Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T10:26:40.720Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  22 February 2016

Masahiko Egami  and
Kazutoshi Yamazaki
Masahiko Egami*
Affiliation:
Kyoto University
Kazutoshi Yamazaki*
Affiliation:
Kansai University
*
∗ 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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).

Keywords

Optimal stoppingspectrally negative Lévy processscale functioncontinuous and smooth fit

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J75: Jump processes
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 139 - 167
Copyright
© Applied Probability Trust 

References

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, 2062–2080.Google Scholar
Alvarez, L. H. R. (2003). On the properties of r-excessive mappings for a class of diffusions. Ann. Appl. Prob. 13, 1517–1533.Google Scholar
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, 75–107.Google Scholar
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, 215–238.Google Scholar
Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156–180.Google Scholar
Baurdoux, E. and Kyprianou, A. E. (2008). The McKean stochastic game driven by a spectrally negative Lévy process. Electron. J. Prob. 13, 173–197.Google Scholar
Baurdoux, E. and Kyprianou, A. E. (2009). The Shepp–Shiryaev stochastic game driven by a spectrally negative Lévy process. Theory Prob. Appl. 53, 481–499.Google Scholar
Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2013). On optimal dividends in the dual model. ASTIN Bull. 43, 359–372.Google Scholar
Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2014). Optimal dividends in the dual model under transaction costs. Insurance Math. Econom. 54, 133–143.Google Scholar
Beibel, M. and Lerche, H. R. (2002). A note on optimal stopping of regular diffusions under random discounting. Theory Prob. Appl. 45, 547–557.Google Scholar
Bertoin, J. (1996). Lévy Processes, (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.Google Scholar
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, 85–91.Google Scholar
Boyarchenko, S. and Levendorskiī, S. (2007). Irreversible Decisions Under Uncertainty. Optimal Stopping Made Easy (Stud. Econom. Theory 27). Springer, Berlin.Google Scholar
Chan, T., Kyprianou, A. E. and Savov, M. (2011). Smoothness of scale functions for spectrally negative Lévy processes. Prob. Theory Relat. Fields 150, 691–708.Google Scholar
Chang, M.-C. and Sheu, Y.-C. (2013). Free boundary problems and perpetual American strangles. Quant. Finance 13, 1149–1155.Google Scholar
Chen, N. and Kou, S. G. (2009). Credit spreads, optimal capital structure, and implied volatility with endogenous default and Jump risk. Math. Finance 19, 343–378.Google Scholar
Christensen, S. and Irle, A. (2011). A harmonic function technique for the optimal stopping of diffusions. Stochastics 83, 347–363.CrossRefGoogle Scholar
Christensen, S., Irle, A. and Novikov, A. (2011). An elementary approach to optimal stopping problems for AR(1) sequences. Sequential Anal. 30, 79–93.Google Scholar
Christensen, S., Salminen, P. and Ta, B. Q. (2013). Optimal stopping of strong Markov processes. Stoch. Process. Appl. 123, 1138–1159.Google Scholar
Cissé, M., Patie, P. and Tanré, E. (2012). Optimal stopping problems for some Markov processes. Ann. Appl. Prob. 22, 1243–1265.Google Scholar
Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173–212.Google Scholar
Dynkin, E. B. (1965). Markov Processes, Vol. II. Academic Press, New York.Google Scholar
Egami, M. and Yamazaki, K. (2014). Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 1–22.Google Scholar
Egami, M. and Yamazaki, K. (2013). Precautionary measures for credit risk management in Jump models. Stochastics 85, 111–143.Google Scholar
Egami, M., Leung, T. and Yamazaki, K. (2013). Default swap games driven by spectrally negative Lévy processes. Stoch. Process. Appl. 123, 347–384.Google Scholar
Hilberink, B. and Rogers, L. C. G. (2002). Optimal capital structure and endogenous default. Finance Stoch. 6, 237–263.Google Scholar
Kou, S. G. and Wang, H. (2003). First passage times of a Jump diffusion process. Adv. Appl. Prob. 35, 504–531.Google Scholar
Kou, S. G. and Wang, H. (2004). Option pricing under a double exponential Jump diffusion model. Manag. Sci. 50, 1178–1192.Google Scholar
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. 97–186.Google Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
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, 428–443.Google Scholar
Kyprianou, A. E. and Surya, B. A. (2007). Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. Finance Stoch. 11, 131–152.Google Scholar
Leland, H. E. (1994). Corporate debt value, bond covenants, and optimal capital structure. J. Finance 49, 1213–1252.Google Scholar
Leland, H. E. and Toft, K. B. (1996). Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads. J. Finance 51, 987–1019.Google Scholar
Leung, T. and Yamazaki, K. (2013). American step-up and step-down default swaps under Lévy models. Quant. Finance 13, 137–157.Google Scholar
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, 1669–1680.Google Scholar
Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493.Google Scholar
Mordecki, E. and Salminen, P. (2007). Optimal stopping of Hunt and Lévy processes. Stochastics 79, 233–251.Google Scholar
Øksendal, B. and Sulem, A. (2005). Applied Stochastic Control of Jump Diffusions. Springer, Berlin.Google Scholar
Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
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. 30–41.Google Scholar
Salminen, P. (1985). Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85–101.Google Scholar
Salminen, P. (2011). Optimal stopping, Appell polynomials, and Wiener–Hopf factorization. Stochastics 83, 611–622.Google Scholar
Surya, B. A. (2007). An approach for solving perpetual optimal stopping problems driven by Lévy processes. Stochastics 79, 337–361.Google Scholar
Surya, B. A. (2008). Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Prob. 45, 135–149.Google Scholar
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.Google Scholar
Yamazaki, K. (2013). Contraction options and optimal multiple-stopping in spectrally negative Lévy models. Preprint. Available at http://uk.arxiv.org/abs/1209.1790.Google Scholar
Yamazaki, K. (2013). Inventory control for spectrally positive Lévy demand processes. Preprint. Available at http://uk.arxiv.org/abs/1303.5163.Google Scholar