Large deviations and portfolio optimization

TL;DR

This paper explores risk management and portfolio optimization using large deviation theory, emphasizing the importance of the full loss distribution and time horizon, and extending classical models to non-Gaussian assets.

Contribution

It introduces a comprehensive framework for portfolio optimization that incorporates large deviations and non-Gaussian asset distributions using functional integral methods.

Findings

Large deviations significantly impact risk assessment and portfolio strategies.

Exponential law models effectively describe daily price variations.

Extended mean-variance theory accounts for non-Gaussian asset correlations.

Abstract

Risk control and optimal diversification constitute a major focus in the finance and insurance industries as well as, more or less consciously, in our everyday life. We present a discussion of the characterization of risks and of the optimization of portfolios that starts from a simple illustrative model and ends by a general functional integral formulation. A major theme is that risk, usually thought one-dimensional in the conventional mean-variance approach, has to be addressed by the full distribution of losses. Furthermore, the time-horizon of the investment is shown to play a major role. We show the importance of accounting for large fluctuations and use the theory of Cram\'er for large deviations in this context. We first treat a simple model with a single risky asset that examplifies the distinction between the average return and the typical return, the role of large deviations in multiplicative processes, and the different optimal strategies for the investors depending on their size. We then analyze the case of assets whose price variations are distributed according to exponential laws, a situation that is found to describe reasonably well daily price variations. Several portfolio optimization strategies are presented that aim at controlling large risks. We end by extending the standard mean-variance portfolio optimization theory, first within the quasi-Gaussian approximation and then using a general formulation for non-Gaussian correlated assets in terms of the formalism of functional integrals developed in the field theory of critical phenomena.