IJPAM: Volume 53, No. 4 (2009)


Na Song$^1$, Yue Jiao$^2$, Wai-Ki Ching$^3$, Tak-Kuen Siu$^4$, Zhen-Yu Wu$^5$
$^{1,2,3}$Advanced Modeling and Applied Computing Laboratory
Department of Mathematics
The University of Hong Kong
Pokfulam Road, HONG KONG, P.R. CHINA
$^1$e-mail: smynmath@163.com
$^2$e-mail: jiaoyue@hkusua.hku.hk
$^3$e-mails: wching@hkusua.hku.hk, wkc@maths.hku.hk
$^4$Department of Mathematics and Statistics
Curtin University of Technology
Perth, W.A. 6845, AUSTRALIA
e-mail: ktksiu2005@gmail.com
$^5$Department of Finance and Management Science
N. Murray Edwards School of Business
University of Saskatchewan
25 Campus Drive, Saskatoon, SK S7N 5A7, CANADA
e-mail: wu@edwards.usask.ca

Abstract.This paper develops a valuation model for a perpetual convertible bond when the price dynamics of the underlying share are governed by continuous-time Markovian regime-switching models. We suppose that the appreciation rate and the volatility of the underlying share are modulated by a continuous-time, finite-state, observable Markov chain. The states of this chain are interpreted as the states of an economy. Here the valuation problem of the perpetual convertible bond can be viewed as that of valuing a perpetual stock loan, or a perpetual American option with time-dependent strike price. With the presence of the regime-switching effect, the market in the model is, in general, incomplete. To provide a convenient method to determine a price kernel for valuation, we employ the regime-switching Esscher transform introduced in Elliott, Chan and Siu (2005) [#!Elliott05!#]. We then adopt the differential equation approach in Guo and Zhang (2004) [#!Guo04!#] to solve the optimal stopping problem associated with the valuation of the perpetual convertible bond. Numerical examples are presented to illustrate the practical implementation of the proposed model.

Received: May 20, 2009

AMS Subject Classification: 93A30, 62L15, 91B28

Key Words and Phrases: perpetual convertible bonds, Esscher transform, regime-switching, Markov chain, incomplete market, optimal stopping

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 53
Issue: 4