Economy – Quantitative Finance – Portfolio Management
Scientific paper
2011-10-28
Economy
Quantitative Finance
Portfolio Management
24 pages
Scientific paper
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 (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 but for a 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. Our approach is very general but has an important limitation in that we assume all admissible wealth processes have a continuous running maximum. This allows us to use Azema-Yor processes. The proofs also rely on convergence properties, in the utility function, of the unconstrained problem which are of independent interest.
Cherny Vladimir
Obloj Jan
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