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Optimal liquidation of an asset under drift uncertainty
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
2016 (English)In: SIAM Journal on Financial Mathematics, ISSN 1945-497X, E-ISSN 1945-497XArticle in journal (Refereed) Published
Abstract [en]

We study a problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. Taking a Bayesian approach, we model the initial beliefs of an individual about the drift by allowing an arbitrary probability distribution to characterize the uncertainty about the drift parameter. Filtering theory is used to describe the evolution of the posterior beliefs about the drift once the price process is being observed. An optimal stopping time is determined as the first passage time of the posterior mean below a monotone boundary, which can be characterized as the unique solution to a nonlinear integral equation. We also study monotonicity properties with respect to the prior distribution and the asset volatility.

Place, publisher, year, edition, pages
2016.
Keyword [en]
optimal liquidation, incomplete information, sequential analysis
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-283523DOI: 10.1137/15M1033265ISI: 000391850000013OAI: oai:DiVA.org:uu-283523DiVA: diva2:919255
Available from: 2016-04-13 Created: 2016-04-13 Last updated: 2017-11-30Bibliographically approved
In thesis
1. Optimal Sequential Decisions in Hidden-State Models
Open this publication in new window or tab >>Optimal Sequential Decisions in Hidden-State Models
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This doctoral thesis consists of five research articles on the general topic of optimal decision making under uncertainty in a Bayesian framework. The papers are preceded by three introductory chapters.

Papers I and II are dedicated to the problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. In Paper I, the price is modelled by the classical Black-Scholes model with unknown drift. The first passage time of the posterior mean below a monotone boundary is shown to be optimal. The boundary is characterised as the unique solution to a nonlinear integral equation. Paper II solves the same optimal liquidation problem, but in a more general model with stochastic regime-switching volatility. An optimal liquidation strategy and various structural properties of the problem are determined.

In Paper III, the problem of sequentially testing the sign of the drift of an arithmetic Brownian motion with the 0-1 loss function and a constant cost of observation per unit of time is studied from a Bayesian perspective. Optimal decision strategies for arbitrary prior distributions are determined and investigated. The strategies consist of two monotone stopping boundaries, which we characterise in terms of integral equations.

In Paper IV, the problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. Besides a few general properties established, structural properties of an optimal strategy are shown to be sensitive to the prior. A general condition for a one-sided optimal stopping region is provided.

Paper V deals with the problem of detecting a drift change of a Brownian motion under various extensions of the classical Wiener disorder problem. Monotonicity properties of the solution with respect to various model parameters are studied. Also, effects of a possible misspecification of the underlying model are explored.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, Uppsala University, 2017. 26 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 101
Keyword
sequential analysis, optimal stopping, optimal liquidation, drift uncertainty, incomplete information, stochastic filtering
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-320809 (URN)978-91-506-2641-4 (ISBN)
Public defence
2017-06-09, 80101, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:00 (English)
Opponent
Supervisors
Available from: 2017-05-18 Created: 2017-04-26 Last updated: 2017-05-18

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Ekström, ErikVaicenavicius, Juozas

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