Portfolio Optimization in Incomplete Markets and Price Constraints Determined by Maximum Entropy in the Mean

Abstract

A solution to a portfolio optimization problem is always conditioned by constraints on the initial capital and the price of the available market assets. If a risk neutral measure is known, then the price of each asset is the discounted expected value of the asset's price under this measure. But if the market is incomplete, the risk neutral measure is not unique, and there is a range of possible prices which can be identified as the bid-ask range. We present in this paper an effective method to determine the prices of a set of assets in incomplete markets, or when the class of risk neutral measures are not known, and such that these prices comply with the cost constraints of a portfolio optimization problem. Our workhorse is the method of maximum entropy in the mean to adjust a distortion function from bid-ask market data. This distortion function plays the role of a risk neutral measure, which is used to price the assets, and the distorted probability that it determines reproduces bid-ask market values. We carry out numerical examples to study the effect on portfolio returns of the computation of prices of the assets conforming the portfolio with the proposed methodology.

Keywords: Econophysics; distorsion function; maximum entropy in mean; portfolio optimization; risk neutral measures; asset pricing

JEL Classification: G11, G12, G13