This paper extends the deep hedging framework to manage index option portfolios by incorporating the full implied volatility (IV) surface and multiple hedging instruments, including options. The authors build on prior work by François et al. (2024) and introduce a realistic market simulator based on the JIVR model, which captures the joint dynamics of S&P 500 returns and the IV surface. The hedging strategy is optimized via deep reinforcement learning using a recurrent neural network with LSTM cells and feedforward layers, trained to minimize risk measures such as MSE, SMSE, and CVaR while accounting for transaction costs and a soft constraint to prevent speculative behavior.

Numerical experiments focus on hedging a short ATM straddle position with a 63-day maturity, using the underlying asset and a longer-maturity ATM call as hedging instruments. The RL agent is trained on simulated data from 1996-2020 and tested out-of-sample on historical data from 2021-2023. Results show that the RL approach consistently and substantially outperforms traditional delta and delta-gamma hedging strategies across various market conditions and transaction cost levels. The key drivers of this outperformance are the RL agent's ability to make smaller, more gradual trades and to adapt to the time-varying variance risk premium embedded in option prices, which myopic Greeks-based methods ignore.

The study also highlights the critical role of IV surface information: RL agents without access to the full surface fail to outperform delta-gamma hedging, whereas those with full information achieve significant improvements. This demonstrates that leveraging the entire IV surface enables more informed and adaptive risk management, making the proposed framework a powerful tool for hedging index option portfolios in realistic market settings.