Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T09:35:25.107Z Has data issue: false hasContentIssue false
  • English
  • Français

AN ANALYTICAL APPROXIMATION FOR CONVERTIBLE BONDS

Part of: Mathematical finance

Published online by Cambridge University Press:  20 June 2022

JOANNA GOARD Open the ORCID record for JOANNA GOARD [Opens in a new window]
JOANNA GOARD*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW2522, Australia
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper looks at adapting the method of Medvedev and Scaillet for pricing short-term American options to evaluate short-term convertible bonds. However unlike their method, we provide explicit formulae for the coefficients of our series solution. This means that we do not need to solve complicated recursive systems, and can efficiently provide fast solutions. We also compare the method with numerical solutions, and find that it performs extremely well, giving accurate bond prices as well as accurate optimal conversion prices.

Keywords

convertible bondsanalytical approximationsoptimal conversion pricefree boundary problems

MSC classification

Primary: 91G20: Derivative securities
Type
Research Article
Information
The ANZIAM Journal , Volume 64 , Issue 2 , April 2022 , pp. 135 - 148
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions (Dover Publications, New York, 1965).Google Scholar
Ammann, M., Kind, A. and Wilde, C., “Simulation-based pricing of convertible bonds”, J. Empir. Finance 15 (2008) 310–331; doi:10.2139/ssrn.762804.CrossRefGoogle Scholar
Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Polit. Econ. 81 (1973) 637–654; doi:10.1086/260062.CrossRefGoogle Scholar
Brennan, M. J. and Schwartz, E. S., “Convertible bonds: valuation and optimal strategies for call and conversion”, J. Finance 32 (1977) 1699–1715; doi:10.1111/j.1540-6261.1977.tb03364.x.CrossRefGoogle Scholar
Brennan, M. J. and Schwartz, E. S., “Analyzing convertible bonds”, J. Financ. Quant. Anal. 15 (1980) 907–929; doi:10.1111/mafi.12218.CrossRefGoogle Scholar
Chambers, D. R. and Lu, Q., “A tree model for pricing convertible bonds with equity, interest rate, and default risk”, J. Deriv. 14 (2007) 25–46; doi:10.3905/jod.2007.686421.CrossRefGoogle Scholar
Chan, L. and Zhu, S.-P., “An analytic formula for pricing American-style convertible bonds in a regime-switching model”, IMA J. Manag. Math. 26 (2015) 403–428; doi:10.1093/imaman/dpu005.CrossRefGoogle Scholar
Ingersoll, J. E., “A contingent-claims valuation of convertible securities”, J. Financ. Econ. 4 (1977) 289–321; doi:10.1016/0304-405X(77)90004-6.CrossRefGoogle Scholar
Lvov, D., Yigitsbasioglu, A. B. and El Bachir, N., “Pricing convertible bonds by simulation”, in: Proceedings of the Second IASTED Int. Conf. on Financial Engineering and Applications, (IASTED/ACTA Press, Cambridge, MA, 2004) ISBN: 9780889864177.CrossRefGoogle Scholar
Maplesoft, Maple 12 users manual (Maplesoft, Waterloo, Ontario, Canada, 2008).Google Scholar
McConnell, J. J. and Schwartz, E. S., “Lyon taming”, J. Finance 41 (1986) 561–576; doi:10.2307/2328484.CrossRefGoogle Scholar
Medvedev, A. and Scaillet, O., “Pricing American options under stochastic volatility and stochastic interest rates”, J. Financ. Econ. 98 (2010) 145–159; doi:10.1016/j.jfineco.2010.03.017.CrossRefGoogle Scholar
Murugaboopathy, P., “Global companies raise massive cash through convertible bonds at start of 2021”, Reuters (17 February 2021). https://www.reuters.com/world/china/global-markets-convertibles-graphics-2021-02-17/.Google Scholar
Nyborg, K. G., “The use and pricing of convertible bonds,” Appl. Math. Finance 3 (1996) 167–190; doi:10.1080/13504869600000009.CrossRefGoogle Scholar
Refinitiv Financial Solutions, An LSEG business. https://www.refinitiv.com/en/financial-data/economic-data.Google Scholar
Shapiro, J., “Why the zero coupon bond market is booming”, Australian Financial Review (25 March 2021). https://www.afr.com/markets/equity-markets/preaching-the-zero-coupon-converted-bonds-20210322-p57d0h.Google Scholar
Tao, L. N., “On free boundary problems with arbitrary initial and flux conditions”, Z. Angew. Math. Phys. (ZAMP) 30 (1979) 416–426; doi:10.1007/BF01588886.CrossRefGoogle Scholar
Tavella, D. and Randall, C., Pricing financial instruments: the finite difference method (Wiley, New York, 2000).Google Scholar
Wilmott, P., Derivatives: the theory and practice of financial engineering (Wiley, Chichester, 1999).Google Scholar
Zhu, S.-P., “A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield”, ANZIAM J. 47 (2006) 477–494; doi:10.1017/S1446181100010087.CrossRefGoogle Scholar
Zhu, S.-P. and Zhang, J., “How should a convertible bond be decomposed?”, Decis. Econ. Finance 35 (2012) 113–149; doi:10.1007/s10203-011-0118-y.CrossRefGoogle Scholar