Pricing of Convertible Bonds with Credit Risk and Stochastic Interest Rate

Abstract

The convertible bond is an interesting security with its hybrid nature from both debt and equity. Complications in pricing convertible bonds arise due to additional contractual features such as callability and puttability, soft call provision.

Since 1991, most practitioners have used the binomial tree models to evaluate convertibles bonds. In this thesis, a partial differential equation is formulated from the Two-Factor model, attempting a consistent treatment of equity, interest rate and credit risk as well as the incorporation of the call and put provisions. I shall present a general framework for valuing convertible bonds, with a Black-Scholes stock price, and the Hull White model for the interest rate.

By no-arbitrage, the Hull-White model is calibrated to fit the initial term structure of interest rates as well as the volatility surface of European swaptions, which are readily quoted from the financial market. The closed form formula of the European swaption under the Hull-White model is deduced. With the Levenberg-Marquardt algorithm, I seek to find model parameters that lead to a least-square fit to its market prices.

The approach for solving the PDE is based on the numerical solution of linear complementarity problems brought up in E Ayache, P Forsyth, K Vertzal (2003) and the penalty method. A convergence study is conducted in the report. I hope that this report will impart on the subject of convertible bond pricing, calibration of Hull-White model and the structure of convertible bonds to students and others readers interested in financial products pricing with PDE approach.