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Finance and Stochastics
Published in: Finance and Stochastics 4/2013

01-10-2013

Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model

Authors: Vladimir Cherny, Jan Obłój

Published in: Finance and Stochastics | Issue 4/2013

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Abstract

A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (Math. Finance 3:241–276, 1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function and the optimal investment policy for the drawdown problem are given explicitly in terms of their counterparts in the unconstrained problem.
previous article Drift dependence of optimal trade execution strategies under transient price impact
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Appendix
Available only for authorised users
Footnotes
1
More precisely, we can take w to be piecewise constant on [v 0,∞) for any v 0>0, and for the drawdown problem we only consider w(x) for x greater than the initial capital.
 
2
We note, however, that it may fail to provide strategies which are optimal on a finite time horizon, as discussed by Klass and Nowicki [23] in the context of drawdown constraints.
 
3
Having a hurdle rate r means that if a new drawdown constraint is set at time t, then it grows at the rate r for u>t until a new constraint level is achieved; see Guasoni and Obłój [16] for more details.
 
4
Recall from Example 3.4 that \(\tilde{V}\) is also an Azéma–Yor process \(\tilde{V}= M^{F}(V)\) corresponding to an affine F.
 
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Metadata
Title
Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model
Authors
Vladimir Cherny
Jan Obłój
Publication date
01-10-2013
Publisher
Springer Berlin Heidelberg
Published in
Finance and Stochastics / Issue 4/2013
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-013-0209-4

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