Robust Portfolio Optimization in an Illiquid Market in Discrete-Time

Nikolay Andreev

doi:10.3390/math7121147

Abstract

We present a robust dynamic programming approach to the general portfolio selection problem in the presence of transaction costs and trading limits. We formulate the problem as a dynamic infinite game against nature and obtain the corresponding Bellman-Isaacs equation. Under several additional assumptions, we get an alternative form of the equation, which is more feasible for a numerical solution. The framework covers a wide range of control problems, such as the estimation of the portfolio liquidation value, or portfolio selection in an adverse market. The results can be used in the presence of model errors, non-linear transaction costs and a price impact.

Highlights

We consider a robust approach to the constrained portfolio optimization problem in the presence of transaction costs and a price impact on the market
The discrete-time approach has been implemented in Bertsimas & Lo [7] and Almgren [8,9] for the optimal execution problem when an investor chooses the optimal strategy of the portfolio liquidation over a given number of consecutive periods
We present a robust approach to a general portfolio optimization problem

Summary

Introduction

We consider a robust approach to the constrained portfolio optimization problem in the presence of transaction costs and a price impact on the market. Optimal portfolio selection has become a popular topic in the mathematical finance literature since the pioneering works of Samuleson [1] and Merton [2] Both papers consider a stochastic dynamic programming approach to the portfolio optimization problem on an efficient frictionless market with no trading limits. One of the main topics of discussion was the VAR risk metric, which assumed the Samuelson price model based on geometric Brownian motion and underestimated the risk of sudden price movements Another question concerned the significance of market liquidity and transaction costs. We provide a framework for the multi-period portfolio optimization in the presence of model error by introducing the ambiguity in the market price distribution.

Overview of the Main Results

The Robust Portfolio Optimization Problem

Simplified Forms of the Bellman-Isaacs Equation

Conclusions