Time-Inconsistency with Rough Volatility
Motivated by recent advances in rough volatility, this paper investigates the impact of roughness on equilibrium feedback strategies for time-inconsistent objectives. Under a general framework embracing non-Markovian and non-semimartingale models, we develop an extended path-dependent Hamilton-Jacobi-Bellman (PHJB) equation system. A verification theorem is provided. By deriving explicit solutions to three problems, including mean-variance portfolio problem (MVP) with constant risk aversion, MVP for log-returns, and an investment/consumption problem with non-exponential discounting, we present that volatility roughness adjusts the equilibrium strategies considerably, up to 40% in certain settings. Since rough volatility models capture the near-term downside risk by fitting the volatility skews, we interpret the adjustments as a hedge for this risk.