Valuing Convertible Bonds with Stock Price, Volatility, Interest Rate, and Default Risk

This paper develops a computational framework to value convertible bonds in general multi-factor Markovian models with credit risk. We show that the convertible bond value function satisfies a variational inequality formulation of the stochastic game between the bondholder and the issuer. We approximate the variational inequality by a penalized nonlinear partial differential equation (PDE). We solve the penalized PDE formulation numerically by applying a finite element spatial discretization and an adaptive time integrator. To provide specific examples, we value and study convertible bonds in affine, as well as nonaffine, models with four risk factors, including stochastic interest rate, stock price, volatility, and default intensity