Advances in Difference Equations
Volume 2011 (2011), Article ID 309678, 21 pages
Research Article

A Variational Inequality from Pricing Convertible Bond

School of Mathematics, South China Normal University, Guangzhou 510631, China

Received 30 December 2010; Accepted 11 February 2011

Academic Editor: Jin Liang

Copyright © 2011 Huiwen Yan and Fahuai Yi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The model of pricing American-style convertible bond is formulated as a zero-sum Dynkin game, which can be transformed into a parabolic variational inequality (PVI). The fundamental variable in this model is the stock price of the firm which issued the bond, and the differential operator in PVI is linear. The optimal call and conversion strategies correspond to the free boundaries of PVI. Some properties of the free boundaries are studied in this paper. We show that the bondholder should convert the bond if and only if the price of the stock is equal to a fixed value, and the firm should call the bond back if and only if the price is equal to a strictly decreasing function of time. Moreover, we prove that the free boundaries are smooth and bounded. Eventually we give some numerical results.