Robust and Efficient Deep Hedging via Linearized Objective Neural Network

TL;DR

The paper tackles the practical bottlenecks of deep hedging—computational load, sensitivity to noisy data, and optimization difficulty—by introducing DHLNN, a nested optimization framework with a linearized objective and periodic fixed-gradient updates. It integrates trajectory-wide optimization and Black-Scholes Delta anchoring to align neural hedging with financial theory, while enabling stable inner updates and reduced training time. The approach yields faster convergence, improved stability, and superior hedging performance on both synthetic and real market data, including path-dependent Lookback options under varying transaction costs and volatilities. This framework offers a scalable, interpretable, and theoretically grounded solution for robust risk management in dynamic markets, with potential extensions to multi-asset portfolios and reinforcement-learning–based policy optimization.

Abstract

Deep hedging represents a cutting-edge approach to risk management for financial derivatives by leveraging the power of deep learning. However, existing methods often face challenges related to computational inefficiency, sensitivity to noisy data, and optimization complexity, limiting their practical applicability in dynamic and volatile markets. To address these limitations, we propose Deep Hedging with Linearized-objective Neural Network (DHLNN), a robust and generalizable framework that enhances the training procedure of deep learning models. By integrating a periodic fixed-gradient optimization method with linearized training dynamics, DHLNN stabilizes the training process, accelerates convergence, and improves robustness to noisy financial data. The framework incorporates trajectory-wide optimization and Black-Scholes Delta anchoring, ensuring alignment with established financial theory while maintaining flexibility to adapt to real-world market conditions. Extensive experiments on synthetic and real market data validate the effectiveness of DHLNN, demonstrating its ability to achieve faster convergence, improved stability, and superior hedging performance across diverse market scenarios.

Figures (15)

• Figure 1: Illustration of the hedging procedure for a European call option. The option has a strike price of $100$, a maturity of $10$ days, and an initial stock price of $90$, starting out-of-the-money. Using the Black-Scholes formula, the hedger dynamically adjusts the hedge delta at each step based on the stock price, time to maturity, and volatility (set at $0.2$). The final stock price exceeds the strike price, creating a liability of approximately $99.52$, which is offset to $11.81$ through the hedging adjustments.
• Figure 2: Comparison of convergence performance for deep hedging models across varying training epochs $\{10, 20, 30, 40\}$ on a European option with a strike price of $1.2$. The experiment is conducted under a fixed volatility of $0.1$ and transaction costs of $2 \times 10^{-3}$. The optimal PNL distribution should be narrow and centered around zero, indicating minimal liability discrepancy, robust convergence, and efficient hedging performance with low sensitivity to noise and transaction costs.
• Figure 3: Performance comparison of deep hedging models based on Entropic Loss (a) and Expected Shortfall (b) metrics across training epochs 10, 20, 30, and 40. The experiments are conducted on a European call option with a strike price of 1.2, a transaction cost rate of $2 \times 10^{-3}$, and a fixed volatility of 0.1. Entropic Loss quantifies overall risk, with lower values indicating better uncertainty management. Expected Shortfall measures tail risk, with smaller values reflecting reduced exposure to extreme losses. DHLNN consistently outperforms baseline methods DHNTB, DHMLP, and BSDH across both metrics, demonstrating faster convergence, superior robustness, and more effective hedging under dynamic market conditions.
• Figure 4: Evaluation of hedging performance across different transaction costs ($2 \times 10^{-3}$ and $4 \times 10^{-3}$) for a European option with a strike price of $1.0$. The analysis is conducted over $50$ training epochs, focusing on the distribution of hedging PNL under varying transaction cost rates, with a fixed volatility of $0.1$. A narrower and more concentrated PNL distribution around zero indicates better risk-neutrality and robustness against market frictions.
• Figure 5: Comparison of hedging performance across different transaction costs ${2 \times 10^{-3}, 4 \times 10^{-3}}$ for a European option with a strike price of $1.2$. The analysis, conducted over $50$ training epochs, evaluates the distribution of hedging PNL to assess the robustness of different deep hedging models, with volatility set at $0.1$. The narrower and more centered PNL distribution reflects better hedging efficiency and risk neutrality.