A Closed-form Analytical Solution for the Valuation of Convertible Bonds With Constant Dividend Yield

In this paper, a closed-form analytical solution for pricing convertible bonds on a single underlying asset with constant dividend yield is presented. To the author’s best knowledge, never has a closed-form analytical formula been found for American-style convertible bonds (CBs) of finite maturity time although there have been quite a few approximate solutions and numerical approaches proposed. The solution presented here is written in the form of a Taylor’s series expansion, which 1 contains infinitely many terms, and thus is completely analytical and in a closed form. Although it is only for simplest CBs without call or put features, it is nevertheless the first closed-form solution that can be utilized to discuss the convertibility analytically. The solution is based on the homotopy analysis method, with which the optimal converting price has been elegantly and temporarily removed in the solution process of each order, and consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical solution for the optimal converting price and the CB’s price.