1. Buehler et al. Deep Hedging Framework

Classical Black-Scholes continuous delta-hedging assumes frictionless markets. Under discrete trading and proportional transaction costs, continuous delta hedging guarantees infinite bankruptcy. Deep Hedging (Buehler et al., 2019) parameterizes discrete hedge ratios $\delta_t^\theta$ using neural networks optimized under coherent convex risk measures.

2. Expected Shortfall (CVaR) Loss Objective

Let $L_T(\theta)$ be the terminal hedging PnL shortfall at maturity $T$. We minimize Conditional Value-at-Risk ($\text{CVaR}_\alpha$):

$$\text{CVaR}_\alpha(L_T) = \inf_{z \in \mathbb{R}} \left\{ z + \frac{1}{1 - \alpha} \mathbb{E}\left[ \max(L_T - z, 0) \right] \right\}$$
import torch
import torch.nn as nn

class DeepHedgingPolicyNet(nn.Module):
    """
    Neural Policy determining discrete delta-hedge ratio delta_t given option state.
    """
    def __init__(self, hidden_dim: int = 64):
        super().__init__()
        self.net = nn.Sequential(
            nn.Linear(3, hidden_dim), # [Asset Px, Time Rem, Current Holding]
            nn.SiLU(),
            nn.Linear(hidden_dim, 1)
        )

    def forward(self, state: torch.Tensor) -> torch.Tensor:
        return self.net(state)

3. Institutional Summary

  • Optimize CVaR over Variance: Minimize tail loss Expected Shortfall ($\text{CVaR}_{95\%}$) rather than MSE when hedging exotic barrier options.