Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T13:43:47.142Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Barone-Adesi, G., Bermudez, A. and Hatgioannides, J., “Two-factor convertible bonds valuation using the method of characteristics/finite elements”, J. Econ. Dyn. Control 27 (2003) 1801–1831.CrossRefGoogle Scholar
[2]Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Polit. Econ. 81 (1973) 637–654.CrossRefGoogle Scholar
[3]Brennan, M. J. and Schwartz, E. S., “Convertible bonds: valuation and optimal strategies for call and conversion”, J. Finance 32 (1977) 1699–1715.CrossRefGoogle Scholar
[4]Brennan, M. J. and Schwartz, E. S., “Analyzing convertible bonds”, J. Fin. Quant. Anal. 15 (1980) 907–929.CrossRefGoogle Scholar
[5]Carslaw, H. S. and Jaeger, J. C., Conduction of heat in solids (Oxford Scientific Publications, Clarendon Press, 1959).Google Scholar
[6]Gukhal, C. R., “Analytical valuation of American options on jump diffusion process”, Math. Finance 11 (2001) 97–115.CrossRefGoogle Scholar
[7]Hill, J. M., One dimensional Stefan problems: an introduction, Pitman Monographs and Surveys in Pure and Applied Mathematics 31 (Longman Scientific & Technical, New York, 1987).Google Scholar
[8]Ingersoll, J. E., “A contingent claims valuation of convertible securities”, J. Fin. Econ. 4 (1977) 289–322.CrossRefGoogle Scholar
[9]Liao, S.-J., “Numerically solving non-linear problems by the homotopy analysis method”, Comput. Mech. 20 (1997) 530–540.CrossRefGoogle Scholar
[10]Liao, S.-J. and Campo, A., “Analytic solutions of the temperature distribution in Blasius viscous flow problems”, J. Fluid Mech. 453 (2002) 411–425.CrossRefGoogle Scholar
[11]Liao, S.-J. and Zhu, J.-M., “A short note on high-order streamfunction-vorticity formulations of 2D steady-state Navier-Stokes equations”, Internat. J. Numer. Methods Fluids 22 (1996) 1–9.Google Scholar
[12]Liao, S.-J. and Zhu, S.-P., “Solving the Liouville equation with the general boundary element method approach”, Boundary Element Technology 13 (1999) 407–416.Google Scholar
[13]McConnel, J.-J. and Schwartz, E. S., “LYON taming”, J. Finance 41 (1986) 561–576.CrossRefGoogle Scholar
[14]Nyborg, K., “The use and pricing of convertible bonds”, Appl. Math. Finance 3 (1996) 167–190.CrossRefGoogle Scholar
[15]Ortega, J. M. and Rheinboldt, W. C., iterative solution of nonlinear equations in several variables (Academic Press, New York, 1970).Google Scholar
[16]Tavella, D. and Randall, C., Pricing financial instruments: the finite difference method (Wiley, New York, 2000).Google Scholar
[17]Zhu, S.-P., “A closed-form exact solution for the value of American put and its optimal exercise boundary”, in Proceedings of the 3rd SPIE international Symposium, May, Austin, Texas, USA, (2005), 186–199.Google Scholar
[18]Zhu, S.-P. and Hung, T.-P., “A new numerical approach for solving high-order non-linear ordinary differential equations”, Comnun. Num. Meth. Engng 19 (2003) 601–614.CrossRefGoogle Scholar
[19]Zvan, R., Forsyth, P. A. and Vetzal, K. R., “A finite volume approach for contingent claims valuation”, IMA J. Numer. Anal. 21 (2001) 703–721.CrossRefGoogle Scholar