ANZIAM J.
47 (2006), 477494

A closedform analytical solution for the valuation of convertible bonds with constant dividend yield

SongPing Zhu
School of Mathematics and Applied Statistics
University of Wollongong
Wollongong
NSW 2522
Australia
spz@uow.edu.au



Abstract

In this paper, a closedform analytical solution
for pricing convertible bonds on a single
underlying asset with constant dividend yield is
presented. A closedform analytical formula has
apparently never been found for Americanstyle
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
closedform 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.

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