Abstract
We present explicit solutions to the perpetual American compound option pricing problems in the Black-Merton-Scholes model.
The method of proof is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. The latter are solved through their associated one-sided free-boundary problems and the subsequent martingale verification. We also obtain a closed form solution to the perpetual American chooser option pricing problem, by means of the analysis of the equivalent two-sided free-boundary problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Carmona, R., Dayanik, S.: Optimal multiple-stopping of linear diffusions. Math. Oper. Res. 33(2), 446–460 (2008)
Carmona, R., Touzi, N.: Optimal multiple stopping and valuation of swing options. Math. Finance 18(2), 239–268 (2008)
Chiarella, C., Kang, B.: The evaluation of American compound option prices under stochastic volatility using the sparse grid approach. In: Research Centre Research Paper No. 245. University of Technology, Sydney, Quantitative Finance (2009). http://www.business.uts.edu.au/qfrc/research/research_papers/rp245.pdf
Dayanik, S., Karatzas, I.: On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173–212 (2003)
Dixit, A.K., Pindyck, R.S.: Investment Under Uncertainty. Princeton University Press, Princeton (1994)
Egami, M., Xu, M.: A continuous-time search model with job switch and jumps. Math. Methods Oper. Res. 70(2), 241–267 (2008)
Duckworth, J.K., Zervos, M.: An investment model with entry and exit decisions. J. Appl. Probab. 37, 547–559 (2000)
Gapeev, P.V.: The spread option optimal stopping game. In: Kyprianou, A., Schoutens, W., Wilmott, P. (eds.) Exotic Option Pricing and Advanced Levy Models, pp. 293–305. Wiley, Chichester (2005)
Gapeev, P.V., Lerche, H.R.: On the structure of discounted optimal stopping problems for one-dimensional diffusions. In: Stochastics: An International Journal of Probability and Stochastic Processes. CDAM Research Report LSE-CDAM-2009-03 (2010). http://www.maths.lse.ac.uk/Personal/pavel/PDF/Publications/Gapeev-Lerche-SSR.pdf
Geske, R.: The valuation of corporate liabilities as compound options. J. Financ. Quant. Anal. 12, 541–552 (1977)
Geske, R.: The valuation of compound options. J. Financ. Econ. 7, 63–81 (1979)
Hodges, S.D., Selby, M.J.P.: On the evaluation of compound options. Manag. Sci. 33, 347–355 (1987)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)
Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes I, 2nd edn. Springer, Berlin (2001)
McKean, H.P.: Appendix: a free boundary problem for the heat equation arising from a problem of mathematical economics. Ind. Manage. Rev. 6, 32–39 (1965)
Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 5th edn. Springer, Berlin (1998)
Peskir, G.: A change-of-variable formula with local time on curves. J. Theor. Probab. 18, 499–535 (2005)
Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel (2006)
Shiryaev, A.N.: Optimal Stopping Rules. Springer, Berlin (1978)
Shiryaev, A.N.: Essentials of Stochastic Finance. World Scientific, Singapore (1999)
Acknowledgements
The authors are grateful to Mihail Zervos for many useful discussions. The authors thank the Editor and two anonymous Referees for their careful reading of the manuscript and helpful suggestions. The second author gratefully acknowledges the scholarship of the Alexander Onassis Public Benefit Foundation for his doctoral studies at the London School of Economics and Political Science.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Gapeev, P.V., Rodosthenous, N. (2014). On the Pricing of Perpetual American Compound Options. In: Kabanov, Y., Rutkowski, M., Zariphopoulou, T. (eds) Inspired by Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-02069-3_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-02069-3_13
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02068-6
Online ISBN: 978-3-319-02069-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)