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

Abstract

Pricing convertible bonds poses numerical challenges that are not easily overcome. We present a quasi five-factor model with interest rate, equity, credit, currency, and local volatility risk. This is implemented using unconditionally stable PDE methods. We extend a method to address credit risk, and propose a means to deal with cross-currency convertibles. A procedure for extracting the price of vanilla options struck on foreign stock in domestic currency is employed to obtain local volatility. This is useful for pricing primary issue and secondary market convertibles whose maturities may be spanned by the currency and equity options markets. We facilitate numerical convergence with Wolfe's (1959) quadratic minimization algorithm, which assists in smoothening local volatility. Coupons, dividends, calls/puts, and reset clauses are easily accommodated. We allow a functional relationship between stock price and credit spread, to capture the negative correlation between spreads and equity.

Keywords: cross-currency convertibles, credit spread, interest rate risk, American feature, local volatility, Crank-Nicolson, quadratic programming, Wolfe's algorithm

JEL Classification: C63, G13, G15