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Optimal Stopping Problems for Asset Management

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  04 January 2016

Savas Dayanik  and
Masahiko Egami
Savas Dayanik*
Affiliation:
Bilkent University
Masahiko Egami*
Affiliation:
Kyoto University
*
∗ Postal address: Departments of Industrial Engineering and Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey. Email address: [email protected]
∗∗ Postal address: Graduate School of Economics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan. Email address: [email protected]
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Abstract

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An asset manager invests the savings of some investors in a portfolio of defaultable bonds. The manager pays the investors coupons at a constant rate and receives a management fee proportional to the value of the portfolio. He/she also has the right to walk out of the contract at any time with the net terminal value of the portfolio after payment of the investors' initial funds, and is not responsible for any deficit. To control the principal losses, investors may buy from the manager a limited protection which terminates the agreement as soon as the value of the portfolio drops below a predetermined threshold. We assume that the value of the portfolio is a jump diffusion process and find an optimal termination rule of the manager with and without protection. We also derive the indifference price of a limited protection. We illustrate the solution method on a numerical example. The motivation comes from the collateralized debt obligations.

Keywords

Optimal stoppingjump diffusionasset management

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J60: Diffusion processes 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60J75: Jump processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 655 - 677
Copyright
© Applied Probability Trust 

References

Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Boyarchenko, S. I. and Levendorskiıˇ, S. Z. (2002). Perpetual American options under Lévy processes. SIAM J. Control Optimization 40, 1663–1696.Google Scholar
Çinlar, E. (2006). Jump-diffusions. Blackwell-Tapia Conference, 3–4 November 2006. Avalaible at http://www.ima.umn.edu/2006-2007/SW11.3-4.06/abstracts.html#Cinlar-Erhan.Google Scholar
Chan, T. (1999). Pricing contingent claims on stocks driven by Lévy processes. Ann. Appl. Prob. 9, 504–528.CrossRefGoogle Scholar
Colwell, D. B. and Elliott, R. J. (2006). Discontinuous asset prices and non-attainable contingent claims. Math. Finance 3, 295–308.Google Scholar
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. Ser. B 46, 353–388.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman & Hall, London.Google Scholar
Dayanik, S. (2008). Optimal stopping of linear diffusions with random discounting. Math. Operat. Res. 33, 645–661.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
Dayanik, S. and Sezer, S. (2009). Multisource Bayesian sequential hypothesis testing. Preprint.Google Scholar
Dayanik, S., Poor, H. V. and Sezer, S. O. (2008). Multisource Bayesian sequential change detection. Ann. Appl. Prob. 18, 552–590.Google Scholar
Di Nunno, G., Øksendal, B. and Proske, F. (2009). Malliavin Calculus for Lévy Processes with Applications to Finance. Springer, Berlin.Google Scholar
Duffie, D. (1996). Dynamic Asset Pricing Theory. Princeton University Press.Google Scholar
Duffie, D. and Gârleanu, N. (2001). Risk and valuation of collateralized debt obligations. Financial Anal. J. 57, 41–59.CrossRefGoogle Scholar
Duffie, D. and Singleton, K. J. (1999). Modeling term structures of defaultable bonds. Rev. Financial Stud. 12, 687–720.Google Scholar
Egami, M. and Esteghamat, K. (2006). An approximation method for analysis and valuation of credit correlation derivatives. J. Banking Finance 30, 341–364.Google Scholar
Hull, J. and White, A. (2004). Valuation of a CDO and an n th to default CDS without Monte Carlo simulation. J. Derivatives 12, 8–23.Google Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.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
Leland, H. E. (1994). Corporate debt value, bond covenants, and optimal capital structure. J. Finance 49, 1213–1252.Google Scholar
Lucas, D. J., Goodman, L. S. and Fabozzi, F. J. (2006). Collateralized Debt Obligations: Structures and Analysis. John Wiley, Hoboken, NJ.Google Scholar
Mordecki, E. (1999). Optimal stopping for a diffusion with Jumps. Finance Stoch. 3, 227–236.Google Scholar
Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493.CrossRefGoogle Scholar
Pham, H. (1997). Optimal stopping, free boundary, and American option in a Jump-diffusion model. Appl. Math. Optimization 35, 145–164.Google Scholar
Schoutens, W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives. John Wiley, Chichester.Google Scholar