A from-scratch implementation of deep hedging that directly optimizes CVaR of the terminal hedging error under realistic trading frictions. Built as a 4-notebook, end-to-end project you can run and extend.
- 1_Theory_and_Background — problem setup, notation, CVaR/OCE risk, objective.
- 2_Data_Simulation — Monte Carlo paths under GBM or Bates (Heston & Merton Diffusion) (toggle), plus feature engineering used by the policy.
- 3_Deep_Hedging_Model-Keras — policy network, RU/OCE head for CVaR, training loop with callbacks, metrics & logging.
-
4_Backtesting_and_Validation — out-of-sample evaluation and plots: tail histograms with VaR/CVaR markers, ES curves across
$α$ , left-tail QQ, ECDF, and effectiveness metrics.
Numbers latest run with
- CVaR@90: -49.31 (hedged) vs -102.53 (zero-hedge)
- VaR@90: -36.57 (hedged) vs -65.79 (zero-hedge)
-
Variance reduction of
$X$ : ≈ 82.6% -
MAE reduction of
$X$ : - 8.5% (we optimize tails, not MAE, consequently MAE suffers) - Mean trading cost per path: ≈ 0.2
- Average turnover (sum over time, per path): ≈ 4.02
Interpretation: the network dominates the left tail (much less negative losses), which is the stated objective. MAE is being reduced—this is expected when optimizing CVaR rather than
$L^1$ . In this run, Monte Carlo simulations were generated intentionally with relatively high volatility through Bates (Merton diffuse & Heston) to assimilate worse tail conditions.
-
Wealth distribution with VaR/CVaR markers (α=0.90)
Shows histograms of$X$ for zero-hedge vs hedged with vertical lines for VaR and CVaR.
- What to look for: hedged histogram remains left of the big zero-spike but has a much lighter extreme left tail; hedged VaR/CVaR lines sit to the right (less negative).
-
Expected Shortfall curve across α (CVaR$_α$ vs
$α$ )
- What to look for: the hedged curve sits above (less negative) the zero-hedge for all
$α∈[0.80,0.99]$ , meaning smaller expected shortfall everywhere in the tail.
- Left-tail QQ plot (hedged quantiles vs zero-hedge, 0–30% tail)
- What to look for: the curve lies below the 45° line in the far left region → hedged left tail is thinner than baseline.
- Empirical CDF (zoomed to left/center)
- What to look for: hedged ECDF dominates (is to the right) near loss regions → better outcomes for adverse quantiles.
-
Stakeholder view (premium-shifted)
Overlay$V_T^{true} := P_0 + V_T$ against$Z_T$ . This reframes wealth with a premium proxy$P_0≈E[Z_T]$ so stakeholders can compare “price-paid vs payoff”.
- Simulate market under GBM or Heston (stochastic vol).
- Features: price/scaled returns, moneyness, realized vol proxy, moneyness, time-to-maturity, call/put (example set of six).
- Policy network outputs trades over time; proportional costs apply.
- Compute terminal hedging error
$X=V_T-Z_T$ . -
RU/OCE head with parameter
$τ$ implements CVaR utility:- Loss per path:
$\ell(X)=\frac{(τ-X)^+}{1-α}-τ$ . - Train
$τ$ jointly with the policy.
- Loss per path:
- Callbacks: ReduceLROnPlateau, EarlyStopping, ModelCheckpoint (best CVaR).
- Out-of-sample eval: VaR/CVaR, ES curves, tail QQ/ECDF, cost & turnover.
# 1) Create env
python -m venv .venv
source .venv/bin/activate # (Windows: .venv\Scripts\activate)
# 2) Install
pip install -r requirements.txt
# core: numpy, pandas, matplotlib, seaborn, tensorflow>=2.12, tqdm
# 3) Execute notebooks in order
# 1_Theory_and_Background.ipynb (read-only)
# 2_Data_Simulation.ipynb (choose GBM/Batses, path counts, seeds)
# 3_Deep_Hedging_Model-Keras.ipynb (train; saves weights + npz eval)
# 4_Backtesting_and_Validation.ipynb (plots + metrics)Artifacts
- Trained weights:
results/best_tail_by_cvar.weights_x.h5 - Test evaluation arrays:
results/hedging_eval_test_keras_x.npzwithV_T, Z_T
-
Risk level:
$α = 0.90$ (CVaR$_{90}$) -
Objective: RU/OCE CVaR of
$X$ (+ tiny MAE regularizer with$β=0.01$ ) - Optimizer: Adam (lr ≈ 1e-3 with ReduceLROnPlateau → min 1e-5)
- Epochs: up to 150 with early stopping (best around ~100–150)
- Batch: 400
- Costs: proportional; turnover tracked as a metric
-
Baseline: zero-hedge
$X_0=-Z_T$
Note on MAE: pushing MAE down meaningfully conflicts with tail optimization. Experimented with large
$β$ and saw CVaR get worse. For a CVaR-first portfolio, keeping$β$ tiny (or even$β = 0 $ ) is the right call.
- Thinner left tail: hedged CVaR and VaR less negative than baseline.
-
ES curve dominance: hedged CVaR$_α$ above baseline across
$α$ grid. -
Stable
$τ$ (RU head) and smooth validation CVaR. - Reasonable costs/turnover: tail gains shouldn’t come from infinite churning.
-
Portfolio of options: vectorized
$Z_T$ and multi-asset underlying. - Richer features: implied/realized vol, term-structure snippets, skew.
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Risk target sweeps: train separate policies for
$α∈{0.85,0.90,0.95}$ and plot the Pareto frontier between CVaR and MAE/variance. - Alternate markets: rough volatility, jumps, or historical resampling.
- Real Market data: Use the trained model to assess hedging real options data.
- Deep Hedging (Buehler et al., 2019), and the OCE formulation used for CVaR-style objectives.
- JPM/industry decks that showcase ES curves, tail QQ, and VaR/CVaR overlays as the right diagnostics for tail-risk control.
MIT
Georgios Drosogiannis MSc Applied Math