A dynamic programming approach to path-dependent constrained portfolios

This paper introduces a methodology to produce analytical solutions to an expected utility optimization problem with path-dependent constraints on wealth. This is achieved via a combination of dynamic programming and financial derivatives. The paper focuses on solving the case of a Value at Risk constraint on the running minimum of the wealth process. The optimal wealth is shown to be a barrier-type contingent claim on the unconstrained optimal wealth; the optimal investment strategy and value function follow similarly. A comparison of Value at Risk constraints between terminal wealth and the running minimum of wealth demonstrates a difference of up to 30% on risky asset allocation. Other meaningful examples of interest for investment managers are briefly described.