  This paper proposes a deep delta hedging framework for options, utilizing
neural networks to learn the residuals between the hedging function and the
implied Black-Scholes delta. This approach leverages the smoother properties of
these residuals, enhancing deep learning performance. Utilizing ten years of
daily S&P 500 index option data, our empirical analysis demonstrates that
learning the residuals, using the mean squared one-step hedging error as the
loss function, significantly improves hedging performance over directly
learning the hedging function, often by more than 100%. Adding input features
when learning the residuals enhances hedging performance more for puts than
calls, with market sentiment being less crucial. Furthermore, learning the
residuals with three years of data matches the hedging performance of directly
learning with ten years of data, proving that our method demands less data.
