×

zbMATH — the first resource for mathematics

Selling a stock at the ultimate maximum. (English) Zbl 1201.60037
The authors consider the problem of selling a stock at the ultimate maximum where the stock price follows geometric Brownian motion. They use two different formulations for this predictive problem and derive optimal strategies for both problems. The predictive problems are reduced to optimal stopping problems which are solved by using the methods of free boundary problems and local space-time calculus. Both solutions support the financial view that one should sell bad stocks and keep good ones.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
62L15 Optimal stopping in statistics
91G80 Financial applications of other theories
60J65 Brownian motion
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Doob, J. L. (1949). Heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 20 393-403. · Zbl 0035.08901
[2] du Toit, J. and Peskir, G. (2007). The trap of complacency in predicting the maximum. Ann. Probab. 35 340-365. · Zbl 1120.60044
[3] Du Toit, J. and Peskir, G. (2008). Predicting the time of the ultimate maximum for Brownian motion with drift. In Proc. Math. Control Theory Finance ( Lisbon 2007) 95-112. Springer, Berlin. · Zbl 1149.60311
[4] du Toit, J., Peskir, G. and Shiryaev, A. N. (2008). Predicting the last zero of Brownian motion with drift. Stochastics 80 229-245. · Zbl 1145.60025
[5] Graversen, S. E., Peskir, G. and Shiryaev, A. N. (2000). Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Theory Probab. Appl. 45 41-50. · Zbl 0982.60082
[6] Graversen, S. E. and Shiryaev, A. N. (2000). An extension of P. Lévy’s distributional properties to the case of a Brownian motion with drift. Bernoulli 6 615-620. · Zbl 0965.60077
[7] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Applications of Mathematics ( New York ) 39 . Springer, New York. · Zbl 0941.91032
[8] Malmquist, S. (1954). On certain confidence contours for distribution functions. Ann. Math. Statist. 25 523-533. · Zbl 0056.37802
[9] Pedersen, J. L. (2003). Optimal prediction of the ultimate maximum of Brownian motion. Stoch. Stoch. Rep. 75 205-219. · Zbl 1032.60038
[10] Peskir, G. (2005). A change-of-variable formula with local time on curves. J. Theoret. Probab. 18 499-535. · Zbl 1085.60033
[11] Peskir, G. (2006). On reflecting Brownian motion with drift. In Proc. Symp. Stoch. Syst. ( Osaka 2005) 1-5. ISCIE, Kyoto, Japan. · Zbl 1196.93074
[12] Peskir, G. and Shiryaev, A.N. (2006). Optimal Stopping and Free-Boundary Problems . Birkhäuser, Basel. · Zbl 1115.60001
[13] Shiryaev, A. N. (2002). Quickest detection problems in the technical analysis of the financial data. In Mathematical Finance-Bachelier Congress , 2000 ( Paris ) 487-521. Springer, Berlin. · Zbl 1001.62038
[14] Shiryaev, A. N. (2007). On the conditionally extremal problems of the quickest detection of non-predictable times for the observable Brownian motion. Theory Probab. Appl.
[15] Shiryaev, A. N., Xu, Z. and Zhou, X. Y. (2008). Thou shalt buy and hold. Working Paper 21/5/2008, University of Oxford. · Zbl 1154.91478
[16] Urusov, M. A. (2004). On a property of the time of attaining the maximum by Brownian motion and some optimal stopping problems. Theory Probab. Appl. 49 169-176. · Zbl 1090.60072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.