Neural Stochastic Differential Equations for Model-Free Option Hedging: Convergence, Calibration, and Risk Bounds

Mysore supreeth, Manish Mehta

Keywords: Neural SDEs, option hedging, quantitative finance, Black-Scholes, SABR, deep hedging, implied volatility, SVI, Value-at-Risk, Wasserstein regularization

TL;DR: Neural SDE Hedger learns drift/volatility from data with $O(n^{-1/2}\log n)$ convergence to minimal-variance hedge and provable SVI arbitrage-free and VaR bounds.

Abstract: We develop a mathematically rigorous framework for model-free option hedging using neural stochastic differential equations (Neural SDEs). Traditional parametric approaches like Black-Scholes assume specific dynamics, while pure deep hedging lacks theoretical guarantees. Our Neural SDE Hedger learns the drift and volatility functions directly from market data while providing provable risk bounds. We establish three main theoretical results: (1) the Neural SDE hedge converges to the minimal-variance hedge at rate $O(n^{-1/2} \log n)$ in mean-square hedging error as training samples $n$ grow; (2) the implied volatility surface generated by the Neural SDE satisfies Gatheral's SVI arbitrage-free conditions with probability approaching 1; (3) a novel Value-at-Risk bound showing the worst-case hedging loss under the learned model is within a factor $(1 + \epsilon)$ of the true minimal risk for $\epsilon = O(n^{-1/4})$. We implement NeuralSDEHedge using adjoint-based SDE solvers with Wasserstein-regularized training. On S&P 500 options (2015--2024), our method achieves 31\% lower hedging P&L variance than Black-Scholes delta hedging, 18\% lower than SABR, and 12\% lower than deep hedging baselines, while providing the first provable risk guarantees for neural hedging strategies.

Submission Number: 151