# Time-inconsistency with rough volatility

## TL;DR

This paper develops a new approach to equilibrium strategies in portfolio optimization with rough volatility, using functional Itô calculus to handle non-Markovian dynamics and deriving explicit solutions for several problems.

## Contribution

It introduces a novel functional Itô calculus method to analyze time-inconsistent portfolio strategies under rough volatility, providing explicit solutions and insights into volatility effects.

## Key findings

- Rough volatility improves trading rule performance over classic models.
- Explicit solutions for mean-variance problems with Volterra processes.
- Volatility roughness significantly impacts equilibrium strategies.

## Abstract

In this paper, we consider equilibrium strategies under Volterra processes and time-inconsistent preferences embracing mean-variance portfolio selection (MVP). Using a functional It\^o calculus approach, we overcome the non-Markovian and non-semimartingale difficulty in Volterra processes. The equilibrium strategy is then characterized by an extended path-dependent Hamilton-Jacobi-Bellman equation system under a game-theoretic framework. A verification theorem is provided. We derive explicit solutions to three problems, including MVP with constant risk aversion, MVP for log returns, and a mean-variance objective with a linear controlled Volterra process. We also thoroughly examine the effect of volatility roughness on equilibrium strategies. Numerical experiments demonstrate that trading rules with rough volatility outperform the classic counterparts.