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An optimal stopping problem with a reward constraint

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Abstract

This article studies an optimal stopping problem with an endogenous constraint on the set of admissible stopping times. The constraint stipulates that continuation is permitted, at any given date t, only if the endogenous reward achieved exceeds a prespecified threshold. Characterizations of the value function and the optimal stopping time are presented. An application to the pricing of corporate claims, when the capital structure of the firm includes equity-trigger debt, is carried out.

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References

  1. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., Ku, H.: Coherent multiperiod risk adjusted values and Bellman’s principle. Ann. Oper. Res. 152, 5–22 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bensoussan, A.: On the theory of option pricing. Acta Appl. Math. 2, 139–158 (1984)

    MathSciNet  MATH  Google Scholar 

  3. Bhanot, K., Mello, A.S.: Should corporate debt include a rating trigger? J. Financ. Econ. 79, 69–98 (2006)

    Article  Google Scholar 

  4. Black, F., Cox, J.C.: Valuing corporate securities: some effects of bond indenture provisions. J. Finance 31, 351–367 (1976)

    Google Scholar 

  5. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)

    Article  Google Scholar 

  6. Brennan, M.J., Schwartz, E.: Evaluating natural resource investments. J. Bus. 58, 135–157 (1985)

    Article  Google Scholar 

  7. Dayanik, S., Karatzas, I.: On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173–212 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Detlefsen, K., Scandolo, G.: Conditional and dynamic convex risk measures. Finance Stoch. 9, 539–561 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dorobantu, D.: Optimal strategies in a risky debt context. Stochastics 81, 269–277 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Dynkin, E.: Optimal choices of the stopping moment of a Markov process. Dokl. Akad. Nauk SSSR 150, 238–240 (1963)

    MathSciNet  Google Scholar 

  11. Dynkin, E., Yushkevich, A.: Markov Processes: Theorems and Problems. Plenum, New York (1969)

    Google Scholar 

  12. Jacka, S.: Optimal stopping and the American put. Math. Finance 1(2), 1–14 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Karatzas, I.: On the pricing of American options. Appl. Math. Optim. 17, 37–60 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)

    Book  MATH  Google Scholar 

  15. Karatzas, I., Shreve, S.: Methods of Mathematical Finance. Applications of Mathematics, vol. 39. Springer, Berlin (1998)

    MATH  Google Scholar 

  16. Kifer, Y.: Game options. Finance Stoch. 4, 443–463 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leland, H.: Corporate debt value, bond covenants, and optimal capital structure. J. Finance 49, 1213–1252 (1994)

    Google Scholar 

  18. Leland, H., Toft, K.: Optimal capital structure, endogenous bankruptcy, and the structure of credit spreads. J. Finance 51, 987–1019 (1996)

    Google Scholar 

  19. McDonald, R., Siegel, D.: The value of waiting to invest. Q. J. Econ. 101, 707–727 (1986)

    Article  Google Scholar 

  20. Merton, R.: On the pricing of corporate debt: the risk structure of interest rates. J. Finance 29, 449–470 (1974)

    Google Scholar 

  21. Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel (2006)

    MATH  Google Scholar 

  22. Powers, W.: Report of Investigation by the Special Investigation Committee of the Board of Directors of Enron Corp, William Powers Committee Report (2002). Available from http://news.findlaw.com/wsj/docs/enron/sicreport/index.html

  23. Riedel, F.: Optimal stopping with multiple priors. Econometrica 77, 857–908 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Snell, J.L.: Application of martingale system theorems. Trans. Am. Math. Soc. 73, 293–312 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shreve, S., Sîrbu, M.: A two-person game for pricing convertible bonds. SIAM J. Control Optim. 45, 1508–1539 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Villeneuve, S.: On threshold strategies and the smooth-fit principle for optimal stopping problems. J. Appl. Probab. 44, 181–198 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank the editor, the co-editor, the associate editor and two anonymous referees for their constructive comments and insightful suggestions that improved the paper. Jie Xiong gratefully acknowledges research support from National Science Foundation grant DMS-0906907. Address correspondence to Weidong Tian, Belk College of Business, University of North Carolina at Charlotte, NC, 28223, USA.

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Authors and Affiliations

  1. School of Management, Boston University, Boston, MA, 02215, USA

    Jérôme Detemple

  2. Belk College of Business, University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    Weidong Tian

  3. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996-1300, USA

    Jie Xiong

Authors
  1. Jérôme Detemple
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  2. Weidong Tian
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  3. Jie Xiong
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Correspondence to Weidong Tian.

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Detemple, J., Tian, W. & Xiong, J. An optimal stopping problem with a reward constraint. Finance Stoch 16, 423–448 (2012). https://doi.org/10.1007/s00780-012-0173-4

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  • DOI: https://doi.org/10.1007/s00780-012-0173-4

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