Survey_KS_2012

Abstract

We make a methodological contribution by solving these constrained portfolio problems applying dynamic programming. The standard approach to dynamic portfolio optimization with constraints on wealth is the so-called martingale method. The martingale method was developed by Karatzas et al. (1987) and Cox and Huang (1989) as an alternative to dynamic programming. The method decomposes the dynamic optimization problem into a static optimization problem and a dynamic hedging problem where the latter one is usually involved. We study the decisions of an investor (asset manager) operating in a standard Black-Scholes financial market with two assets, a bond (B) and a stock (S). The goal of our paper is to study portfolio problems where additional constraints on wealth and/or consumption such as (1) are imposed. In these cases, the above separation breaks down and thus finding the right conjecture for V is involved. For this reason, we suggest a different approach that reduces the dimension of the problem. It turns out that in many relevant applications involving constraints the investor’s optimal terminal wealth can be expressed as an option-like contract on his unconstrained optimal wealth Y . Hence, we introduce an option (syn. claim) f on Y and relate its price ? to the solution of the HJB equation. We show that the problem simplifies to finding the one-dimensional function f (instead of finding the two-dimensional function V ). To make the reader familiar with our approach, we first study constraints on terminal wealth. The most prominent ones are the ones stemming from CPPI-strategies or VaR-constraints. CPPI (constant proportion portfolio insurance) as well as OBPI (option-based portfolio insurance) are actively used by asset managers who need down-side protection.