| A Convertible Bond (CB) is a corporate debt security that gives the holder the right to exchange future coupon payments and principal repayment for a prescribed number of shares of equity. Thus, it has both an equity part and a fixed-income part, and may contain some additional features, such as callability and puttability.; In this paper, we study the model for valuing Convertible Bonds with credit risk originally developed by Kostas Tsiveriotis and Chris Fernandes (TF). The Convertible Bond is a derivative of the stock price, and the pricing model developed by TF is based on a free boundary value problem associated with a pair of parabolic Partial Differential Equations (PDEs) with discontinuities at the time points when there is a coupon payment, or when the bond is converted, or when it is called back (purchased) by the issuer, or when it is put (sold) to the issuer. We explore the possible derivation of the TF model and study the convergence of several numerical methods for solving the free boundary value problem associated with the TF model. In particular, we consider the Successive Over-Relaxation (SOR) iteration and a penalty method for solving the linear complementarity problem used to handle the free boundary. Special emphasis is given to the effectiveness of the numerical scheme as well to the treatment of discontinuities. |
