Dynamic Hedging Under Stochastic Volatility and Model Uncertainty: PDE Characterization and Regime-Based Evidence

We study dynamic hedging in an incomplete market where the underlying asset follows a stochastic-volatility process and the hedger trades only the stock and the money-market account. The hedging problem is formulated as a multi-stage stochastic control problem with a quadratic terminal-loss objective and is solved through a Hamilton–Jacobi–Bellman framework. For the Heston model, the resulting mean-variance hedge specializes to the Galtchouk–Kunita–Watanabe projection and can be written as the sum of the spot delta and a volatility-risk correction term. We emphasize that this representation is used in the paper as an implementation theorem for our setting, rather than as a new general result. On the numerical side, we compare a finite-difference alternating-direction implicit solver with a Deep Galerkin Method, providing full implementation details for both. The finite-difference solver is the preferred method for the two-state Heston problem because it is faster and more accurate on low-dimensional grids, whereas the neural solver becomes attractive only for higher-dimensional extensions where mesh-based methods become computationally burdensome. In backtests across major S&P 500 market regimes from 2006 to 2022, the stochastic-volatility-aware hedge modestly improves on Black–Scholes hedging during stress episodes, while differences are negligible in calm markets. Across the reported experiments, the PDE-optimal mean-variance hedge is numerically indistinguishable from the recalibrated Heston hedge, indicating that the main value of the framework is theoretical unification and implementation guidance rather than a materially different trading rule in the tested setting. Fixed worst-case robust hedging is overly conservative in the historical sample, although adaptive robustness remains a promising conceptual extension. The main contribution of the paper is therefore a rigorous and implementable unification of multi-stage PDE optimization with stochastic-volatility-aware hedging, together with evidence that the economic value of model sophistication is concentrated in stressed markets.