Time Deep Gradient Flow Method for pricing American options

Jasper Rou

REVIEW 3 major objections 5 minor 22 references

Time Deep Gradient Flow Method for pricing American options

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Time Deep Gradient Flow method is extended to American options by training only where the price exceeds the payoff, matching Monte Carlo accuracy in up to five dimensions while cutting evaluation time by orders of magnitude.

desk verdict The claimed free-boundary self-selection is vacuous because the architecture forces f > Ψ everywhere, so the paper's central extension does not actually work as described. read the letter →

arxiv 2507.17606 v1 pith:U7HALXIC submitted 2025-07-23 q-fin.CP cs.LGmath.PRq-fin.MF

Jasper Rou This is my paper · ORCID

classification q-fin.CPcs.LGmath.PRq-fin.MF MSC 91G2091G6068T07

keywords Americanoptionpricingfree-boundaryPDETimeDeepGradientFlowGalerkinMethodneuralnetworksolverBlack-ScholesmodelHestonhigh-dimensionalfinance

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 extends the Time Deep Gradient Flow (TDGF) method, a neural-network solver for high-dimensional parabolic PDEs, to price American put options. American options introduce a free boundary because early exercise means the price satisfies the pricing PDE only where it exceeds the payoff $\Psi$ and equals the payoff elsewhere. The extension trains the network only on points where the current network output is above the payoff, and builds the payoff directly into the network output, so the free boundary is handled without a separate boundary-tracking step. In numerical tests for two- and five-dimensional Black-Scholes and Heston models, the method matches the accuracy of the Deep Galerkin Method (DGM) and of least-squares Monte Carlo, while running much faster at evaluation time; TDGF also trains faster than DGM.

What carries the argument

The load-bearing mechanism is the time-stepped energy minimization of TDGF. The time interval is split into $K$ equally spaced steps; at step $k$ the network $f^k(x;\theta)$ minimizes a Monte Carlo approximation of $$I^k(u)=\tfrac12\|u-$U^{{k-1}}$\|^2_{$L^{2}$(\$\Omega$)} + h\int_\$\Omega$ \left(\tfrac12(\nabla u)^T A \nabla u + r $u^{2}$ + (b\cdot\nabla $U^{{k-1}}$) u\right)dx,$$ the discrete gradient-flow formulation of the pricing PDE. For American options, the new ingredient is that only sample points satisfying $f^k(x)>\Psi(x)$ enter the loss, which implements the complementary condition of the variational inequality, while the payoff is encoded in the architecture as $\Psi + \sigma_2(\cdot)$, so the network learns the continuation value. The combination lets the free boundary emerge from the training data instead of being tracked explicitly.

What would settle it

Run the algorithm on a one-dimensional American put with a deliberately poor initial network, such as a constant zero continuation value, and compare the final price and the set of selected training points to the binomial reference: if the selected region and the price depend on the initial guess, the self-selection fixed point is not stable; if they match the reference, the assumption holds.

Watch

Extended reading notes

Core claim

The central claim is that the TDGF time-stepping scheme can be adapted to the free-boundary variational inequality of American options by restricting the PDE loss to the region where the network price exceeds the payoff. At every time slice, the algorithm samples points, keeps only those with $f^k(x)>\Psi(x)$, and minimizes a discretized energy that measures the $L^2$ distance from the previous time step plus a weighted residual of the diffusion operator. The network architecture outputs $\Psi + \sigma_2(W X^{L+1}+b)$ with $\sigma_2$ the softplus function, so the no-arbitrage bound $u\ge\Psi$ holds by construction and the network only learns the continuation value. With box-stratified sampling to keep extreme moneyness values represented, the method gives accurate prices for up to five underlying assets (ten state variables in the Heston case) and its training time is consistently lower than DGM's, for example 6,583 versus 16,174 seconds for the five-dimensional Black-Scholes case.

Load-bearing premise

The load-bearing premise is that training only on points where the network's current price already exceeds the payoff converges to the true continuation region, even though the network's early guesses may be far from correct.

Editorial extensions

If this is right

American put prices in up to five underlying assets under Black-Scholes and Heston dynamics can be evaluated about three orders of magnitude faster than least-squares Monte Carlo in the reported tests.
TDGF's training-time advantage over DGM grows with dimension, so for high-dimensional free-boundary problems it is the cheaper neural-network training choice.
The box-sampling strategy is part of the method's performance: it keeps samples at high moneyness where the continuation value is small, preventing the network from ignoring the boundary of the domain.
Because the architecture enforces $u\ge\Psi$ by construction, the method prices American options without penalty approximations or front-tracking for the free boundary.

Reading between the lines

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

A self-consistency check on the selected training region would tell whether the method has converged: track the set $\{x: f^k(x)>\Psi(x)\}$ over training iterations; if it is still changing near the end of training, the network has not settled on a stable continuation region.
The same time-stepped energy formulation should transfer to Bermudan options, where exercise is allowed at discrete dates, by applying the $\Psi$-threshold selection only at exercise dates.
Since the network separates the continuation value from the payoff, a single trained network might be re-priced across different strikes by rescaling the payoff term, although the paper does not test this.

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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

Colleague,

The short version: the paper's central methodological claim doesn't survive contact with its own architecture. Section 3.2 defines the network output as f = Ψ + softplus(...), so f > Ψ for every input, at every parameter value. Algorithm 1, line 11 ('select the points where f^k > Ψ') therefore never filters anything. The 'self-selection' of the continuation region is a no-op. What the method actually does is minimize the time-discretized PDE energy over the whole domain, including the stopping region where the PDE equality in (2.1) does not hold. This is not the free-boundary extension advertised in the abstract and Section 3.1; it's a softplus-constrained penalty method with the claimed justification gone.

The paper isn't without merit. The box-sampling scheme for maintaining moneyness coverage in high dimensions is a sensible practical contribution. The training-time comparison against DGM is clearly presented and the factor-2–3 speed advantage looks credible. The variational reformulation inherited from prior TDGF work is cleanly summarized.

Soft spots beyond the main flaw: accuracy is supported only by visual curve overlap in Figures 3–6, with no error norms, no tables of numbers, no code or data. Running times are single runs without variance. The Monte Carlo reference uses only 1,000 paths, which is thin for a 5-dimensional Heston problem.

The internal inconsistency is the load-bearing issue. It's probably fixable — for instance, parameterize the continuation value with an unconstrained network and select points based on that network's sign, or honestly state a penalty method — but as written, the method does not do what it claims, and the convergence concern the authors would need to address is even more basic than the missing fixed-point proof.

Who is this for? Computational finance people working on neural PDE solvers. It's not a must-read, but it's a useful cautionary example of an architecture-level bug defeating a method's stated mechanism.

Recommendation: send it to referees, because the underlying idea is worth developing and a careful review should catch exactly this kind of flaw. Expect a major revision and a request for quantitative error metrics and code.

Referee Report

3 major / 5 minor

Summary. The paper considers neural-network pricing of American basket puts under Black-Scholes and Heston dynamics in up to five dimensions. It proposes an extension of the Time Deep Gradient Flow (TDGF) method in which the variational inequality is discretized by time stepping and each time step is solved by minimizing an energy functional with a neural network ansatz; the free boundary is supposedly handled by restricting training samples to the region where the network value exceeds the payoff. The price ansatz uses a DGM-style architecture with a softplus output so that the network value is always at least the payoff. The authors compare TDGF with the Deep Galerkin Method (DGM) and a Longstaff-Schwartz Monte Carlo reference, reporting that both are accurate and that TDGF trains faster.

Significance. If the reported accuracy and speed held, a fast deep solver for multidimensional American options would be a useful contribution. The paper's divergence-form derivations for the generator coefficients in the Black-Scholes and Heston models (§2.2-2.3) are careful, and the box-sampling idea (§3.3) is sensible. However, the central methodological claim is undermined by an architectural inconsistency, and the numerical evidence is only qualitative; the paper as it stands does not establish that the proposed method solves the free-boundary problem.

major comments (3)

[§3.1-3.2, Algorithm 1] The self-selection step in Algorithm 1 (line 11) is vacuous. In §3.2 the network is defined as f(x;θ)=Ψ(x)+σ2(WX^{L+1}+b) with σ2 the softplus, which is strictly positive on R. Therefore f^k(x)>Ψ(x) holds for every x at every parameter value, and line 11 never discards any sample. The statement in §3.1 that 'we only train the PDE on the part of the domain where the solution is above the payoff' is therefore not realized by the implemented algorithm; the method actually minimizes the time-discretized PDE energy over the entire domain, including the stopping region, and it contains no term enforcing u=Ψ in that region. The claimed extension of TDGF to free-boundary problems is thus not supported by the stated algorithm. This needs to be fixed (for example by parameterizing the continuation value separately and masking or penalizing the stopping region) and the fixed-point behavior of the resulting selection rule must be demonstrated.
[§4.1, Figures 3-6] The claim of 'high accuracy' is supported only by visual coincidence of curves in Figures 3-6; no quantitative error measure (e.g., RMSE or maximum absolute error against the Monte Carlo reference) is reported, and no confidence intervals or off-diagonal test points are given. The reference Longstaff-Schwartz estimate itself is based on only 1,000 paths and 1,000 time steps, so its own statistical error should be quantified before 'similar values' can be interpreted as accuracy. Please provide error tables or error-vs-moneyness plots with error bars.
[§4.2, Tables 1 and 2] The speed comparisons rest on unreplicated measurements: Table 1 reports a single training time per configuration with no variance, and Table 2 reports averages over 34 evaluations without standard deviations or statistical tests. Since both methods use stochastic optimization, a single run does not establish that 'TDGF tends to be faster during training than DGM.' Please report mean and standard deviation over multiple independent runs, and describe precisely what the numbers in Table 2 measure (evaluation time per price? time for the 47-point grid?).

minor comments (5)

[§2.2] The term 'dWi(t)t' appears to be a typo for 'dWi(t)'.
[§3.1] The step size h is defined as 1/K although the time interval is [0,T]; it should be T/K. The experiments use T=1, so the numbers coincide, but the definition should be general.
[§4] The duplicate phrase 'In the DGM approach, In the DGM approach,' at the start of Section 4 should be corrected.
[§3.3] The statement that for TDGF 'during the time steps we apply uniform sampling' is hard to reconcile with the claimed restriction to the continuation region; please clarify how uniform sampling interacts with the selection step in Algorithm 1.
[Table 2] The caption of Table 2 should state whether the reported times are training times, evaluation times, or end-to-end times, and the units should be consistent with Table 1.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial by-construction vacuity: softplus architecture makes Algorithm 1's free-boundary sample selection a tautology, so the claimed extension reduces to whole-domain PDE training.

other [Section 3.2 network architecture f(x;θ)=Ψ+σ2(W X^{L+1}+b) interacting with Algorithm 1, line 11.]
"f (x; θ) = Ψ + σ2 (W XL+1 + b) ... the softplus function, σ2(x) = log (ex + 1) ... Select the points xm where f k(xm) > Ψ(xm)."

Softplus is strictly positive on R, so f^k(x)=Ψ(x)+σ2(...)>Ψ(x) for every x and every parameter value. The selection predicate in Algorithm 1 line 11 is therefore true everywhere; no sample is ever excluded. The paper's justification, 'we only train the PDE on the part of the domain where the solution is above the payoff,' is never operative, and the free-boundary mechanism that is the paper's claimed extension reduces by construction to whole-domain TDGF training with a softplus penalty. The central methodological step is thus a no-op as written.

full rationale

The numerical experiments are not circular in the fitted-prediction sense: reference prices come from an independent Longstaff-Schwartz Monte Carlo simulation, the TDGF energy functional is restated in Section 3.1 rather than merely imported from the self-citation [19], and no fitted parameter is later relabeled as an out-of-sample prediction. The self-citation to Papapantoleon and Rou is not load-bearing because the discretized cost functional and Algorithm 1 are given explicitly. However, the architecture in Section 3.2 forces f^k>Ψ everywhere, so the sample-selection step that is supposed to restrict training to the continuation region never filters any point. This makes the claimed free-boundary extension vacuous by construction and reduces the method to whole-domain training, a significant by-construction failure even though the external benchmark comparison retains independent content.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The paper does not introduce new physical or mathematical entities. Its main free parameters are neural network hyperparameters and domain sampling bounds, all chosen by hand or from prior work. The axiomatic assumptions are standard for the problem, except for the self-selection convergence assumption, which is a significant unproven premise.

free parameters (7)

Sampling domain upper bound for moneyness (Black-Scholes) = 3.0
Chosen based on numerical experiments; continuation value can be positive beyond 2.
Sampling domain upper bound for moneyness (Heston) = 2.0
Chosen because continuation value is observed to be zero above 1.5.
Number of sampling boxes = 19
Hand-picked to improve edge coverage of the domain.
Samples per box per dimension = 30 per box per dimension (60d for Heston)
Hand-chosen sampling budget.
Sampling stages per time step = 2000
Fixed training budget; DGM uses 200,000 total stages.
Network architecture (layers, neurons) = 3 layers, 50 neurons
Standard DGM architecture, chosen without sensitivity analysis.
Number of time steps K = 100
Time discretization resolution, chosen for accuracy.

assumptions (4)

domain assumption The TDGF discretization produces a valid approximation to the linear evolution PDE (from [19]).
The paper's cost functional L^k is taken from prior work and not re-derived.
standard math The variational inequality (2.1) is the correct formulation for American option prices.
Widely accepted formulation; cited from [12].
ad hoc to paper The self-selection rule {x : f^k(x) > Ψ(x)} converges to the true continuation region.
No proof or numerical analysis; central to the free-boundary extension.
domain assumption The Longstaff-Schwartz Monte Carlo with 1000 paths and 1000 time steps is accurate enough as a reference.
Used to validate the networks, but no standard errors are reported.

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Cite this review

Pith. "Pith review of Time Deep Gradient Flow Method for pricing American options." pith.science (2026). https://pith.science/paper/U7HALXIC

@misc{pith2026250717606,
  author       = {Pith},
  title        = {Pith review of: Time Deep Gradient Flow Method for pricing American options},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7HALXIC}},
  note         = {Machine review of arXiv:2507.17606}
}

read the original abstract

In this research, we explore neural network-based methods for pricing multidimensional American put options under the BlackScholes and Heston model, extending up to five dimensions. We focus on two approaches: the Time Deep Gradient Flow (TDGF) method and the Deep Galerkin Method (DGM). We extend the TDGF method to handle the free-boundary partial differential equation inherent in American options. We carefully design the sampling strategy during training to enhance performance. Both TDGF and DGM achieve high accuracy while outperforming conventional Monte Carlo methods in terms of computational speed. In particular, TDGF tends to be faster during training than DGM.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 18 canonical work pages

[1]

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
[2]

Becker, P

S. Becker, P. Cheridito, and A. Jentzen. Pricing and hedging American-style options with deep learning. Journal of Risk and Financial Management, 13(7):158, 2020. 10 J. ROU FIGURE 4. Difference between the option price and the payoff in the five-dimensional Black– Scholes model against the moneyness of the stock, compared to the DGM and Monte Carlo with L...

work page 2020
[3]

Becker, P

S. Becker, P. Cheridito, A. Jentzen, and T. Welti. Solving high-dimensional optimal stopping problems using deep learning. European Journal of Applied Mathematics, 32(3):470–514, 2021

work page 2021
[4]

Belomestny and J

D. Belomestny and J. Schoenmakers. Advanced Simulation-Based Methods for Optimal Stopping and Control: With Applications in Finance. Springer, 2018

work page 2018
[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]

Clarke and K

N. Clarke and K. Parrott. Multigrid for American option pricing with stochastic volatility. Applied Math- ematical Finance, 6(3):177–195, 1999

work page 1999
[7]

J. C. Cox, S. A. Ross, and M. Rubinstein. Option pricing: A simplified approach. Journal of financial Economics, 7(3):229–263, 1979

work page 1979
[8]

DelftBlue Supercomputer (Phase 2)

Delft High Performance Computing Centre (DHPC). DelftBlue Supercomputer (Phase 2). https: //www.tudelft.nl/dhpc/ark:/44463/DelftBluePhase2, 2024. TIME DEEP GRADIENT FLOW METHOD FOR PRICING AMERICAN OPTIONS 11 FIGURE 5. Difference between the option price and the payoff in the two-dimensional Hes- ton model against the moneyness of the stock, compared to t...

work page 2024

Show all 22 references

[9]

E. H. Georgoulis, M. Loulakis, and A. Tsiourvas. Discrete gradient flow approximations of high dimen- sional evolution partial differential equations via deep neural networks. Communications in Nonlinear Science and Numerical Simulation, 117:106893, 2023

2023
[10]

Herrera, F

C. Herrera, F. Krach, P. Ruyssen, and J. Teichmann. Optimal stopping via randomized neural networks. arXiv preprint arXiv:2104.13669, 2021

2021 arXiv
[11]

S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The review of financial studies, 6(2):327–343, 1993

1993
[12]

Hilber, O

N. Hilber, O. Reichmann, C. Schwab, and C. Winter. Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing. Springer Science & Business Media, 2013

2013
[13]

Ikonen and J

S. Ikonen and J. Toivanen. Efficient numerical methods for pricing American options under stochastic volatility. Numerical Methods for Partial Differential Equations: An International Journal , 24(1):104– 126, 2008. 12 J. ROU FIGURE 6. Difference between the option price and t...

2008
[14]

D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014

2014 arXiv
[15]

F. A. Longstaff and E. S. Schwartz. Valuing American options by simulation: a simple least-squares approach. The review of financial studies, 14(1):113–147, 2001

2001
[16]

Musiela and M

M. Musiela and M. Rutkowski. American options. In Martingale Methods in Financial Modelling, pages 205–228. Springer, 2005

2005
[17]

R. Myneni. The pricing of the American option. The Annals of Applied Probability, pages 1–23, 1992

1992
[18]

Nwankwo, N

C. Nwankwo, N. Umeorah, T. Ware, and W. Dai. Deep learning and American options via free boundary framework. Computational Economics, 64(2):979–1022, 2024

2024
[19]

Papapantoleon and J

A. Papapantoleon and J. Rou. A time-stepping deep gradient flow method for option pricing in (rough) diffusion models. arXiv preprint arXiv:2403.00746, 2024. TIME DEEP GRADIENT FLOW METHOD FOR PRICING AMERICAN OPTIONS 13

2024 arXiv
[20]

Y . Peng, P. Wei, and W. Wei. Deep penalty methods: A class of deep learning algorithms for solving high dimensional optimal stopping problems. arXiv preprint arXiv:2405.11392, 2024

2024 arXiv
[21]

Sirignano and K

J. Sirignano and K. Spiliopoulos. DGM: A deep learning algorithm for solving partial differential equa- tions. Journal of computational physics, 375:1339–1364, 2018

2018
[22]

W. Wadman. An advanced Monte Carlo method for the multi-asset Heston model. Inst. of Appl. Math., 2010. DELFT INSTITUTE OF APPLIED MATHEMATICS , EEMCS, TU D ELFT , 2628CD D ELFT , THE NETHERLANDS Email address: J.G.Rou@tudelft.nl

2010

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

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

[2] [2]

Becker, P

S. Becker, P. Cheridito, and A. Jentzen. Pricing and hedging American-style options with deep learning. Journal of Risk and Financial Management, 13(7):158, 2020. 10 J. ROU FIGURE 4. Difference between the option price and the payoff in the five-dimensional Black– Scholes model against the moneyness of the stock, compared to the DGM and Monte Carlo with L...

work page 2020

[3] [3]

Becker, P

S. Becker, P. Cheridito, A. Jentzen, and T. Welti. Solving high-dimensional optimal stopping problems using deep learning. European Journal of Applied Mathematics, 32(3):470–514, 2021

work page 2021

[4] [4]

Belomestny and J

D. Belomestny and J. Schoenmakers. Advanced Simulation-Based Methods for Optimal Stopping and Control: With Applications in Finance. Springer, 2018

work page 2018

[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]

Clarke and K

N. Clarke and K. Parrott. Multigrid for American option pricing with stochastic volatility. Applied Math- ematical Finance, 6(3):177–195, 1999

work page 1999

[7] [7]

J. C. Cox, S. A. Ross, and M. Rubinstein. Option pricing: A simplified approach. Journal of financial Economics, 7(3):229–263, 1979

work page 1979

[8] [8]

DelftBlue Supercomputer (Phase 2)

Delft High Performance Computing Centre (DHPC). DelftBlue Supercomputer (Phase 2). https: //www.tudelft.nl/dhpc/ark:/44463/DelftBluePhase2, 2024. TIME DEEP GRADIENT FLOW METHOD FOR PRICING AMERICAN OPTIONS 11 FIGURE 5. Difference between the option price and the payoff in the two-dimensional Hes- ton model against the moneyness of the stock, compared to t...

work page 2024