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A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield

Published online by Cambridge University Press:  17 February 2009

Song-Ping Zhu
Song-Ping Zhu
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia; e-mail: spz@uow.edu.au.
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Abstract

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In this paper, a closed-form analytical solution for pricing convertible bonds on a single underlying asset with constant dividend yield is presented. A closed-form analytical formula has apparently never 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 contains infinitely many terms, and thus is completely analytical and in a closed form. Although it is only for the simplest CBs without call or put features, it is nevertheless the first closed-form solution that can be utilised to discuss 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 CBs' price.

Keywords

convertible bondsclosed-form analytical formulaehomotopy analysis method.
Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 4 , April 2006 , pp. 477 - 494
Copyright
Copyright © Australian Mathematical Society 2006

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