Optimal Portfolio under Fast Mean-reverting Fractional Stochastic Environment

TL;DR

This paper investigates optimal portfolio strategies in a financial model where volatility exhibits long-range dependence, modeled by a fractional Ornstein-Uhlenbeck process with fast mean reversion, providing asymptotic approximations and optimality results.

Contribution

It introduces a rigorous analysis of portfolio optimization under a fractional OU volatility model with fast mean reversion, extending results to general utility functions.

Findings

First order approximations of value and strategy using martingale distortion

Asymptotic optimality of zeroth order trading strategies

Extension of results to general utility functions

Abstract

Empirical studies indicate the existence of long range dependence in the volatility of the underlying asset. This feature can be captured by modeling its return and volatility using functions of a stationary fractional Ornstein--Uhlenbeck (fOU) process with Hurst index $H \in (\frac{1}{2}, 1)$. In this paper, we analyze the nonlinear optimal portfolio allocation problem under this model and in the regime where the fOU process is fast mean-reverting. We first consider the case of power utility, and rigorously give first order approximations of the value and the optimal strategy by a martingale distortion transformation. We also establish the asymptotic optimality in all admissible controls of a zeroth order trading strategy. Then, we extend the discussions to general utility functions using the epsilon-martingale decomposition technique, and we obtain similar asymptotic optimality results within a specific family of admissible strategies.