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Applied Mathematics & Optimization

, Volume 67, Issue 1, pp 97–122 | Cite as

Discrete-Time Pricing and Optimal Exercise of American Perpetual Warrants in the Geometric Random Walk Model

  • Robert J. Vanderbei
  • Mustafa Ç. PınarEmail author
  • Efe B. Bozkaya
Article
  • 207 Downloads

Abstract

An American option (or, warrant) is the right, but not the obligation, to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual American option (both call and put) in discrete time using linear programming duality. Under the assumption that the underlying stock price follows a discrete time and discrete state Markov process, namely a geometric random walk, we formulate the pricing problem as an infinite dimensional linear programming (LP) problem using the excessive-majorant property of the value function. This formulation allows us to solve complementary slackness conditions in closed-form, revealing an optimal stopping strategy which highlights the set of stock-prices where the option should be exercised. The analysis for the call option reveals that such a critical value exists only in some cases, depending on a combination of state-transition probabilities and the economic discount factor (i.e., the prevailing interest rate) whereas it ceases to be an issue for the put.

Keywords

American perpetual warrants Pricing Optimal stopping Optimal exercise Random walk Linear programming Duality 

References

  1. 1.
    Çınlar, E.: Introduction to Stochastic Processes. Prentice Hall, New York (1975) zbMATHGoogle Scholar
  2. 2.
    Darling, D., Liggett, T., Taylor, H.: Optimal stopping for partial sums. Ann. Math. Stat. 43, 1363–1368 (1972) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Deligiannidis, G., Le, H., Utev, S.: Optimal stopping for processes with independent increments, and applications. J. Appl. Probab. 46, 1130–1145 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Diermann, F.: Optimal stopping for processes under ambiguity via measure transformation. Master’s thesis, Bielefeld University, Germany. http://erasmus-mundus.univ-paris1.fr/fichiers_etudiants/3160_dissertation.pdf (2010)
  5. 5.
    Dynkin, E.B., Yushkevich, A.A.: Markov Processes: Theorems and Problems. Plenum, New York (1969). Translated from Russian by James S. Wood Google Scholar
  6. 6.
    Helmes, K., Stockbridge, R.: Construction of the value function and stopping rules in optimal stopping of one-dimensional diffusions. Adv. Appl. Probab. 42, 158–182 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hobson, D.: A survey of mathematical finance. Proc. R. Soc. A, Math. Phys. Eng. Sci. 460(2052), 3369–3401 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Karatzas, I.: On the pricing of american options. Appl. Math. Optim. 17, 37–60 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Myneni, R.: The pricing of the American option. Ann. Appl. Probab. 2(1), 1–23 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Samuelson, P.A.: Rational theory of warrant pricing. Ind. Manage. Rev. 6, 13–31 (1965). Appendix by H.P. McKean, pp. 32–39 Google Scholar
  11. 11.
    Shreve, S.E.: Stochastic Calculus fr Finance II Continuous-Time Models. Springer, New York (2004) CrossRefGoogle Scholar
  12. 12.
    Sødal, S.: The stochastic rotation problem: a comment. J. Econ. Dyn. Control 26, 509–515 (2002) CrossRefGoogle Scholar
  13. 13.
    Sødal, S.: Entry and exit decisions based on a discount factor approach. J. Econ. Dyn. Control 30, 1963–1968 (2006) CrossRefGoogle Scholar
  14. 14.
    Taksar, M.: Infinite dimensional linear programming approach to singular stochastic control. SIAM J. Control Optim. 35(2), 604–625 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    van Moerbeke, P.: On optimal stopping and free boundary problems. Arch. Ration. Mech. Anal. 60, 101–148 (1976) zbMATHCrossRefGoogle Scholar
  16. 16.
    Vanderbei, R., Pınar, M.Ç.: Pricing american perpetual warrants by linear programming. SIAM Rev. 51(4), 767–782 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Willassen, Y.: The stochastic rotation problem: A generalization of Faustmann’s formula to stochastic growth. J. Econ. Dyn. Control 22, 573–596 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Wong, D.: Generalized Optimal Stopping Problems and Financial Markets. Addison-Wesley, Longman (1996) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Robert J. Vanderbei
    • 1
  • Mustafa Ç. Pınar
    • 2
    Email author
  • Efe B. Bozkaya
    • 3
  1. 1.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA
  2. 2.Department of Industrial EngineeringBilkent UniversityAnkaraTurkey
  3. 3.Faculty of Administrative SciencesSabancı UniversityIstambulTurkey

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