Portfolio Optimization With Downside Constraints

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.

Optimal Consumption and Investment with Labor Income and European/American Capital Guarantee

“…The paper by Lakner and Nygren [7] considers a similar problem. They maximize expected utility from consumption and terminal wealth when both the consumption rate and terminal wealth must not fall below a given minimum consumption level and a minimum wealth level, respectively.…”

A Dynamic Programming Approach to Constrained Portfolios

“…Korn and Wiese (2008) (2006) where not only the terminal wealth but also a continuous consumption rate is restricted downwards in a strict sense. As all other above-mentioned papers with constraints on wealth, Lakner and Nygren (2006) use the martingale approach. We distinguish ourselves by using dynamic programming and by allowing for non-strict constraints.…”

“…We now decompose this two-period problem into two one-period problems and a one-dimensional maximization problem. The line of arguments is adapted from Lakner and Nygren (2006), but since our constraints are not strict we need to deal with ε 1 and ε 2 in the right way. We start with an admissible pair (π, C) and define…”

“…Here, the fundamental ideas are adapted from Lakner and Nygren (2006), but since we allow for non-strict constraints the results are new. We introduce several ways to relax constraints on intermediate consumption.…”

“…claims of policy holders) that are modeled via constraints on consumption and/or wealth. Problems with no constraints/utility on/from intermediate consumption (Basak and Shapiro (2001)) and with strict constraints on intermediate consumption and/or wealth (Lakner and Nygren (2006)) are included as special cases. There exists an extensive literature on various types of constrained portfolio problems.…”