Robust Deep Hedging Across Market Regimes

# Practical Applications of Deep Hedging


# Practical Applications of Deep Hedging












In the study "Deep Hedging of Long-Term Financial Derivatives" by Alexandre Carbonneau, the author presents a framework for constructing an optimal hedging strategy for a lookback put option with payoff defined as: $$ \Phi(S_T, Z_T) := \max(Z_T - S_T, 0), \quad Z_T = \max(S_0, \dots, S_T), $$ where a trainable parameter set $\theta$ (comprising weight matrices and bias vectors) is optimized within a model $\mathcal{M}(\theta)$. This model outputs the hedging portfolio positions, including holdings in the underlying asset and $m$ available options, at each rebalancing date. The feature vector at each time step $t_n$ is defined as: $$ X_{t_n} := \big(\log(\Bar{H}_{t_n}), \log(Z_{t_n}), V_{t_n}/V_{t_0}\big), $$ where $\Bar{H}_{t_n}$ represents the vector of log-transformed prices of hedging instruments, and $V_{t_n}$ is the value of the hedging portfolio at time $t_n$: $$ V_{t_n} = \Bar{H}_{t_n} \cdot \Bar{w}_{t_n}^h + B_{t_n} \cdot \Bar{w}_{t_n}^b, $$ composed of hedging instruments and a risk-free asset, weighted by $\Bar{w}_{t_n}^h$ and $\Bar{w}_{t_n}^b$ as determined by the model. The optimal hedging strategy minimizes a loss function, where the error measure at each path $i$ of a mini-batch is: $$ \pi_i = \Phi(S_{T,i}, Z_{T,i}) - V_{T,i}. $$ This error is used to update the parameter set $\theta$ via the Adam algorithm, refining the model to achieve an optimal hedging strategy tailored to the specified loss function.

Questions on Practical Implications and Adaptability

While this approach is insightful for long-term derivatives, I am exploring its practical application and raise the following questions:

- Adaptation to Changing Market Regimes: The sensitivities generated by $\mathcal{M}(\theta)$ are inherently influenced by the volatility surface dynamics implied by the sample paths used during model training. In conventional models like Heston, parameter recalibration is required to adapt to shifts in the volatility surface, ensuring stability in the sensitivities and hedging strategies produced. However, neural networks like $\mathcal{M}(\theta)$ lack the interpretability of Heston parameters, making it challenging to assess the stability of the learned sensitivities. 1) How can we ensure that $\mathcal{M}(\theta)$ produces stable hedging strategies under varying market conditions? 2) Is there a method to evaluate or constrain the stability of the deep model's parameters $\theta$ in response to changes in market regimes?

- Impact of Loss Function and Derivative Maturity: If the loss function penalizes only negative differences (i.e., losses), the author notes that the model may increase equity risk exposure when the sample paths exhibit positive expected log-returns of the underlying. This behavior aligns with the concept of time diversification of risk, which assumes that long-term investment horizons reduce the likelihood of significant losses compared to short-term horizons. 1) Does this behavior change when hedging mid/short-term derivatives, where time diversification may be less relevant? 2) Is the propensity to increase equity risk exposure primarily driven by the maturity of the derivative or by the underlying's average log-returns in the sample paths used for training? 3) Would mixing sample paths from different market regimes improve the robustness of the hedging strategy for shorter-term derivatives?

## Answer by Adam C. Jones (score 2)

https://quant.stackexchange.com/a/84083

I can answer 1: There is a lot of literature on the robustification of deep hedging, since standard deep hedging is only as good as the training data you use: the hedging strategy learnt will be optimal for that training data distribution, but that alone doesn't guarantee that the hedging strategy will be good in different distributions. A few papers that come to mind are "Robust Hedging GANs", "Distributional Adversarial Attacks and Training in Deep Hedging", and one of mine which came out recently "Ambiguity-Averse Deep Hedging with Feature Clustering". The two former, like most of the literature, aim to robustify against any distributional shift up to a certain amount, the latter aims to target specific shifts. Re: constraints, the original paper by Buehler et al simply titled "Deep Hedging" discusses this a bit, see their Example 2.1 for imposing vega limits.