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On the Continuous and Smooth Fit Principle for Optimal Stopping Problems in Spectrally Negative Lévy Models

  • Masahiko Egami (a1) and Kazutoshi Yamazaki (a2)
Abstract

We consider a class of infinite time horizon optimal stopping problems for spectrally negative Lévy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshold levels. We obtain and show the equivalence of the continuous/smooth fit condition and the first-order condition for maximization over threshold levels. As examples of its applications, we give a short proof of the McKean optimal stopping problem (perpetual American put option) and solve an extension to Egami and Yamazaki (2013).

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Corresponding author
Postal address: Graduate School of Economics, Kyoto University, Sakyo-Ku, Kyoto, 606-8501, Japan. Email address: egami@econ.kyoto-u.ac.jp
∗∗ Postal address: Department of Mathematics, Faculty of Engineering Science, Kansai University, Suita-shi, Osaka, 564-8680, Japan. Email address: kyamazak@kansai-u.ac.jp
References
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[1] Alili L. and Kyprianou A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Prob. 15, 20622080.
[2] Alvarez L. H. R. (2003). On the properties of r-excessive mappings for a class of diffusions. Ann. Appl. Prob. 13, 15171533.
[3] Avram F., Chan T. and Usabel M. (2002). On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr's approximation for American puts. Stoch. Process. Appl. 100, 75107.
[4] Avram F., Kyprianou A. E. and Pistorius M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215238.
[5] Avram F., Palmowski Z. and Pistorius M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156180.
[6] Baurdoux E. and Kyprianou A. E. (2008). The McKean stochastic game driven by a spectrally negative Lévy process. Electron. J. Prob. 13, 173197.
[7] Baurdoux E. and Kyprianou A. E. (2009). The Shepp–Shiryaev stochastic game driven by a spectrally negative Lévy process. Theory Prob. Appl. 53, 481499.
[8] Bayraktar E., Kyprianou A. E. and Yamazaki K. (2013). On optimal dividends in the dual model. ASTIN Bull. 43, 359372.
[9] Bayraktar E., Kyprianou A. E. and Yamazaki K. (2014). Optimal dividends in the dual model under transaction costs. Insurance Math. Econom. 54, 133143.
[10] Beibel M. and Lerche H. R. (2002). A note on optimal stopping of regular diffusions under random discounting. Theory Prob. Appl. 45, 547557.
[11] Bertoin J. (1996). Lévy Processes, (Camb. Tracts Math. 121). Cambridge University Press.
[12] Bertoin J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156169.
[13] Biffis E. and Kyprianou A. E. (2010). A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Math. Econom. 46, 8591.
[14] Boyarchenko S. and Levendorskiī S. (2007). Irreversible Decisions Under Uncertainty. Optimal Stopping Made Easy (Stud. Econom. Theory 27). Springer, Berlin.
[15] Chan T., Kyprianou A. E. and Savov M. (2011). Smoothness of scale functions for spectrally negative Lévy processes. Prob. Theory Relat. Fields 150, 691708.
[16] Chang M.-C. and Sheu Y.-C. (2013). Free boundary problems and perpetual American strangles. Quant. Finance 13, 11491155.
[17] Chen N. and Kou S. G. (2009). Credit spreads, optimal capital structure, and implied volatility with endogenous default and Jump risk. Math. Finance 19, 343378.
[18] Christensen S. and Irle A. (2011). A harmonic function technique for the optimal stopping of diffusions. Stochastics 83, 347363.
[19] Christensen S., Irle A. and Novikov A. (2011). An elementary approach to optimal stopping problems for AR(1) sequences. Sequential Anal. 30, 7993.
[20] Christensen S., Salminen P. and Ta B. Q. (2013). Optimal stopping of strong Markov processes. Stoch. Process. Appl. 123, 11381159.
[21] Cissé M., Patie P. and Tanré E. (2012). Optimal stopping problems for some Markov processes. Ann. Appl. Prob. 22, 12431265.
[22] Dayanik S. and Karatzas I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173212.
[23] Dynkin E. B. (1965). Markov Processes, Vol. II. Academic Press, New York.
[24] Egami M. and Yamazaki K. (2014). Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 122.
[25] Egami M. and Yamazaki K. (2013). Precautionary measures for credit risk management in Jump models. Stochastics 85, 111143.
[26] Egami M., Leung T. and Yamazaki K. (2013). Default swap games driven by spectrally negative Lévy processes. Stoch. Process. Appl. 123, 347384.
[27] Hilberink B. and Rogers L. C. G. (2002). Optimal capital structure and endogenous default. Finance Stoch. 6, 237263.
[28] Kou S. G. and Wang H. (2003). First passage times of a Jump diffusion process. Adv. Appl. Prob. 35, 504531.
[29] Kou S. G. and Wang H. (2004). Option pricing under a double exponential Jump diffusion model. Manag. Sci. 50, 11781192.
[30] Kuznetsov A., Kyprianou A. E. and Rivero V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061), Springer, Heidelberg, pp. 97186.
[31] Kyprianou A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
[32] Kyprianou A. E. and Palmowski Z. (2007). Distributional study of de Finetti's dividend problem for a general Lévy insurance risk process. J. Appl. Prob. 44, 428443.
[33] Kyprianou A. E. and Surya B. A. (2007). Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. Finance Stoch. 11, 131152.
[34] Leland H. E. (1994). Corporate debt value, bond covenants, and optimal capital structure. J. Finance 49, 12131252.
[35] Leland H. E. and Toft K. B. (1996). Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads. J. Finance 51, 9871019.
[36] Leung T. and Yamazaki K. (2013). American step-up and step-down default swaps under Lévy models. Quant. Finance 13, 137157.
[37] Loeffen R. L. (2008). On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 16691680.
[38] Mordecki E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473493.
[39] Mordecki E. and Salminen P. (2007). Optimal stopping of Hunt and Lévy processes. Stochastics 79, 233251.
[40] Øksendal B. and Sulem A. (2005). Applied Stochastic Control of Jump Diffusions. Springer, Berlin.
[41] Peskir G. and Shiryaev A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.
[42] Pistorius M. R. (2005). A potential-theoretical review of some exit problems of spectrally negative Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 3041.
[43] Salminen P. (1985). Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85101.
[44] Salminen P. (2011). Optimal stopping, Appell polynomials, and Wiener–Hopf factorization. Stochastics 83, 611622.
[45] Surya B. A. (2007). An approach for solving perpetual optimal stopping problems driven by Lévy processes. Stochastics 79, 337361.
[46] Surya B. A. (2008). Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Prob. 45, 135149.
[47] Surya B. A. and Yamazaki K. (2013). Optimal capital structure with scale effects under spectrally negative Lévy models. Preprint. Available at http://uk.arxiv.org/abs/1109.0897.
[48] Yamazaki K. (2013). Contraction options and optimal multiple-stopping in spectrally negative Lévy models. Preprint. Available at http://uk.arxiv.org/abs/1209.1790.
[49] Yamazaki K. (2013). Inventory control for spectrally positive Lévy demand processes. Preprint. Available at http://uk.arxiv.org/abs/1303.5163.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
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