Henderson, Vicky and Hobson, David (David G.). (2008) An explicit solution for an optimal stopping/optimal control problem which models an asset sale. Annals of Applied Probability, Vol.18 (No.5). pp. 1681-1705. ISSN 1050-5164

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Abstract

In this article we study an optimal stopping/optimal control problem which models the decision facing a ask-averse agent over when to sell an asset. The market is incomplete so that the asset exposure cannot be hedged. In addition to the decision over when to sell, the agent has to choose a control strategy which corresponds to a feasible wealth process. We formulate this problem as one involving the choice of a stopping time and a martingale. We conjecture the form of the solution and verify that the candidate solution is equal to the value function. The interesting features of the solution are that it is available in a very explicit form, that for some parameter values the optimal strategy is more sophisticated than might originally be expected, and that although the setup is based on continuous diffusions, the optimal martingale may involve a jump process. One interpretation of the solution is that it is optimal for the risk-averse agent to gamble.

Item Type:	Journal Article
Subjects:	H Social Sciences > HG Finance Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Optimal stopping (Mathematical statistics), Control theory, Mathematical optimization, Utility theory, Capital market -- Mathematical models, Local times (Stochastic processes)
Journal or Publication Title:	Annals of Applied Probability
Publisher:	Institute of Mathematical Statistics
ISSN:	1050-5164
Date:	October 2008
Volume:	Vol.18
Number:	No.5
Number of Pages:	25
Page Range:	pp. 1681-1705
Identification Number:	10.1214/07-AAP511
Status:	Peer Reviewed
Publication Status:	Published
Funder:	Engineering and Physical Sciences Research Council (EPSRC)
References:	[1] DIXIT, A. K. and PINDYCK, R. S. (1994). Investment under Uncertainty. Princeton Univ. Press. [2] EVANS, J. D., HENDERSON, V. and HOBSON, D. (2007). Optimal timing for an asset sale in an incomplete market. Math. Finance. To appear. [3] HENDERSON, V. (2007). Valuing the option to invest in an incomplete market. Math. Financ. Econ. 1 103–128. MR2365685 [4] JACKA, S. D. (1991). Optimal stopping and best constants for Doob-like inequalities. I. The case p = 1. Ann. Probab. 19 1798–1821. MR1127729 [5] KARATZAS, I. and KOU, S. G. (1998). Hedging American contingent claims with constrained portfolios. Finance Stoch. 2 215–258. MR1809521 [1] DIXIT, A. K. and PINDYCK, R. S. (1994). Investment under Uncertainty. Princeton Univ. Press. [2] EVANS, J. D., HENDERSON, V. and HOBSON, D. (2007). Optimal timing for an asset sale in an incomplete market. Math. Finance. To appear. [3] HENDERSON, V. (2007). Valuing the option to invest in an incomplete market. Math. Financ. Econ. 1 103–128. MR2365685 [4] JACKA, S. D. (1991). Optimal stopping and best constants for Doob-like inequalities. I. The case p = 1. Ann. Probab. 19 1798–1821. MR1127729 [5] KARATZAS, I. and KOU, S. G. (1998). Hedging American contingent claims with constrained portfolios. Finance Stoch. 2 215–258. MR1809521
URI:	http://wrap.warwick.ac.uk/id/eprint/29106

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