Robust deep hedging
Robust deep hedging
This paper addresses pricing and hedging under parameter uncertainty for a class of Markov processes called nonlinear generalized affine (NGA) processes, which include Black-Scholes and CEV models as special cases. The authors first develop a theoretical framework for robust pricing by establishing a dynamic programming principle and linking the nonlinear expectation to a variational form of the Kolmogorov equation. This allows for fast numerical pricing using standard PDE methods. The main contribution is a novel deep hedging approach that efficiently solves the hedging problem under parameter uncertainty. The method trains neural networks on simulated paths where parameters are randomly sampled from specified intervals at each time step, reflecting the ambiguity set of possible models. Numerical experiments on simulated data show that robust deep hedging significantly outperforms classical hedging strategies when parameter uncertainty is present, with smaller relative hedging errors. The approach is also applied to real market data from the COVID-19 period, where it demonstrates remarkable robustness and reduced hedging errors compared to traditional delta hedging. The paper provides a sliding-window maximum-likelihood method to estimate parameter intervals for practical implementation. Overall, the work offers a practical and computationally efficient solution for hedging under model uncertainty, with potential applications in risk management and derivative pricing.
Highlights
- 1Introduces nonlinear generalized affine (NGA) processes to model parameter uncertainty in a broad class of Markov processes including Black-Scholes and CEV models.
- 2Establishes a dynamic programming principle and nonlinear Kolmogorov equation for robust pricing, enabling fast numerical solutions.
- 3Proposes a novel deep hedging approach that efficiently solves the hedging problem under parameter uncertainty by training neural networks on paths with randomly sampled parameters.
- 4Demonstrates through simulated and real data (COVID-19 period) that robust deep hedging significantly reduces hedging errors compared to classical methods.
- 5Provides a sliding-window maximum-likelihood method to estimate parameter intervals for real-world applications.
Methods
- MNonlinear generalized affine (NGA) processes: a class of diffusions with parameter uncertainty in drift and volatility coefficients.
- MDynamic programming principle and viscosity solutions for nonlinear Kolmogorov equations to compute robust prices.
- MDeep hedging with feed-forward neural networks (ReLU activation, 4 layers, 256 neurons) trained via stochastic gradient descent to minimize quadratic hedging error.
- MEuler-Maruyama discretization with uniform random sampling of parameters at each time step to approximate the set of possible measures.
Results
- RRobust deep hedging outperforms non-robust hedging strategies under parameter uncertainty, with substantially smaller relative hedging errors.
- RFor at-the-money call options, the robust strategy reduces both mean and standard deviation of hedging errors compared to fixed-parameter hedging.
- RFor butterfly options, robust hedging also shows improved performance, highlighting the method's effectiveness for non-monotone payoffs.
- RIn real-data application during the COVID-19 period, robust deep hedging yields remarkably smaller hedging errors than classical delta hedging.
- RThe computational overhead of robust deep hedging is modest (only about 1.24 times slower than non-robust deep hedging).
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