Boxed In, Cashed Out: Deep Gradient Flows for Fast American Option Pricing

TL;DR for operators

American options are awkward because the model must decide not only what the contract is worth, but also where exercise becomes optimal. That turns pricing into a free-boundary problem, which is exactly the kind of thing that makes high-dimensional PDE methods start sweating through their nice academic shirts.

Jasper Rou’s paper extends Time Deep Gradient Flow (TDGF) to American basket put options under multidimensional Black-Scholes and Heston models.¹ The useful trick is not “throw a neural network at finance”. We have tried that spell before; it produces PowerPoint before it produces risk control. The actual mechanism is more specific: TDGF trains a neural PDE solver through time steps, only applies the PDE loss in the continuation region, and builds the payoff floor into the network so the model cannot price below the intrinsic value.

The headline result is practical but narrow. Across two- and five-asset Black-Scholes and Heston tests, TDGF and DGM produce curves close to a Longstaff-Schwartz Monte Carlo reference. TDGF trains materially faster than DGM: from 4,543 vs 8,293 seconds in the two-asset Black-Scholes case to 12,881 vs 41,718 seconds in the five-asset Heston case. After training, both neural methods compute prices in milliseconds, while the paper’s Monte Carlo reference takes several seconds per computation.

The operator’s interpretation: this is a candidate pattern for offline training and fast repeated valuation, not a permission slip to delete Monte Carlo, validation, or model risk governance. The paper does not test calibration to market data, hedging Greeks, stressed regimes, dividends, transaction costs, or production-grade portfolios. Still, it shows a credible path: if the pricing surface is stable enough for a product family, neural PDE methods may help move repeated scenario valuation out of the “go make coffee” category.

The real problem is not the payoff; it is the exercise boundary

An American option is not just a European option with a more assertive personality. A European option asks for value at maturity. An American option asks a harder question at every point before maturity: exercise now, or continue holding?

For a basket put, the payoff in the paper is:

That payoff is easy to write. The difficulty is deciding where the option value equals the payoff and where it exceeds the payoff. In the exercise region, the holder should cash out. In the continuation region, the option value follows the pricing PDE. The boundary between those regions is not given in advance. It must be discovered.

That is why American-option pricing is usually framed as a variational inequality or a free-boundary PDE. The solver must satisfy two conditions at once:

The paper’s main contribution is to adapt TDGF to this structure. The important word is “adapt”. TDGF was already proposed for option-pricing PDEs, but American options add the awkward free-boundary condition. The extension has to answer a practical modelling question: how do you train a neural PDE solver when the PDE is not supposed to hold across the whole domain?

The answer is selective training. In TDGF, the method trains the PDE only on points where the current approximation is above the payoff. That is the continuation region. In plain English: do not punish the model for failing to satisfy a continuation PDE where the contract should simply be exercised. Revolutionary? No. Necessary? Yes. Much of applied machine learning is just remembering not to optimise the wrong thing at scale.

TDGF wins its speed argument through mechanism, not magic

The paper compares TDGF with the Deep Galerkin Method (DGM). Both are neural approaches to solving PDEs. The difference is in how they train.

DGM minimises a residual of the free-boundary PDE over the time-space domain. That residual involves the differential operator directly, which means the neural network training must deal with second derivatives. This is not computationally free. Autodifferentiation is useful, not charitable.

TDGF instead first discretises time, then reformulates the PDE step as an energy minimisation problem. It advances the solution through time, using the previous time step as the starting point for the next one. The paper attributes TDGF’s faster training to this time stepping and to the absence of a second derivative in the cost function.

The method can be read as a sequence:

Known payoff at maturity / zero time-to-maturity
        ↓
Network learns continuation value above payoff
        ↓
Payoff floor is embedded through softplus architecture
        ↓
PDE loss is applied only where continuation is active
        ↓
Time-stepped optimisation builds the pricing surface
        ↓
Fast valuation after training

That is the article’s central point. The speed result is not merely a benchmark table. It follows from design choices that reduce the training burden and make the problem better aligned with the American-option structure.

Three choices matter most.

First, TDGF uses time stepping. Instead of learning the whole object in one large residual-fitting problem, it advances through time slices. This gives the next optimisation step a useful starting point: the previous network parameters.

Second, TDGF avoids second derivatives in the loss. DGM’s PDE residual requires second-order derivative information. TDGF’s energy formulation works with gradients in a way that reduces this burden. In high-dimensional settings, that difference can become material.

Third, the network does not learn the full price from scratch. It learns the continuation value on top of the intrinsic value. The architecture is written so that:

which keeps the price above the payoff. That is a clean example of financial structure being built into the model rather than left to the optimiser to discover while wandering around the loss landscape with a tiny lantern.

The box-sampling trick is an implementation detail with strategic consequences

The paper’s sampling section looks minor until you think about what happens in five dimensions.

For an equal-weight basket, the relevant moneyness is the average of the individual asset moneynesses. If each asset is sampled uniformly, the average concentrates near the middle. The model then sees plenty of examples around the centre and relatively few near the edges. Unfortunately, option-pricing surfaces often become interesting precisely near boundaries, kinks, and exercise regions. A sampler that ignores those regions is not neutral; it is a quiet saboteur.

Rou addresses this by splitting the sampling domain into boxes and sampling from each box separately. Figure 1 is best read as an implementation diagnostic, not as main performance evidence. It shows why naive uniform sampling underrepresents edge moneyness in five dimensions and how box sampling spreads samples more evenly. Figure 2 illustrates the box construction.

The paper then uses this sampling logic differently for TDGF:

This is more than a neat sampling footnote. In production pricing, bad sampling is one way a model becomes confidently wrong in the exact region where traders care. The paper does not run a formal ablation proving box sampling is the decisive factor. Still, it identifies the right operational anxiety: a high-dimensional solver is only as good as the domain it actually sees.

The evidence says “similar accuracy”, not “universal replacement”

The numerical experiments focus on American basket put options. The paper excludes dividend-paying American calls because, with no dividends and non-negative rates, the optimal exercise strategy for a call is to wait until maturity, making the American call price equal to the European call price. That is not a minor setup detail. It tells us the experiments are aimed at the early-exercise problem where the free boundary actually matters.

The comparison set is:

For accuracy, the paper uses 1,000 Monte Carlo paths with 1,000 time steps and Longstaff-Schwartz regression with polynomial order 4 on paths where intrinsic value is positive. Evaluation is shown by plotting continuation value against moneyness on an equidistant grid of 47 points, with moneyness and volatility equal in each dimension.

That last sentence matters. The accuracy evidence is not a full error table over the entire state space. It is a set of plotted slices through the pricing surface. In the figures, TDGF, DGM, and Monte Carlo produce similar curves across two- and five-dimensional Black-Scholes and Heston settings. That supports the paper’s claim of high accuracy in the tested scenarios. It does not support the stronger claim that the method is validated across all states, all calibrations, or all exotic payoffs.

The figures have a clear evidentiary hierarchy:

The chart evidence is persuasive for the paper’s defined experiment. It is not a production validation report. That distinction is not pedantry. It is the difference between “this method deserves attention” and “please put this in tomorrow’s risk stack and see what the regulator thinks”. One of those is reasonable.

The speed result is the business-relevant part

The strongest numerical result is not the visual similarity of the curves. It is the training and computational time comparison.

The paper reports the following training times in seconds:

The pattern is important: TDGF’s advantage grows in the harder tested cases. The five-asset Heston case is the most computationally expensive, and it is also where TDGF shows the largest relative reduction versus DGM.

Post-training computation is even more striking. The paper reports average computational times over 34 computations at different time points:

This is the business pathway. Training is paid upfront. Once trained, the neural solver can evaluate very quickly. For a desk that needs repeated scenario runs, intraday risk views, portfolio sweeps, or what-if analysis, milliseconds matter more than they sound like they should. Pricing one contract once is not the problem. Pricing many contracts under many shocks while everyone pretends the spreadsheet is still “temporary” is the problem.

The practical value is therefore not “AI prices options”. The practical value is:

This is the part a quant platform team should care about. Not because the paper proves production readiness. It does not. But because it shows a plausible architecture for turning a difficult pricing problem into a reusable valuation surface.

Where the method would fit in a derivatives workflow

A sensible deployment interpretation would not replace the existing pricing stack overnight. It would start as a parallel engine.

A practical workflow might look like this:

1. Use conventional methods to build reference values for a defined product family.
2. Train TDGF-style neural pricing surfaces offline for that family and model setup.
3. Validate against reference methods across a richer state grid than the paper reports.
4. Stress the surface near exercise boundaries, low-liquidity regions, and extreme moneyness.
5. Use the neural surface for rapid scenario sweeps where full Monte Carlo is too slow.
6. Keep conventional methods for independent checks, model governance, and periodic recalibration.

The attractive use case is repeated valuation under controlled assumptions. Think internal scenario analysis, exploratory risk dashboards, and pre-trade approximation for structures that later receive full pricing treatment. The least attractive use case is pretending the neural network has magically solved model risk because the curves in a paper line up nicely. They do line up nicely. That is not the same as a governance framework.

The paper’s approach is especially relevant when the pricing task has three characteristics:

If the product changes constantly, if calibration moves violently, or if the payoff has discontinuities the solver has not been designed to handle, the economics become less obvious. A fast wrong surface is still wrong. It just gets there with impressive punctuality.

The boundary of the paper is narrow, and that is fine

The most tempting misconception is that this paper proves neural methods can replace Monte Carlo for American-option pricing. It does not. It tests TDGF and DGM against Longstaff-Schwartz Monte Carlo in controlled American basket put cases under Black-Scholes and Heston assumptions, with up to five assets.

That is a useful result. It is not a universal theorem disguised as a benchmark.

The key boundaries are specific:

These limitations do not weaken the paper’s contribution. They keep it in its proper box. Conveniently, the paper already likes boxes.

The contribution is methodological and computational: TDGF can be adapted to American free-boundary problems, and in the tested settings it delivers similar plotted accuracy with faster training than DGM and very fast computation after training. That is enough to be interesting. It is not enough to be unmonitored infrastructure.

What Cognaptus infers for business use

The paper directly shows that TDGF can be extended to American basket put pricing under the chosen Black-Scholes and Heston experiments, that it produces plotted values similar to DGM and Longstaff-Schwartz Monte Carlo, and that it trains faster than DGM in the reported cases.

Cognaptus infers that the practical value lies in amortised computation. A desk or platform team might spend time training a neural pricing surface, then use it repeatedly for scenario analysis or rapid approximation. The more repeated the valuation task, the more attractive the trade-off becomes.

What remains uncertain is whether this approach survives contact with production requirements: calibration, stress testing, explainability, hedging Greeks, monitoring, model approval, and unpleasant market days. The usual things, in other words. Markets have a gift for finding the part of your model you described as “outside scope”.

The best business reading is therefore neither sceptical dismissal nor AI enthusiasm. It is a targeted question:

Which American-style products in our workflow are stable enough, repeated enough, and slow enough under current methods that an offline-trained neural PDE surface would be worth validating?

That is the right level of ambition. It turns the paper from an academic benchmark into a practical evaluation agenda.

Conclusion: the box is the point

TDGF’s American-option extension is not interesting because it uses a neural network. By now, that phrase has been stapled onto everything from portfolio optimisation to toaster firmware. It is interesting because the method respects the structure of the problem.

American options require a payoff floor, a continuation region, and an exercise boundary. Rou’s TDGF adaptation handles those through architecture, selective PDE training, time stepping, and sampling design. The results are credible within the tested space: similar accuracy curves, faster TDGF training than DGM, and millisecond-scale computation after training.

For derivatives teams, the message is practical. Do not ask whether neural PDE solvers are “better than Monte Carlo” in the abstract. That question is too vague to be useful and too useful to consultants. Ask where training cost can be amortised, where fast repeated valuation matters, and where reference methods can still police the neural approximation.

The option is boxed in by its exercise boundary. The business case is boxed in by validation. Inside that box, TDGF looks worth taking seriously.

Cognaptus: Automate the Present, Incubate the Future.