Solving high-dimensional optimal stopping problems using deep learning

Arnulf Jentzen; Patrick Cheridito; Sebastian Becker; Timo Welti

REVIEW 3 major objections 4 minor 2 cited by

Solving high-dimensional optimal stopping problems using deep learning

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A deep learning algorithm prices high-dimensional American and Bermudan options by training neural networks to represent stopping times directly.

desk verdict A solid and honest deep-optimal-stopping paper whose low-dimensional validation is convincing, but whose 5000-dimensional headline number is self-referential and should not be read as an independent accuracy claim. read the letter →

arxiv 1908.01602 v3 pith:AYI6HYED submitted 2019-08-05 cs.CE cs.LGmath.PRq-fin.CP

Sebastian Becker , Patrick Cheridito , Arnulf Jentzen , Timo Welti This is my paper · ORCID

classification cs.CEcs.LGmath.PRq-fin.CP MSC 60G4065C0591G20

keywords optimalstoppingAmericanoptionsBermudandeeplearningcurseofdimensionalityneuralnetworkstimefactorisationstochasticgradientascent

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

The paper proposes a deep learning algorithm that turns a high-dimensional optimal stopping problem, such as pricing an American or Bermudan option on many underlying assets, into a single neural network training problem. The central move is to represent a discrete stopping time through measurable factors that decide, at each time step, whether to stop, and to relax those binary factors into smooth network outputs. Training then maximizes the expected discounted payoff by stochastic gradient ascent, producing both an approximate optimal exercise strategy and a low-biased estimate of the option price. The numerical results suggest the method can handle problems with hundreds or thousands of assets, including a Bermudan max-call option in 5000 dimensions with a reported price of about 165.430 and a runtime of about 2899 seconds. If the method holds up beyond the tested examples, it would offer a practical route around the curse of dimensionality for early-exercise derivatives.

What carries the argument

The load-bearing identity is the factorisation lemma for stopping times (Lemma 2.2), which expresses any discrete stopping time as a sum of measurable functions of the observed path, together with the recursive construction of approximate stopping-time factors $U_{n,\theta}$ that always sum to one. The paper pairs this with a fixed feedforward architecture: one network per time step, with two hidden layers of width equal to the problem dimension, logistic output activation, batch normalisation, and the Adam optimiser. This machinery converts optimal stopping into an unconstrained parameter search and yields the explicit exercise rule that stops at the first time the current stopping factor outweighs all later factors.

What would settle it

Run the same algorithm on a 100-asset payoff whose optimal exercise region is fragmented into many disjoint zones, for example a basket payoff with several separate in-the-money bands, and compare the low-biased estimate with a dual upper bound; if the gap exceeds the Monte Carlo error, the chosen network architecture is too small to represent the optimal stopping time.

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Extended reading notes

Core claim

The paper's central claim is that the supremum over all stopping times in a discrete-time optimal stopping problem can be approximated by a supremum over neural network parameters. This is made possible by a factorisation lemma: any stopping time adapted to a Markov process can be written as a sum over time indices of indicator functions of the observed path, and these indicators can be relaxed into neural-network outputs that define a randomised stopping time. The resulting objective function, an expectation of the payoff weighted by these randomised stopping factors, is maximized by stochastic gradient ascent; a final threshold rule converts the trained network into a genuine stopping time. The paper reports that this single-training-pass procedure matches reference values across many benchmark problems, including American puts, geometric average options, max-call options, basket options under local volatility, and a path-dependent derivative recast as a 100-dimensional problem.

Load-bearing premise

The method assumes that a feedforward network with two hidden layers, each as wide as the number of underlying assets, can represent the optimal stopping rule closely enough for the tested payoff structures and dynamics; this is supported by numerical examples but not by a convergence proof.

Editorial extensions

If this is right

Bermudan and American options on hundreds or thousands of underlying assets can be priced without any grid or mesh in the asset dimension, because the networks scale with the dimension of the state rather than the size of a state space.
The method outputs an approximate exercise strategy, not only a price, so it can be used to guide early-exercise decisions in practice.
The same formulation applies to any simulable Markov process, including path-dependent derivatives once they are embedded in a higher-dimensional Markovian state.
Because the final Monte Carlo estimate uses a genuine stopping time, the reported price is a low-biased lower bound; reference values are needed to judge how close it is to the true price.
The 5000-dimensional max-call result suggests that many high-dimensional exercise problems have enough low-dimensional structure for a moderately sized network to capture.

Reading between the lines

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

Beyond the paper: the factorisation construction could be reused for related discrete decision problems, such as optimal stopping with multiple exercise rights or swing options, by replacing the single stopping indicator with a sequence of exercise decisions.
Beyond the paper: pairing this low-biased primal estimate with a dual upper bound method would turn the reported point estimates into proper confidence intervals, which is a natural next step the paper does not pursue.
Beyond the paper: a testable extension is to apply the same architecture to payoffs with fragmented or discontinuous exercise regions, such as barrier-like payoffs, where the smooth network representation may need more layers or width to stay accurate.

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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 one thing to know: this is a genuinely useful reformulation of the deep optimal stopping approach in Becker–Cheridito–Jentzen. Instead of solving a backward recursion, they optimize a single objective over randomized stopping times, then convert the relaxed factors into a true stopping time via (44). The factorisation lemma (Lemma 2.2) is correct and clean, and the low-bias caveat is stated plainly in (48). I came away with a good opinion of the algorithmic idea and the exposition.

What is new and good: the single-objective construction is a real simplification, not a repackaging. The numerical results for standard Bermudan max-call benchmarks in 2, 3, and 5 dimensions match the Andersen–Broadie and Broadie–Cao confidence intervals well. The geometric-average examples are validated against exact one-dimensional representations from Proposition 4.3; those are genuine checks, since the distributional symmetry is mathematically exact. Runtime scaling up to a few hundred dimensions looks plausible.

The soft spot, and it is a real one: the d=5000 max-call benchmark is not independent. Table 8 replaces the exact price (120) by 165.430, which is itself a realisation of the same algorithm at M=6000. So the relative errors in Figure 1 and Table 8 measure training convergence, not accuracy against any external truth. Inequality (48) guarantees only a lower bound, and no dual upper bound is computed for d=5000, so the evidence cannot rule out a substantial low bias. The non-monotonic training path (M=750 at 103.764 after M=500 at 156.038) reinforces that a single run can stall in a poor local optimum. The authors disclose the self-reference, so this is not a hidden flaw, but the advertised claim of “highly effective in 5000 dimensions” is stronger than the evidence supports.

A second, smaller concern is the unproven expressiveness assumption: two hidden layers of width d, plus the recursive factorisation in (60), may not represent optimal stopping times for all payoff structures. That is typical for this literature and is mitigated by the breadth of examples, so I treat it as minor.

Who is this for? Researchers in quantitative finance and machine learning who want a practical, scalable algorithm for high-dimensional early-exercise problems. It deserves a serious referee, not a desk reject. My recommendation: send it to peer review, and ask the authors to either (a) compute dual upper bounds for the high-dimensional cases, (b) release the code, or (c) at minimum re-label the d=5000 comparison as a training consistency check rather than an independent accuracy result.

Referee Report

3 major / 4 minor

Summary. This paper proposes a deep-learning algorithm for high-dimensional optimal stopping problems arising in American and Bermudan option pricing. The method represents randomized stopping times through neural-network stopping-time factors, maximizes the expected discounted payoff over network parameters by stochastic gradient ascent (Adam), and then converts the trained randomized rule into a true stopping time whose payoff is evaluated by Monte Carlo. The authors prove a factorisation lemma for discrete stopping times (Lemma 2.2), observe that the final price estimator is low-biased (inequality (48)), and report numerical experiments for Brownian-motion stopping, geometric-average options, Bermudan max-call options up to 5000 dimensions, basket options, and a path-dependent derivative. Most lower-dimensional results are compared with binomial-tree or literature reference values.

Significance. If the numerical claims hold, this is a useful contribution: it offers a single-objective, non-recursive training formulation, avoids the temporal recursion of earlier deep optimal stopping algorithms, and shows empirically stable pricing in hundreds of dimensions. The theoretical lemmas are clearly proved, and the low-bias caveat is stated explicitly. The lower-dimensional comparisons against Longstaff-Schwartz, Andersen-Broadie, Broadie-Cao, and other literature benchmarks are genuine strengths. However, the headline high-dimensional claim rests on the d=5000 max-call experiment whose reference value is generated by the algorithm itself; without an independent upper bound or benchmark, the accuracy in high dimension is not established.

major comments (3)

[§4.4.1.2, Table 8] The reference value 165.430 is not independent: the text states that the exact value of the price (120) has been replaced by a realisation of P with M=6000. Consequently, the relative errors in Table 8 and Figure 1 compare the algorithm to another run of itself and measure training stability, not proximity to the true optimal stopping value. Moreover, the 95% confidence interval for that reference is [165.378, 165.483], so the reported errors below 10^-3 are within the Monte Carlo noise of the reference. The row M=750 (103.764) also shows that training can stall in poor local optima, so a single self-referential baseline cannot certify reliability. This is load-bearing because the abstract and introduction claim effectiveness in 5000 dimensions.
[§2.7, Eq. (48)] Since the paper only establishes a lower bound and no dual upper bound or other independent bound is provided for d=5000, the numerical evidence cannot rule out a substantial low bias. The authors should either compute a dual upper bound (for example, an Andersen-Broadie type bound) for the max-call example or clearly restrict the high-dimensional effectiveness claim to the validated lower-dimensional cases. This is fixable, but as it stands the central high-dimensional claim is not fully supported.
[§2.4 and §3.2, Eq. (60)] The approximation power of the network class is an assumption rather than a theorem: no expressiveness or trainability result is given for the two-hidden-layer architecture of width d with the recursive factorisation (60). This is acceptable for a numerical paper only if independent benchmarks carry the validation. Since the main high-dimensional benchmark is self-referential, the high-dimensional claim remains conjectural. A convergence or approximation result, even for a simplified setting, or a broader set of independently benchmarked high-dimensional examples would strengthen the paper considerably.

minor comments (4)

[§4.4.1.2, text after Table 8] The sentence 'the runtime in seconds need or calculating the realisation of P' contains a typo; it should read 'needed for calculating'. In addition, J0 appears as '2 20' and should be typeset as 2^20.
[§2.6] The phrase 'under suitable hyptheses' contains a typo; it should read 'under suitable hypotheses'.
[§4.4.1.1, Table 7] The table lists no reference values for d=10 to 500. The caption should state explicitly that these rows are unbenchmarked, so that readers do not mistake them for validated accuracy claims.
[§4.1, Proposition 4.3 proof] The formula 'Itˆ o's formula' has a formatting artifact in the proof; it should be 'Itô's formula'. The same artifact appears in the text before equation (81).

Circularity Check

1 steps flagged · score 6.0 of 10

The d=5000 benchmark in §4.4.1.2 uses the algorithm's own M=6000 output as the reference value, so the reported relative approximation errors measure convergence to one self-generated realisation, not accuracy against the true optimal stopping value.

fitted input called prediction [Section 4.4.1.2, Table 8 and Figure 1 (high-dimensional Bermudan max-call benchmark)]
"In the approximative calculations of the relative approximation error the exact value of the price (120) has been replaced by the value 165.430, which corresponds to a realisation of P with M = 6000."

The reference value used to define the relative approximation error is itself a realisation of P, the algorithm's own price estimator, at M=6000. Consequently the row M=6000 has relative error exactly 0, and the other rows report the distance of earlier training steps to that single later run. They do not measure the distance to the true optimal stopping price (120). Since (48) only ensures P is a lower bound and no dual upper bound is computed for d=5000, the table cannot substantiate the claimed accuracy; the reported tiny errors are forced by the choice of benchmark.

full rationale

The derivation of the algorithm in Sections 2–3 is mathematically self-contained: Lemma 2.2 gives a factorisation of stopping times, equation (60) defines the randomised stopping-time architecture, and the objective (39) is a direct relaxation of the optimal stopping supremum. None of these steps reduces to its own conclusion. The lower-bound property (48) is correctly stated. The validation examples with one-dimensional representations (Subsections 4.3.1–4.3.2) use independent binomial-tree references from Smirnov's website or closed-form results, and the low-dimensional max-call benchmarks in §4.4.1.1 cite external confidence intervals from Andersen–Broadie and Broadie–Cao, so those comparisons are not circular. The problematic step is confined to the 5000-dimensional example in §4.4.1.2, where the 'reference value' 165.430 is a realisation of the algorithm's own output P with M=6000. The reported relative error at M=6000 is 0 by construction, and the non-monotone training behaviour (e.g., M=750 gives 103.764 while M=500 gives 156.038) shows that a single self-generated reference is fragile as evidence of accuracy. Because the paper's headline high-dimensional effectiveness claim leans heavily on this table, the circularity is partial but real. There is no load-bearing self-citation or imported uniqueness theorem; the issue is a self-referential numerical benchmark.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The algorithm introduces no new physical or mathematical entities, but it relies on several unproved modelling and optimisation assumptions: Markovian structure, a specific neural network class, hand-chosen hyperparameters, and convergence of stochastic gradient ascent. These are the main sources of uncertainty in the central claim.

free parameters (3)

learning rate schedule = piecewise constant, e.g., 5e-2, 5e-3, 5e-4
The learning rate is chosen by hand and varied per example; it is a tuning parameter that affects convergence and accuracy.
network width and depth = two hidden layers of width d
The architecture is chosen ad hoc and is assumed sufficient for the tested problems; no systematic justification is given.
number of training steps M = varies from 500 to 6000 per example
M is chosen manually per example and directly controls runtime and accuracy.

assumptions (4)

domain assumption The underlying process is Markovian after time discretisation.
The stopping time factors in (29) depend only on the current state, relying on the Markov property stated in Subsection 2.4.
domain assumption The Euler-Maruyama discretisation accurately approximates the continuous-time optimal stopping problem.
Equation (7) approximates the continuous-time stopping problem by a discrete-time problem, but no discretisation error bound is provided for the stopping problem.
ad hoc to paper The chosen neural network class can approximate the optimal stopping rule well enough.
Expressiveness and trainability of the specific architecture are assumed without proof; the paper appeals to numerical evidence instead.
domain assumption Stochastic gradient ascent with Adam converges to a good local maximum of the objective.
The training success is inferred from numerical results, not from a convergence theorem for the non-concave objective.

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Cite this review

Pith. "Pith review of Solving high-dimensional optimal stopping problems using deep learning." pith.science (2026). https://pith.science/paper/AYI6HYED

@misc{pith2026190801602,
  author       = {Pith},
  title        = {Pith review of: Solving high-dimensional optimal stopping problems using deep learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYI6HYED}},
  note         = {Machine review of arXiv:1908.01602}
}

read the original abstract

Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

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