Portfolio Optimization under Fast Mean-reverting and Rough Fractional Stochastic Environment

TL;DR

This paper investigates portfolio optimization in a fractional stochastic volatility environment, focusing on the fast mean-reverting and rough regimes characterized by Hurst index less than 0.5, revealing unique correction terms in the asymptotic expansion.

Contribution

It extends the analysis of portfolio optimization to the less-studied rough fractional environment with H<0.5, identifying the specific correction term structure in this regime.

Findings

For H<0.5, the first order correction involves a single deterministic term of order √ε.

The correction term differs from the slow and fast regimes with H>0.5, showing unique behavior.

The study provides asymptotic expansions for the value function under rough fractional stochastic volatility.

Abstract

Fractional stochastic volatility models have been widely used to capture the non-Markovian structure revealed from financial time series of realized volatility. On the other hand, empirical studies have identified scales in stock price volatility: both fast-time scale on the order of days and slow-scale on the order of months. So, it is natural to study the portfolio optimization problem under the effects of dependence behavior which we will model by fractional Brownian motions with Hurst index $H$, and in the fast or slow regimes characterized by small parameters $\eps$ or $\delta$. For the slowly varying volatility with $H \in (0,1)$, it was shown that the first order correction to the problem value contains two terms of order $\delta^H$, one random component and one deterministic function of state processes, while for the fast varying case with $H > \half$, the same form holds at order $\eps^{1-H}$. This paper is dedicated to the remaining case of a fast-varying rough environment ($H < \half$) which exhibits a different behavior. We show that, in the expansion, only one deterministic term of order $\sqrt{\eps}$ appears in the first order correction.