PORTFOLIO OPTIMIZATION WITH DOWNSIDE CONSTRAINTS

• Published in: Mathematical finance Vol. 16; no. 2; pp. 283 - 299
• Main Authors: Lakner, Peter, Ma Nygren, Lan
• Format: Journal Article
• Language: English
• Published: 350 Main Street , Malden , MA 02148 , USA , and 9600 Garsington Road , Oxford OX4 2DQ , UK Blackwell Publishing, Inc 01.04.2006; Wiley Blackwell; Blackwell Publishing Ltd
• Series: Mathematical Finance
• Subjects: Assets; Brownian motion; Clark-Ocone formula; Derivation; downside constraint; Financial analysis; Financial economics; Financial market; Financial models; gradient operator; Hedging; Interest rates; Investors; Malliavin calculus; Mathematical economics; Mathematical models; optimal portfolio selection; Optimization; Portfolio management; Portfolio selection; Studies; Utility functions; Utility maximization
• ISSN: 0960-1627; 1467-9965
• DOI: 10.1111/j.1467-9965.2006.00272.x

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.