Pricing Illiquid Assets by Entropy Maximization Through Linear Goal Programming

Abstract

In this contribution we study the problem of retrieving a risk neutral probability (RNP) in an incomplete market, with the aim of pricing non-traded assets and hedging their risk. The pricing issue has been often addressed in the literature finding an RNP with maximum entropy by means of the minimization of the Kullback-Leibler divergence. Under this approach, the efficient market hypothesis is modelled by means of the maximum entropy RNP. This methodology consists of three steps: firstly simulating a finite number of market states of some underlying stochastic model, secondly choosing a set of assets—called benchmarks—with characteristics close to the given one, and thirdly calculating an RNP by means of the minimization of its divergence from the maximum entropy distribution over the simulated finite sample market states, i.e. from the uniform distribution. This maximum entropy RNP must exactly price the benchmarks by their quoted prices. Here we proceed in a different way consisting of the minimization of a different divergence resulting in the total variation distance. This is done by means of a two steps linear goal programming method. The calculation of the super-replicating portfolios (not supplied by the Kullback-Leibler approach) would then be derived as solutions of the dual linear programs.