Model-Free Deep Hedging with Transaction Costs and Light Data Requirements

Gabriel Turinici; Pierre Brugi\`ere

REVIEW 3 major objections 5 minor 14 references

Model-Free Deep Hedging with Transaction Costs and Light Data Requirements

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that 256 asset price trajectories are enough to train a neural network that outperforms Black-Scholes and Leland hedging in a geometric Brownian motion market with transaction costs.

desk verdict The small-data training idea is genuinely worth a look, but the paper's headline claim of significant outperformance over Black-Scholes and Leland is not supported by its own tables, which show the neural network losing at low transaction costs and winning by a gap smaller than sampling noise at high costs. read the letter →

arxiv 2505.22836 v1 pith:X2QV3RNV submitted 2025-05-28 q-fin.MF q-fin.ST

Pierre Brugière , Gabriel Turinici This is my paper

classification q-fin.MFq-fin.ST MSC 91G2091G6068T07

keywords deephedgingtransactioncostsneuralnetworksBlack-ScholesLelandmodelgeometricBrownianmotionsmall-sampletrainingoption

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a neural-network hedging strategy can be trained on as few as 256 asset-price trajectories, and that in a geometric Brownian motion market with proportional transaction costs it outperforms both the discretized Black-Scholes delta hedge and Leland's volatility-adjusted hedge. The training uses real-world drift rather than a risk-neutral measure, and minimizes the standard deviation of the discounted terminal wealth of the hedging account. The paper also shows that 256 overlapping windows cut from a single 285-point price series train the network just as well, and that the trained network lowers both the mean and the dispersion of hedging costs once transaction costs reach 0.5% or more (with 90 hedging points) or 1% or more (with 30 points). If the claim holds, deep hedging becomes calibratable from very small datasets in simple markets, removing the need for the $10^{5}$–$10^{6}$ trajectories used in earlier deep-hedging studies.

What carries the argument

The load-bearing object is the delta network $\mathrm{DeltaNN}$, a feedforward ReLU network mapping $(K/X_t, T-t, H_{t-\Delta t})$ to the next holding $H_t$. Including the previous hedge position makes the rebalancing decision depend on the transaction cost $\alpha |H_t - H_{t-\Delta t}| X_t$, which is the feature that lets the network adapt hedging frequency under costs. The loss is the standard deviation of the discounted terminal wealth $Z_T$ over a batch, and training is done under the real-world measure P with drift $\mu\neq r$, so no risk-neutral simulation is needed. The theoretical anchor is the Leland-type decomposition that splits the discrete hedging error into gamma-weighted realized variance minus its expected value plus transaction costs; this decomposition explains why a fixed implied-volatility hedge is suboptimal and why the learned delta may improve on it.

What would settle it

Re-train the network on 100 different random seeds, each with a fresh 256-path training set and a fresh 256-path test set, at the configuration $\alpha=2\%$, frequency 30; if the network's median test-set standard deviation of $Z_T$ is not below Leland's 0.98%, the claim that 256 trajectories suffice is not supported.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a simple three-input neural network, taking moneyness $K/X_t$, time to maturity $T-t$, and the pre-existing hedge position $H_{t-\Delta t}$, and trained to minimize the standard deviation of the discounted terminal wealth $Z_T$, converges to a good hedging rule from only 256 GBM trajectories. At transaction cost $\alpha=2\%$ with 30 rebalancing points, the test-set mean hedging cost is 6.83% for the network versus 7.09% for Leland and 7.60% for Black-Scholes, with standard deviations 0.97%, 0.98%, and 1.55%; at 90 rebalancing points the advantage appears from $\alpha=0.2\%$ onward on the mean and widens sharply at $\alpha=2\%$ (mean 8.11% vs 8.81%, standard deviation 0.73% vs 0.88%). The paper also establishes that training on 256 overlapping sequences extracted from a single 285-point series reproduces the simulated-data performance, although applying the same protocol to a one-year S&P 500 series fails without volatility-related inputs.

Load-bearing premise

The reported advantage of the neural network over Black-Scholes and Leland is computed from a single training seed and a single 256-path test set, with no confidence intervals, so the small advantages (e.g., 0.97% vs 0.98% standard deviation) could be sampling noise.

Editorial extensions

If this is right

In a GBM market, a deep hedge can be calibrated from a few hundred paths rather than $10^5$–$10^6$, cutting the data requirement by three to four orders of magnitude.
The neural hedge reduces mean hedging cost relative to Leland at transaction costs of 1% and 2% (30 rebalancing points) and at 0.2% and above (90 points), and it reduces the standard deviation of hedging cost relative to Black-Scholes in all cost regimes.
Because training works under the real-world measure and on overlapping windows from a single realized price series, calibration from short windows of historical raw data becomes plausible without synthetic data augmentation.
The failure on the one-year S&P 500 series, attributed to volatility regime changes, indicates that the small-sample benefit is tied to the GBM assumption and to stationary volatility.
The protocol of training on 256 overlapping sequences from one series suggests that even a single episode of intraday data may suffice to calibrate a hedge for simple underlyings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An extension the authors leave implicit: the same 256-path protocol could be tested on other payoffs (digital, barrier, spread) to see whether the small-sample advantage is generic or specific to European calls; a negative result there would bound the scope of the claim.
A practical recipe suggested by the paper: when the training set has few in-the-money paths, train on several strikes at once by minimizing the sum of the standard deviations, as sketched in equation (21); this could make the light-data regime work for out-of-the-money options.
The reliance on a single seed means the headline improvement margins are not statistically quantified; a multi-seed benchmark against Leland would convert the qualitative claim into an operational rule of thumb for when deep hedging cheaply beats explicit formulas.
If the result survives re-sampling, it suggests that for simple underlyings, market data alone (no parametric model, no GAN or VAE augmentation) can drive deep hedging, which would lower the barrier to adoption for small desks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The genuinely new thing here is the empirical demonstration that a simple neural network can be calibrated for delta hedging from as few as 256 trajectories, including overlapping windows. Prior deep hedging work used 10^5 to 10^6 paths, so this is a real data point. The authors also write up the Leland formula and the Soner-Shreve-Cvitanic no-arbitrage result carefully, and the appendix proof is a useful self-contained reminder. They are honest about the real-data experiment failing, which is a point in their favor.\n\nThe soft spots are load-bearing. The abstract claims the network 'significantly outperforms' both Black-Scholes and Leland in the GBM framework, but the paper's own test tables contradict that. At freq=30 and alpha=1%, the network's standard deviation is 0.87% against Leland's 0.79%. At alpha=2%, the gap is 0.97% vs 0.98%, about one basis point. With 256 test paths and an estimated standard deviation near 1%, the standard error of the sample standard deviation is roughly 4 to 5 basis points, so a 1bp gap is pure noise. The results at freq=90 are similarly mixed: the network only beats both benchmarks at alpha=2%, and loses at lower costs. The claim should be restated to the regimes where the network actually wins, and the experiments need multiple seeds and confidence intervals. There is also an internal inconsistency: the model is described as not setting the risk-free rate to zero, but the GBM tables say r=0. The real-data section is an acknowledged failure; that is a limitation, but it undercuts the 'real-time series' motivation.\n\nThe math and the citation pattern look fine; the theory is standard and the relevant literature is cited. The central empirical claim is just statistically unsecured as presented. This is fixable with more compute and a more careful abstract.\n\nThe paper deserves a serious referee: the small-data observation is interesting enough to warrant checking, and the theoretical appendix is solid. But it should not be accepted without major revision. I would not cite it in its current form, but I would read a revised version. Bring it to reading group if you want to discuss what counts as 'sufficient' data in deep hedging; it will generate a good argument.

Referee Report

3 major / 5 minor

Summary. The paper proposes a deep hedging approach that trains a small neural network on as few as 256 simulated Geometric Brownian Motion trajectories, minimizing the standard deviation of the discounted terminal wealth of a short option position, with proportional transaction costs. It compares the learned strategy against discretized Black–Scholes delta hedging and Leland's volatility-adjusted delta hedge, reporting tables of means and standard deviations for transaction costs from 0% to 2% and hedging frequencies of 30 and 90. The paper also investigates calibration on 256 overlapping sequences extracted from a single long simulated path, and attempts an application to S&P 500 data. The central claim is that 256 trajectories suffice to train a network that significantly outperforms both Black–Scholes and Leland in the GBM framework. The theoretical sections revisit continuous-time replication, discrete gamma costs, and Leland's adjustment, and Appendix B proves a version of Soner–Shreve–Cvitanic on infinite transaction costs for fixed implied volatility.

Significance. If the central claim were fully substantiated, the result would be practically interesting: it would show that deep hedging can be calibrated from very small datasets, in contrast to earlier work that used of the order of 10^5–10^6 trajectories, and it would support the idea of calibrating hedge policies from short market windows. The paper has genuine strengths: the theoretical discussion of discrete hedging and gamma costs is informative, the link to Leland's formula is clearly developed, Appendix B provides a self-contained proof of a standard result, and the authors are transparent about the failure of the real-data experiment in Section 6.2. The main weakness is statistical: the headline outperformance claim rests on a single training seed and a single test set, with no confidence intervals or repeated-seed analysis, and the paper's own point estimates do not consistently favor the neural network over Leland across the reported configurations. The significance of the paper therefore depends on whether the small-data advantage survives re-sampling; the present evidence does not establish that.

major comments (3)

[§5.3, Tables 1–2] The central claim that the neural network 'significantly outperforms' both Black–Scholes and Leland is not supported by the reported experiments. All results come from a single training seed (seed_value=42, §5) and a single 256-path test set per configuration, with no confidence intervals or repeated seeds. The point estimates are also not consistently in the neural network's favor: in Table 1, test set, α=1%, the neural network standard deviation is 0.87% versus Leland's 0.79%, and at α=2% the gap is only 0.97% versus 0.98%, about one basis point. For n=256 and a standard deviation near 0.01, the standard error of the sample standard deviation is roughly σ/sqrt(2(n−1)) ≈ 4.4 basis points, larger than most reported gaps. In Table 2, test set, at α=0.5% the neural network is worse than Leland (0.55% vs 0.45%) and at α=1% they are tied (0.60% vs 0.60%), directly contradicting the sentence in §5.3 that the neural network 'further improves the Leland approach for transaction costs at 0.5% and above.' The authors should provide multiple-seed results, confidence intervals or error bars, and statistical tests, and should restrict the abstract's 'significantly outperforms' to configurations where the evidence actually supports it.
[Introduction, §5, Tables 1–4] The paper lists as a contribution that 'we do not write our model using a risk-free rate of zero,' yet every experimental table and the simulation section use r=0. This is a direct inconsistency between a stated novelty and the reported evidence. If nonzero-r experiments exist, they should be reported; otherwise the claim should be reworded, for example to say that the framework accommodates nonzero r but the present experiments set r=0 for simplicity.
[§6.2 and Abstract] The only real-data experiment, on S&P 500 total return data, is reported as a failure: the trained model 'produces unrealistic deltas that are not even close' to Black–Scholes deltas, attributed to changing volatility. The authors honestly acknowledge this in §6.2 and the Conclusion, but the Abstract still claims the work 'suggests the potential for more practical and simple implementation on real-time financial series.' As written, that suggestion is not supported by any successful real-market result, and the negative finding should be reflected in the abstract or the claim should be explicitly restricted to the simulated GBM setting.

minor comments (5)

[Appendix B, Lemma 1 proof] The proof says that a continuous function on a compact set 'is also absolutely continuous'; this is false in general, although the needed conclusion (uniform continuity) does follow from compactness. Please correct the wording.
[Abstract and §1] The phrase 'of the order of 105 or 106' appears without superscripts; please format as 10^5 or 10^6.
[§5.3] The sentence 'with 90 hedging points, its benefits become apparent for costs of 0.5% and above' is not consistent with Table 2, where the neural network has a higher standard deviation than Leland at α=0.5% and an equal standard deviation at α=1% on the test set. Please align the prose with the reported numbers.
[Appendix B] There is a typo 'Cauchy Schwartz' for 'Cauchy–Schwarz'.
[§6.1 and §6.2] Section 6.1 says a sequence of length 285 is used, while Section 6.2 says a sequence of length 286; please reconcile the descriptions of the overlapping-window construction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the neural-network hedge is compared against external benchmarks (Black-Scholes and Leland) on a held-out test set, and no fitted parameter is renamed as a prediction.

full rationale

The paper's derivation chain is self-contained. The neural network is trained to minimize the standard deviation of the discounted terminal wealth ZT, and the same statistic is reported in Tables 1-4; this is a congruence between objective and evaluation metric, not a circular reduction, because the test set is independently generated and the benchmarks (Black-Scholes and Leland) are external, formula-based hedges evaluated under the same metric. Leland's formula is re-derived from standard arguments in Appendix A, and the Soner et al. (1995) result is proved directly in Appendix B rather than merely imported. The only self-citation (Brugiere and Turinici, 2023) appears as contextual mention of variational-autoencoder data augmentation and is not load-bearing. The paper also reports an honest failure case on S&P 500 data (Section 6.2), which is an external falsifiability check rather than a circular confirmation. Concerns about a single training seed and missing confidence intervals are statistical robustness issues, not circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the GBM data-generating process, the transaction cost and settlement model, and the standard-deviation objective. No new physical or financial entities are introduced. The reported hyperparameters are standard choices; the absence of seed variation is the main risk in the ledger.

free parameters (5)

random seed = 42
All simulations use seed_value=42; no multiple seeds are tested, so observed differences could be seed-specific.
batch size = 64
Chosen by hand without ablation; affects gradient noise and final hedge.
learning rate = 0.001
Adam default; not tuned per problem.
training epochs = 500
Fixed stopping point; no convergence criterion or early stopping.
network architecture = 3-64-32-1 MLP
One fixed architecture; no experiments on width or depth.

assumptions (4)

domain assumption Asset prices follow GBM with constant volatility under the real-world measure.
Section 5: dX = mu X dt + sigma X dW with sigma=0.2; all training and test paths are drawn from this model.
domain assumption Transaction costs are proportional to traded value and apply only at rebalancing times, not at inception or physical settlement.
Section 4.2 and the cost_Y_M code; changing this settlement assumption changes the optimal hedge.
domain assumption The hedging objective is to minimize the standard deviation of discounted terminal wealth ZT, with no penalty on its mean.
Section 1 states the choice, following Kolm and Ritter (2019); this is a modeling choice, not a theorem.
standard math The Black-Scholes delta is computed with the true volatility sigma for the BS benchmark, and with Leland's adjusted volatility for the Leland benchmark.
Sections 4 and 5; the Leland adjustment follows Leland (1985).

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Cite this review

Pith. "Pith review of Model-Free Deep Hedging with Transaction Costs and Light Data Requirements." pith.science (2026). https://pith.science/paper/X2QV3RNV

@misc{pith2026250522836,
  author       = {Pith},
  title        = {Pith review of: Model-Free Deep Hedging with Transaction Costs and Light Data Requirements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2QV3RNV}},
  note         = {Machine review of arXiv:2505.22836}
}

abstract

Option pricing theory, such as the Black and Scholes (1973) model, provides an explicit solution to construct a strategy that perfectly hedges an option in a continuous-time setting. In practice, however, trading occurs in discrete time and often involves transaction costs, making the direct application of continuous-time solutions potentially suboptimal. Previous studies, such as those by Buehler et al. (2018), Buehler et al. (2019) and Cao et al. (2019), have shown that deep learning or reinforcement learning can be used to derive better hedging strategies than those based on continuous-time models. However, these approaches typically rely on a large number of trajectories (of the order of $10^5$ or $10^6$) to train the model. In this work, we show that using as few as 256 trajectories is sufficient to train a neural network that significantly outperforms, in the Geometric Brownian Motion framework, both the classical Black & Scholes formula and the Leland model, which is arguably one of the most effective explicit alternatives for incorporating transaction costs. The ability to train neural networks with such a small number of trajectories suggests the potential for more practical and simple implementation on real-time financial series.

Figures

Figures reproduced from arXiv: 2505.22836 by the authors.

**Figure 2.** Calibration of Independent Sequences: standard deviations in basis [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Test Set, as discussed in section 5.2: hedging costs ZT with 30 hedging points and T = 0.25 for the 256 trajectories considered. is expected as the delta tends to be highly unstable in such cases. As a result, the mean hedging values and the standard deviation of the hedging values can be considered to be nearly identical for Black & Scholes and the neural network in these cases [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Realized Black & Scholes deltas, Neural Network deltas, and differ [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: Calibration Overlapping Sequences, as discussed in section [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

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Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

[1]

Black and M

F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81: 0 637--654, 1973

work page 1973
[2]

Brugiere and G

P. Brugiere and G. Turinici. Deep learning of value at risk through generative neural network models: The case of the variational auto encoder. Methods X, 10: 0 102192, 2023

work page 2023
[3]

Buehler, B

H. Buehler, B. Mohan, B. Wood, and J. Teichmann. Deep hedging: from theory to practice, 2018. https://arxiv.org/abs/1802.03042

arXiv 2018
[4]

Buehler, L

H. Buehler, L. Gonon, J. Teichmann, and B. Wood. Deep hedging. Quantitative Finance, 19 0 (8): 0 1271--1291, 2019

work page 2019
[5]

J. Cao, J. Chen, J. Hull, and Z. Poulos. Deep hedging of derivatives using reinforcement learning, 2019. https://arxiv.org/abs/2103.16409

work page Pith review arXiv 2019
[6]

R. M. Dudley. Sample functions on the Gaussian process. The Annals of Probability, 1 0 (1): 0 66--103, 1973. https://www.jstor.org/stable/2959347

work page arXiv 1973
[7]

Föllmer and P

H. Föllmer and P. Leukert. Efficient hedging: Cost versus shortfall risk. Finance and Stochastics, 4: 0 117--146, 2000

work page 2000
[8]

L.S. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6: 0 327--343, 1993

work page 1993

Show all 14 references

[9]

Hirano, K

M. Hirano, K. Minami, and K. Imajo. Adversarial deep hedging: Learning to hedge without price process modeling, 2023. https://arxiv.org/pdf/2307.13217

2023 arXiv
[10]

Kolm and G

P.N. Kolm and G. Ritter. Dynamic replication and hedging: A reinforcement learning approach. The Journal of Financial Data Science, 1: 0 159--171, 2019

2019
[11]

H.E. Leland. Option pricing and replication with transactions costs. The Journal of Finance, 40 0 (5), 1985

1985
[12]

Mikkil \"a and J

O. Mikkil \"a and J. Kanniainen. Empirical deep hedging. Quantitative Finance, 23 0 (1): 0 111--122, 2022. doi:10.1080/14697688.2022.2136037

2022
[13]

Soner, S.E

H.M. Soner, S.E. Shreve, and J. Cvitanic. There is no nontrivial hedging portfolio for option pricing with transaction costs. The Annals of Applied Probability, 5 0 (2): 0 327--355, 1995

1995
[14]

M. Xu. Risk measure pricing and hedging in incomplete markets. Annals of Finance, 2 0 (1): 0 51--71, 2006

2006

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Black and M

F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81: 0 637--654, 1973

work page 1973

[2] [2]

Brugiere and G

P. Brugiere and G. Turinici. Deep learning of value at risk through generative neural network models: The case of the variational auto encoder. Methods X, 10: 0 102192, 2023

work page 2023

[3] [3]

Buehler, B

H. Buehler, B. Mohan, B. Wood, and J. Teichmann. Deep hedging: from theory to practice, 2018. https://arxiv.org/abs/1802.03042

arXiv 2018

[4] [4]

Buehler, L

H. Buehler, L. Gonon, J. Teichmann, and B. Wood. Deep hedging. Quantitative Finance, 19 0 (8): 0 1271--1291, 2019

work page 2019

[5] [5]

J. Cao, J. Chen, J. Hull, and Z. Poulos. Deep hedging of derivatives using reinforcement learning, 2019. https://arxiv.org/abs/2103.16409

work page Pith review arXiv 2019

[6] [6]

R. M. Dudley. Sample functions on the Gaussian process. The Annals of Probability, 1 0 (1): 0 66--103, 1973. https://www.jstor.org/stable/2959347

work page arXiv 1973

[7] [7]

Föllmer and P

H. Föllmer and P. Leukert. Efficient hedging: Cost versus shortfall risk. Finance and Stochastics, 4: 0 117--146, 2000

work page 2000

[8] [8]

L.S. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6: 0 327--343, 1993

work page 1993