery and loss specification. The affect of introducing a stochastic interest rate is quantified, and asset and interest rate delta and gammas are found. In our example we find that the value of the convertible bond is relatively insensitive to the initial asset value. Its sensitivity to interest rate changes is about one fifth less than that of a corresponding defaultable straight bond, chiefly due to the presence of the conversion feature.