Optimal Distributional Trading Gain: Generalizations of Merton's Portfolio Problem with Implications to Bayesian Statistics

Abstract

This paper considers multiple market agents who have distinct distributional opinions about the state price density. We first determine the optimal trading positions of a utility maximizing market taker who trades Arrow-Debreu securities for prices set by the market maker. We use calculus of variations to determine the solution of this problem for a general utility function. The choice of the logarithmic utility function leads to a solution in terms of a likelihood ratio of the densities corresponding to the market taker and the market maker and the resulting optimal utility is the Kullback-Leibler divergence. In particular, we obtain a trivial solution for Merton's portfolio problem in the traditional geometric Brownian motion model and and we show its immediate extension to the multivariate case. A further extension gives a solution for the market driven by a geometric Poisson process. In a market without the market maker, the distributional opinions of market takers reach an equilibrium in the form of the linear mixture of the distributions. We show that when the the result of the outcome is observed, the profit and loss from trading updates agents' bankrolls in a Bayesian fashion, which provides one to one correspondence for the logarithmic utility maximazers' profits and Bayesian statistics. We extend these results to the continuous time setup and show that the bankrolls of agents following Merton optimal portfolio strategy evolve as a posterior distribution.

Keywords: State Price Density, Kullback-Leibler Divergence, Merton's Portfolio Problem, Bayesian Statistics, Utility Maximization, Kelly Criterion, Equilibrium.

JEL Classification: C68, D58, G13