Selling a stock at the ultimate maximum
Jacques du Toit and Goran Peskir
Source: Ann. Appl. Probab.
Volume 19, Number 3
(2009), 983-1014.
Abstract
Assuming that the stock price Z=(Zt)0≤t≤T follows a geometric Brownian motion with drift μ∈ℝ and volatility σ>0, and letting Mt=max 0≤s≤tZs for t∈[0, T], we consider the optimal prediction problems
![\[V_{1}=\inf_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{M_{T}}{Z_{\tau}}\biggr)\]](/DPubS?service=Repository&version=1.0&verb=Disseminate&view=body&handle=euclid.aoap/1245071016&content-type=gif_1)
and ![\[V_{2}=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_{T}}\biggr)\]](/DPubS?service=Repository&version=1.0&verb=Disseminate&view=body&handle=euclid.aoap/1245071016&content-type=gif_2)
,
where the infimum and supremum are taken over all stopping times τ of Z. We show that the following strategy is optimal in the first problem: if μ≤0 stop immediately; if μ∈(0, σ2) stop as soon as Mt/Zt hits a specified function of time; and if μ≥σ2 wait until the final time T. By contrast we show that the following strategy is optimal in the second problem: if μ≤σ2/2 stop immediately, and if μ>σ2/2 wait until the final time T. Both solutions support and reinforce the widely held financial view that “one should sell bad stocks and keep good ones.” The method of proof makes use of parabolic free-boundary problems and local time–space calculus techniques. The resulting inequalities are unusual and interesting in their own right as they involve the future and as such have a predictive element.
Primary Subjects: 60G40, 35R35, 60J65
Secondary Subjects: 91B28, 60G25, 45G10
Keywords: Geometric Brownian motion; optimal prediction; optimal stopping; ultimate maximum; parabolic free-boundary problem; smooth fit; normal reflection; local time–space calculus; curved boundary; nonlinear Volterra integral equation; Markov process; diffusion process
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1245071016
Digital Object Identifier: doi:10.1214/08-AAP566
Zentralblatt MATH identifier:
05580230
Mathematical Reviews number (MathSciNet):
MR2537196
References
[1] Doob, J. L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Statist. 20 393–403.
Mathematical Reviews (MathSciNet):
MR30732
[2] du Toit, J. and Peskir, G. (2007). The trap of complacency in predicting the maximum. Ann. Probab. 35 340–365.
[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.
[4] du Toit, J., Peskir, G. and Shiryaev, A. N. (2008). Predicting the last zero of Brownian motion with drift. Stochastics 80 229–245.
[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.
[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.
[7] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Applications of Mathematics (New York) 39. Springer, New York.
[8] Malmquist, S. (1954). On certain confidence contours for distribution functions. Ann. Math. Statist. 25 523–533.
Mathematical Reviews (MathSciNet):
MR65095
[9] Pedersen, J. L. (2003). Optimal prediction of the ultimate maximum of Brownian motion. Stoch. Stoch. Rep. 75 205–219.
[10] Peskir, G. (2005). A change-of-variable formula with local time on curves. J. Theoret. Probab. 18 499–535.
[11] Peskir, G. (2006). On reflecting Brownian motion with drift. In Proc. Symp. Stoch. Syst. (Osaka 2005) 1–5. ISCIE, Kyoto, Japan.
[12] Peskir, G. and Shiryaev, A.N. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.
[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.
[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. To appear.
[15] Shiryaev, A. N., Xu, Z. and Zhou, X. Y. (2008). Thou shalt buy and hold. Working Paper 21/5/2008, University of Oxford.
[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.
