Deep Gamma Hedging

Paper: arXiv 2409.13567

Abstract

We train neural networks to learn optimal replication strategies for an option when two replicating instruments are available, namely the underlying and a hedging option. If the price of the hedging option matches that of the Black–Scholes model then we find the network will successfully learn the Black-Scholes gamma hedging strategy, even if the dynamics of the underlying do not match the Black–Scholes model, so long as we choose a loss function that rewards coping with model uncertainty. Our results suggest that the reason gamma hedging is used in practice is to account for model uncertainty rather than to reduce the impact of transaction costs.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced mathematics including rough-path theory, Itô calculus, and neural network optimization, yet its empirical approach relies on synthetic data and simulation without real-world backtesting or published code, placing it in the theoretical ‘Lab Rats’ quadrant.

Research Flowchart

  flowchart TD
  A["Research Goal: Why is Gamma Hedging<br>Used in Practice?"] --> B{"Methodology"}
  
  subgraph B ["Key Methodology"]
      B1["Train Neural Networks<br>to Learn Replication Strategies"]
      B2["Use Two Instruments:<br>Underlying & Hedging Option"]
      B3["Define Loss Function<br>Rewarding Model Uncertainty Coping"]
  end
  
  B --> C{"Data & Inputs"}
  
  subgraph C ["Data & Inputs"]
      C1["Black-Scholes<br>Priced Hedging Option"]
      C2["Non-BS Dynamics<br>Underlying Asset"]
  end
  
  C --> D{"Computational Process"}
  
  subgraph D ["Computational Process"]
      D1["Neural Network<br>Optimizes Hedging Strategy"]
      D2["Minimize Loss Function<br>with Uncertainty Reward"]
  end
  
  D --> E{"Key Findings / Outcomes"}
  
  subgraph E ["Key Findings"]
      E1["Network Learns<br>BS Gamma Hedging Strategy"]
      E2["Effective Even When<br>Underlying Dynamics Deviate from BS"]
      E3["Gamma Hedging's Purpose:<br>Manage Model Uncertainty"]
  end
  
  E --> F["Conclusion: Gamma hedging is used<br>to account for model uncertainty"]

Cited References (19)

1. Deep Hedging (2018) — link
2. Lecture notes for 7CCMFM18 , M achine L earning (2022)
3. Deep Hedging: Hedging Derivatives Under Generic Market Frictions Using Reinforcement Learning (2019) — link
4. Gamma Hedging and Rough Paths (2023) — link
5. A Nonparametric Approach to Pricing and Hedging Derivative Securities Via Learning Networks (1994) — link
6. Lecture Notes Learning to Trade III: Deep Hedging with Impact, Deep Bellman Hedging, Open Research Questions (2022) — link
7. Deep Hedging: Learning to Simulate Equity Option Markets (2019) — link
8. Deep Bellman Hedging (2022) — link

0 of 19 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.