Pricing Convertible Bonds with Interest Rate, Equity, Credit and FX Risk

Abstract: Convertible bonds are hybrid securities whose pricing relies on a set of complex inter-dependencies due to the sensitivity to interest rate risk, underlying (equity) risk, FX risk, and credit risk, and due to the convertible bond's early exercise American feature. We present a two factor model of interest rate and equity risk that is implemented using the Crank-Nicholson technique on the discretized pricing equation with projective successive over-relaxation. This paper extends a methodology proposed in the literature (TF[98]) to deal with credit risk in a self-consistent way, and proposes a new methodology to deal with FX sensitive cross-currency convertibles. A technique for extracting the price of vanilla options struck on a synthetic asset, the foreign equity in domestic currency, is employed to obtain the implied volatility for these options. These implied volatilities are then used to obtain the local volatility for use in the numerical routine. The model is designed to deal with most of the usual contractual features such as coupons, dividends, continuous and/or Bermudan call and put clauses. We suggest that credit spread adjustments in the boundary conditions can be made, to account for the negative correlation between spreads and equity. Detailed description of the numerical methods and the discretization schemes, together with their accuracy, are provided.