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 .
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 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 .
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but for a
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. 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