The Batch Size Is Part of the Risk Measure: Entropic Deep Hedging with an Explicit Sample Size

Despite their importance in risk-averse learning, entropic risk objectives remain hard to train correctly when rare losses are huge, as they are in option hedging. Deep hedging libraries such as pfhedge estimate the risk on each minibatch; we prove that each such step is an unbiased stochastic gradient step on ρn, a risk measure that changes with the batch size n, so that at λ = 2 a one-parameter delta hedge of an at-the-money call, trained by minibatch gradient descent, moves by 2.7 volatility points between 32 and 8192 paths per batch. In this work, we propose TRACE (tail risk agents, checked by evidence), a hedging agent trained on ρm, the entropic risk of blocks of m outcomes, with m stated as part of the risk preference. TRACE combines two ingredients: 1. block estimation of ρm, which removes the dependence on the batch size (the delta hedge then moves by 0.05 points, and the hedge of TRACE itself stays within 0.15 points from 512 to 8192 paths per batch, where with the batch estimator it rises by 0.62), and 2. a residual policy that sees the model delta and becomes a no-trade band anchored at Whalley and Wilmott when trading costs money; an ablation finds that its inputs, which include the model delta, carry most of its gain. We implement TRACE in TensorFlow and evaluate it on 4.2 million held-out paths per market that no training used: it lowers the entropic risk ρ512 of deep hedging networks by 2.7% to 4.6% in three markets, or 0.12 to 0.18 volatility points of premium, and matches published band networks. Our main predictions were fixed before the runs, and the failures, under stress and on fifty years of S&P 500 paths, are reported.