Robust Utility Maximization Under Model Uncertainty via a Penalization Approach

34 Pages Posted: 22 Jun 2020 Last revised: 7 Jul 2020

See all articles by Ivan Guo

Ivan Guo

Monash University - School of Mathematical Sciences

Nicolas Langrené

BNU-HKBU United International College

Gregoire Loeper

BNP Paribas; Monash University - School of Mathematical Sciences; Monash University - Monash Centre for Quantitative Finance and Investment Strategies; Ecole Polytechnique, Palaiseau - CMAP CNRS-UMR 7641 and Ecole Polytechnique

Wei Ning

Monash University - School of Mathematical Sciences; Monash University - Monash Centre for Quantitative Finance and Investment Strategies

Date Written: May 28, 2020

Abstract

This paper addresses the problem of utility maximization under uncertain parameters. In contrast with the classical approach, where the parameters of the model evolve freely within a given range, we constrain them via a penalty function. We show that this robust optimization process can be interpreted as a two-player zero-sum stochastic differential game. We prove that the value function satisfies the Dynamic Programming Principle and that it is the unique viscosity solution of an associated Hamilton–Jacobi–Bellman–Isaacs equation. We test this robust algorithm on real market data. The results show that robust portfolios generally have higher expected utilities and are more stable under strong market downturns. To solve for the value function, we derive an analytical solution in the logarithmic utility case and obtain accurate numerical approximations in the general case by three methods: finite difference method, Monte Carlo simulation, and Generative Adversarial Networks.

Keywords: robust portfolio optimization, differential games, HJBI equation, Monte Carlo, GANs

JEL Classification: C45, C61, C72

Suggested Citation

Guo, Ivan and Langrené, Nicolas and Loeper, Gregoire and Ning, Wei, Robust Utility Maximization Under Model Uncertainty via a Penalization Approach (May 28, 2020). Available at SSRN: https://ssrn.com/abstract=3612503 or http://dx.doi.org/10.2139/ssrn.3612503

Ivan Guo

Monash University - School of Mathematical Sciences ( email )

Clayton Campus
Victoria, 3800
Australia

Nicolas Langrené

BNU-HKBU United International College ( email )

Zhuhai
China

HOME PAGE: http://staff.uic.edu.cn/nicolaslangrene/en

Gregoire Loeper

BNP Paribas ( email )

Paris
France

Monash University - School of Mathematical Sciences ( email )

Clayton Campus
Victoria, 3800
Australia

Monash University - Monash Centre for Quantitative Finance and Investment Strategies ( email )

9 Rainforest Walk
Clayton Campus
Monash University, Victoria 3800
Australia

HOME PAGE: http://https://www.monash.edu/science/quantitative-finance

Ecole Polytechnique, Palaiseau - CMAP CNRS-UMR 7641 and Ecole Polytechnique ( email )

Route de Saclay
Palaiseau, 91128
France

Wei Ning (Contact Author)

Monash University - School of Mathematical Sciences ( email )

Clayton Campus
Victoria, 3800
Australia

Monash University - Monash Centre for Quantitative Finance and Investment Strategies ( email )

9 Rainforest Walk
Clayton Campus
Monash University, Victoria 3800
Australia

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