Error Analysis of Deep PDE Solvers for Option Pricing

Jasper Rou

REVIEW 4 major objections 5 minor 21 references

Error Analysis of Deep PDE Solvers for Option Pricing

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Deep PDE error shrinks almost linearly with sampling stages for option pricing

desk verdict Useful hyperparameter benchmark, but the headline convergence rates aren't yet credible: missing fitting details, no uncertainty, and 'almost linear' doesn't match Table 1. read the letter →

arxiv 2505.05121 v1 pith:P4A2BJRZ submitted 2025-05-08 q-fin.CP cs.LGq-fin.MF

Jasper Rou This is my paper · ORCID

classification q-fin.CPcs.LGq-fin.MF MSC 91G2091G6068T07

keywords optionpricingdeepPDEsolversGalerkinMethodTimeGradientFlowempiricalconvergenceBlack-ScholesmodelHestonL2error

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 measures how the accuracy of two deep-learning PDE solvers, the Deep Galerkin Method and the Time Deep Gradient Flow method, changes as their key training parameters are varied. It finds that the $L^2$ error falls almost linearly as the number of sampling stages grows, that adding layers helps but at an unclear rate, and that for the time-stepping method more time steps reduce error faster with second-order discretization than with first-order. The study also records training time, showing it grows linearly for stages, layers, and time steps but barely changes with the number of samples or nodes per layer. If these patterns hold, practitioners can choose training budgets based on a quantitative accuracy-versus-cost trade-off instead of guesswork.

What carries the argument

The central objects are two neural-network PDE solvers applied to option-pricing problems. The Time Deep Gradient Flow (TDGF) method solves the pricing PDE time step by time step, minimizing an energy functional with Monte Carlo estimates of integrals and using either a first- or second-order time discretization. The Deep Galerkin Method (DGM) trains a single network over the whole time-space domain by minimizing the squared PDE residual plus a boundary term. The accuracy measure is the $L^2$ error on a fixed equidistant grid of 47 points per dimension, averaged over five random seeds, and convergence rates are obtained by power-law fits to the error as each parameter is varied one at a time.

What would settle it

Repeat the sampling-stages experiment with the same models and settings but evaluate the $L^2$ error on a much finer or adaptive grid and over a wider range, for example 16 to 2000 stages; if the log-log slope changes substantially or bends downward, the reported near-linear rate is an artifact of the evaluation grid or the chosen range rather than a robust scaling law.

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Extended reading notes

Core claim

The paper's central finding is that empirical convergence of both solvers is driven mainly by sequential training effort: the $L^2$ error decreases almost linearly with the number of sampling stages, with fitted rates between roughly -0.6 and -0.9 across the Black-Scholes and Heston models, and training time rises linearly. Increasing the number of layers also reduces error, but the fitted rates vary and do not follow a clean law. For the Time Deep Gradient Flow method, increasing the number of time steps lowers the error, and the second-order scheme consistently converges faster than the first-order scheme in both test models. In contrast, the number of samples and the number of nodes per layer show no clear or uniform relationship with error, and for the Deep Galerkin Method the error can even rise slightly with more nodes per layer.

Load-bearing premise

The fitted convergence rates are meaningful scaling laws, which assumes that the $L^2$ error on a fixed grid of 47 points per dimension, averaged over five random seeds, is a stable and representative measure of solver accuracy, and that a single power-law fit over the chosen parameter ranges captures the true behavior.

Editorial extensions

If this is right

For both solvers, increasing the number of sampling stages is the most dependable accuracy lever, and the near-linear error decrease means each additional stage buys roughly proportional accuracy until a plateau is reached.
The second-order time-stepping scheme for TDGF measurably outperforms first-order at the same time-step count, so using it is a direct accuracy improvement without changing the network architecture.
Since training time grows linearly with sampling stages, layers, and time steps, choosing these parameters is a clear speed-accuracy trade-off, and the reported rates give a basis for deciding where to stop.
The small training-time cost of additional samples and nodes per layer means, for TDGF, error can be reduced essentially for free by using many samples and many nodes.
The flat or slightly positive node-per-layer rate for DGM suggests that for this solver, network width beyond a modest size does not buy accuracy, so money is better spent on sampling stages.

Reading between the lines

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

The near-linear rate in sampling stages resembles Monte Carlo averaging of the stochastic energy estimates; if so, variance-reduction techniques such as quasi-random sampling or importance sampling could push the empirical rate toward or past the observed -0.9.
The paper's fixed 47-point evaluation grid is coarse relative to the sharp payoff kink near the strike, so the reported rates may depend on the grid; evaluating on a finer or adaptive grid could reveal whether the scaling laws are genuine.
The contrasting behavior of samples and nodes between DGM and TDGF hints that DGM's error is limited by optimization rather than by approximation capacity, so better optimizers or learning-rate schedules might change the flat node-per-layer result.

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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

Rou's paper is the first I've seen that maps L2 error and training time for DGM and TDGF across five hyperparameters in both Black-Scholes and Heston. That is genuinely useful engineering knowledge. The experimental setup is mostly sensible: exact Black-Scholes prices and COS prices for Heston as external references, a fixed 47-point grid, five seeds averaged, and the default parameter set clearly stated. The observations about training time scaling linearly with sampling stages, layers, and time steps are plausible and worth reporting. The citation pattern looks fine; the self-citations point to the actual architecture and the convergence theory used, which is legitimate.

The soft spots are load-bearing rather than cosmetic. The fitted convergence rates in Tables 1-5 are the central quantitative results, but the fitting procedure is never specified -- log-log least squares over which points? on seed-averaged errors? -- and no standard errors or seed spread are reported. The abstract and Section 1 claim the L2-error decreases 'almost linearly' with sampling stages, but Table 1 gives -0.63 for TDGF in Heston, which is not almost -1. Section 5.1 even says all rates are 'slightly larger than -1', which doesn't square with -0.63. Likewise, Section 5.5 says the second-order scheme stops improving after 25 time steps while Table 5 fits one power law over the whole 2-25 range; a decreasing-then-flat curve biases the exponent toward zero and makes the rate depend on the chosen range. Since the practical recommendations are derived from these rates, the missing uncertainty quantification is a real deficiency, not a stylistic one. No code or data is released, so the numbers can't be checked independently.

That said, this isn't a paper with a broken core. It is an honest, under-specified empirical study. The author is appropriately cautious in many spots, and the direction of the findings is probably right. The fix is straightforward: state the fitting method, add error bars or show seed trajectories, fit rates only on ranges where the error is actually decreasing, release code and data, and soften or qualify the 'almost linear' claim. With those changes I would trust the tables.

Who is this for? Quants choosing hyperparameters for DGM or TDGF, and researchers benchmarking deep PDE solvers. I would not desk-reject it. It deserves a serious referee, but with major revision before publication.

Referee Report

4 major / 5 minor

Summary. The paper empirically studies two deep PDE solvers, the Deep Galerkin Method (DGM) and the Time Deep Gradient Flow method (TDGF), for European call option pricing in the Black-Scholes and Heston models. Fixing one hyperparameter at a time, it measures the L2-error on a fixed 47-point grid against exact Black-Scholes prices and COS-based Heston reference prices, averages over five random seeds, and reports fitted 'convergence rates' (power-law exponents) for the number of sampling stages, number of samples, number of layers, number of nodes per layer, and, for TDGF, number of time steps and discretization order. It also reports training-time scaling for each parameter. The main findings are that the L2-error decreases roughly linearly with sampling stages, layers tend to reduce error without a clear rate, more time steps reduce error faster for the second-order scheme, and samples/nodes show no consistent effect.

Significance. If the reported rates were robust, the paper would give practitioners a useful accuracy-cost map for configuring DGM and TDGF in low-dimensional option pricing and would complement existing theory with concrete empirical guidance. The study has clear strengths: it uses external reference solutions (exact Black-Scholes and COS Heston), states its implementation choices in detail, and is appropriately hedged when no clear trend exists. Its main limitation is that the central quantitative claims, the fitted exponents in Tables 1-5, rest on an unspecified fitting procedure, no uncertainty quantification, only five seeds per configuration, and no released code. The practical recommendations in Section 6 are therefore only as strong as those fits allow.

major comments (4)

[Section 5, Tables 1-5] The fitted convergence rates in Tables 1-5 are the central quantitative output of the paper, but the fitting procedure is never specified: the paper does not state whether a log-log least-squares fit was used, which data points were included, whether the fit was performed on the seed-averaged errors, or how the two-decimal precision of the exponents should be interpreted. No standard errors, confidence intervals, or seed-to-seed spreads accompany the rates, although the L2-error is reported only as an average over five seeds. In addition, the abstract and Section 1 claim that the error 'decreases almost linearly' with the number of sampling stages, yet Table 1 reports a TDGF-Heston rate of -0.63, which is far from the -1 implied by 'almost linear' and is not 'slightly larger than -1' as stated in Section 5.1. Because the recommendations in Section 6 are derived directly from these exponents, the missing specification and this internal contradiction are load-bearing.
[Section 5.5 and Section 5.1] Section 5.5 states that after 25 time steps the second-order scheme 'does not improve any further', and Section 5.1 states that the error stops converging beyond some number of sampling stages, yet Table 5 and Table 1 report single power-law exponents fitted over the full ranges (K=2,...,25 and the full stage ranges). A single exponent fitted across a curve that first decreases and then flattens is biased toward zero and its value depends on the chosen range; the reported rates therefore cannot be interpreted as intrinsic convergence rates. The fits should be restricted to the decreasing regime, or piecewise rates should be reported, and the sensitivity of the exponents to the fitting range should be assessed.
[Section 5.3 and Section 5.4] The 'convergence rates' for the number of layers and the number of nodes per layer are fitted from only four and five distinct parameter values, respectively (layers 1-4; nodes 10-50). Table 4 even reports positive rates for the DGM (0.15 for Black-Scholes, 0.07 for Heston), which the text does not reconcile with the claim that more nodes tend to decrease or not affect error. Without uncertainty quantification or a statement about fit stability over such short ranges, these exponents are over-interpreted; the only robust conclusion supported by the text is that there is 'no clear rate'. Please report the underlying data points and a stability check, or drop the rate interpretation for these parameters.
[Section 2.2] The displayed DGM cost functional L(θ;t,x) is written as a sum of un-squared residuals, whereas the preceding text defines the DGM objective as the sum of the squared L2-errors of the PDE residual and the initial condition. As typeset, the formula minimizes a signed residual, which is not the DGM loss and could in principle be driven to negative values. If the implementation actually used the displayed objective, the DGM results correspond to a different training criterion; if the implementation used the squared form, the formula needs correction. Please clarify and fix.

minor comments (5)

[Section 5.4] The first sentence of Section 5.4 says 'We vary the number of layers from 10 to 50', which should read 'the number of nodes per layer'.
[Section 6] In the bullet on sampling stages, 'with the the number of sampling stages' contains a duplicated article and should be corrected.
[Section 6] In the bullet on nodes per layer, 'forty of fifty' should be 'forty or fifty'.
[Section 4] The hardware description 'using one seventh instance of a NVidia Tesla A100 GPU' is unclear; please specify what 'one seventh instance' means (for example, a GPU slice or a specific resource allocation).
[Section 5.2] In Section 5.2, the statement 'For the DGM the rates are larger' is ambiguous; 'less negative' would be clearer for readers comparing the fitted exponents.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the reported error rates are fitted to measurements against external references (Black-Scholes formula, COS Heston); the author's self-citations supply method and architecture provenance but are not load-bearing.

full rationale

The paper's central claims are empirical: it measures L2 errors of DGM and TDGF on a fixed 47-point grid against the exact Black-Scholes solution ('This PDE has an exact solution: u(t,S)=SPhi(d1)-Ke^{-rt}Phi(d2)') and against the Heston COS reference ('the characteristic function ... has an analytical representation [13], from which a reference price can be determined using the COS method'). The convergence rates in Tables 1-5 are exponents fitted to those measured errors; they are outputs of the experiments, not inputs or assumptions. The TDGF formulation is taken from the author's prior work [18], and [16] is cited only to point to a theoretical convergence analysis ('For theoretical convergence analyses, see ... Liu, Papapantoleon, and Rou [16]'); neither citation is used to force an empirical result or to forbid alternatives. The architecture from [18] is a design choice, and the network's payoff term ('f(x;theta)=(S-Ke^{-rt})_+ + sigma_2(W X_{L+1}+b)') enforces a no-arbitrage constraint; it does not define the measured L2 error against external references. Missing fit details or uncertainty quantification, mentioned by a skeptic, concern robustness of the empirical conclusions, not circularity. No step in the paper reduces, by construction or by self-citation chain, to its own inputs.

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

The paper introduces no new mathematical postulates or entities; it empirically measures the behavior of existing methods. The main inputs are experimental settings (learning rate, architecture size, sample counts, seeds) and standard reference solutions. The listed free parameters are hand-chosen experimental settings that influence all reported rates, not parameters fitted to the option pricing data.

free parameters (4)

learning rate 3e-4 = 3e-4
Chosen by hand for Adam optimizer; directly affects convergence and all measured errors and rates.
default network size = 3 layers, 50 nodes per layer
Hand-picked default architecture used as the baseline in all parameter sweeps.
default samples per dimension = 600
Hand-picked sample count per stage in the default configuration.
number of random seeds = 5
Averages over five seeds are used to define the error, but no uncertainty from seed variation is reported.

assumptions (4)

standard math Feynman-Kac formula gives the option price as the solution to the Black-Scholes and Heston PDEs.
Invoked in Section 3.1 to justify solving the PDE for the option price.
domain assumption The exact Black-Scholes formula and the COS method for Heston provide accurate reference prices.
Used in Section 3 and Section 5 to compute the L2 errors; assumes these references are effectively exact for the grid used.
domain assumption Neural network training with Adam reaches a sufficiently converged state within the stated number of sampling stages.
The paper states that after the default number of stages the error does not decrease further (Section 4), which is used to justify comparing different parameters at fixed training budgets.
domain assumption The architecture from Papapantoleon and Rou [18] is an appropriate and sufficiently expressive function class for these pricing problems.
Adopted without proof in Section 4; the error analysis measures the behavior of this specific architecture, not the general method.

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Cite this review

Pith. "Pith review of Error Analysis of Deep PDE Solvers for Option Pricing." pith.science (2026). https://pith.science/paper/P4A2BJRZ

@misc{pith2026250505121,
  author       = {Pith},
  title        = {Pith review of: Error Analysis of Deep PDE Solvers for Option Pricing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P4A2BJRZ}},
  note         = {Machine review of arXiv:2505.05121}
}

read the original abstract

Option pricing often requires solving partial differential equations (PDEs). Although deep learning-based PDE solvers have recently emerged as quick solutions to this problem, their empirical and quantitative accuracy remain not well understood, hindering their real-world applicability. In this research, our aim is to offer actionable insights into the utility of deep PDE solvers for practical option pricing implementation. Through comparative experiments in both the Black--Scholes and the Heston model, we assess the empirical performance of two neural network algorithms to solve PDEs: the Deep Galerkin Method and the Time Deep Gradient Flow method (TDGF). We determine their empirical convergence rates and training time as functions of (i) the number of sampling stages, (ii) the number of samples, (iii) the number of layers, and (iv) the number of nodes per layer. For the TDGF, we also consider the order of the discretization scheme and the number of time steps.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Abi Jaber and O

E. Abi Jaber and O. El Euch. Multifactor approximation of rough volatility models.SIAM Journal on Financial Mathematics, 10(2):309–349, 2019

2019

[2] [2]

Akrivis and Y .-S

G. Akrivis and Y .-S. Smyrlis. Implicit–explicit BDF methods for the Kuramoto–Sivashinsky equation. Applied numerical mathematics, 51(2-3):151–169, 2004

work page 2004

[3] [3]

Assabumrungrat, K

R. Assabumrungrat, K. Minami, and M. Hirano. Error analysis of option pricing via deep PDE solvers: Empirical study. In2024 16th IIAI International Congress on Advanced Applied Informatics (IIAI-AAI), pages 329–336. IEEE, 2024

work page 2024

[4] [4]

Becker, P

S. Becker, P. Cheridito, and A. Jentzen. Deep optimal stopping.Journal of Machine Learning Research, 20(74):1–25, 2019

work page 2019

[5] [5]

Black and M

F. Black and M. Scholes. The pricing of options and corporate liabilities.Journal of Political Economy, 81(3):637–654, 1973

1973

[6] [6]

Buehler, L

H. Buehler, L. Gonon, J. Teichmann, and B. Wood. Deep hedging.Quantitative Finance, 19(8):1271– 1291, 2019. ERROR ANALYSIS OF DEEP PDE SOLVERS FOR OPTION PRICING 9 (A) Linear scale (B) Logarithmic scale FIGURE2.L 2-error of the TDGF for a call option in the Black–Scholes model against number of sampling stages varying from 16 to 500

work page 2019

[7] [7]

DelftBlue Supercomputer (Phase 1).https: //www.tudelft.nl/dhpc/ark:/44463/DelftBluePhase1, 2022

Delft High Performance Computing Centre (DHPC). DelftBlue Supercomputer (Phase 1).https: //www.tudelft.nl/dhpc/ark:/44463/DelftBluePhase1, 2022

work page 2022

[8] [8]

Fang and C

F. Fang and C. W. Oosterlee. A novel pricing method for European options based on Fourier-cosine series expansions.SIAM Journal on Scientific Computing, 31:826–848, 2009

work page 2009