PORTFOLIO OPTIMIZATION WITH DOWNSIDE CONSTRAINTS

Peter Lakner and Lan Ma Nygren

Mathematical Finance, 2006, vol. 16, issue 2, 283-299

Abstract: We consider the portfolio optimization problem for an investor whose consumption rate process and terminal wealth are subject to downside constraints. In the standard financial market model that consists of d risky assets and one riskless asset, we assume that the riskless asset earns a constant instantaneous rate of interest, r > 0, and that the risky assets are geometric Brownian motions. The optimal portfolio policy for a wide scale of utility functions is derived explicitly. The gradient operator and the Clark–Ocone formula in Malliavin calculus are used in the derivation of this policy. We show how Malliavin calculus approach can help us get around certain difficulties that arise in using the classical “delta hedging” approach.

Date: 2006
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