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.

Options hedging is an important problem in financial markets. The prevailing approach to hedging first assumes a parametric stochastic model for the dynamics of the underlying asset. The model is then calibrated to observed option prices from the market, based on which various sensitivities are computed and used to hedge the risk of options. Popular choices include local volatility models ([5]), stochastic volatility models ([15], [12], [8]), jump-diffusions and purejump processes ([4], [18], [20]). Despite the prevalence of the model-based approach, it is well understood that model risk can affect the hedging result significantly. Recently, a data-driven approach that doesn't rely on any stochastic model for the underlying asset is proposed.