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arxiv: 2605.06688 · v1 · submitted 2026-05-01 · 💱 q-fin.CP · math.PR· math.ST· stat.TH

Recognition: 2 theorem links

· Lean Theorem

American Options Pricing under Heston Model via Curriculum Learning in Coupled PINNs

Amit N. Kumar, Rohan, Siddanth Shetty

Pith reviewed 2026-05-11 01:14 UTC · model grok-4.3

classification 💱 q-fin.CP math.PRmath.STstat.TH
keywords American optionsHeston modelphysics-informed neural networkscurriculum learningfree boundarystochastic volatilityoption pricingdeep learning
0
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The pith

Coupled physics-informed neural networks with curriculum learning solve the free-boundary problem for American options under the Heston stochastic-volatility model.

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

American options can be exercised early, which requires finding an unknown moving exercise boundary at the same time as the option price, and the Heston model makes volatility itself random so no closed-form answer exists. The paper trains two linked networks inside a physics-informed framework: one network outputs the option price while satisfying the Heston partial differential equation, and the second outputs the boundary location. Curriculum learning starts with simpler problems and ramps up to the full case, while adaptive resampling adds more training points where the current error is largest. If the training succeeds, the result is a mesh-free solver that returns both price and boundary after a single forward pass. A sympathetic reader would care because traditional grid or Monte Carlo methods become slow or unstable when volatility changes over time and the boundary must be tracked.

Core claim

The central claim is that a pair of coupled physics-informed neural networks, one for the American option price and one for the free boundary, can be trained to satisfy the Heston PDE together with the early-exercise condition; curriculum learning that gradually raises problem difficulty combined with adaptive resampling of high-error points produces stable convergence and yields accurate prices and boundaries on test cases.

What carries the argument

Coupled physics-informed neural networks in which the price network and the boundary network are optimized jointly so that the PDE residual, terminal payoff, and free-boundary conditions are all enforced in the loss function.

If this is right

  • The networks produce option prices and exercise boundaries that agree with established numerical references for typical American put contracts.
  • After training, prices at any time and asset level can be obtained in milliseconds without re-solving a grid.
  • Curriculum learning and adaptive resampling reduce the sensitivity of training compared with plain physics-informed optimization.
  • The method supplies a mesh-free alternative to finite-difference or Monte Carlo schemes for stochastic-volatility American pricing.
  • Rapid evaluation opens the door to repeated pricing inside larger risk or hedging calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same coupled-network structure could be reused for American options under other volatility dynamics such as SABR or rough-volatility models.
  • Fast forward passes would allow real-time calculation of Greeks or portfolio sensitivities that include early-exercise effects.
  • The networks could be placed inside an outer optimization loop to calibrate model parameters to observed American option quotes.
  • The approach might transfer to other free-boundary problems in finance, such as optimal stopping decisions under stochastic control.

Load-bearing premise

The training procedure will converge reliably to the correct free-boundary solution rather than to unstable or inaccurate local minima.

What would settle it

For a standard set of Heston parameters, the network prices or boundaries deviate by more than a few basis points from reference values obtained by a fine-grid finite-difference solver or a binomial tree.

Figures

Figures reproduced from arXiv: 2605.06688 by Amit N. Kumar, Rohan, Siddanth Shetty.

Figure 1
Figure 1. Figure 1: Overview of the proposed coupled PINN architecture. [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The three phases describing how training proceeds. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Power-law distributions used for biased sampling. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scatter plot representing the distribution on a 2d grid of [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: ReLU and GELU activation functions. The solution network consists of 5 hidden layers with the number of neurons in each layer given by [32, 64, 128, 64, 32]. The boundary network uses a different 5-layer structure: [32, 64, 64, 64, 16]. To ensure a stable start to training, we initialize the bias of the final layer of the boundary network to 0.9. To implement the curriculum learning strategy, the training … view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the free boundary surface across time [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Training loss convergence over all epochs. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: illustrates the option price (y-axis) as a function of the asset price (x-axis) for different re￾maining times to maturity. As evident from the graphs, at maturity, the option price follows the payoff function max(K − S, 0). As the time to maturity increases, the option price for a given asset price also increases, reflecting the time value of the option [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 3D surface plot of the option price with respect to asset price and variance at time of maturity. [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 3D surface plots of the Delta and Vega risk sensitivities at [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Absolute PDE residual heatmaps across varying volatilities utilizing adaptive resampling. [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Absolute PDE residual heatmaps demonstrating high boundary errors without adaptive [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
read the original abstract

In American options, the early exercise feature allows the option to be exercised at any time prior to expiration. However, this flexibility introduces a challenge: the pricing model must value the option while simultaneously determining an unknown, time-varying exercise boundary. The Heston model is one of the most popular ways to model real market behavior because it allows volatility to change over time. However, unlike European options, there is no closed-form solution for American options under the Heston model, so we have to use numerical methods. In this paper, we propose a novel approach to solving the stochastic Heston partial differential equation for American options, using coupled physics-informed neural networks (PINNs) to predict both the option price and the free boundary, while employing curriculum learning and adaptive resampling to stabilize model training. Our work builds on recent deep learning methods but introduces a more effective training strategy to address the limitations of these approaches. The numerical results demonstrate the effectiveness of the proposed learning framework, providing a robust and efficient alternative to pricing American options, enabling rapid inference and accurate estimation under stochastic volatility.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a coupled physics-informed neural network (PINN) framework, augmented with curriculum learning and adaptive resampling, to solve the free-boundary PDE arising from American option pricing under the Heston stochastic volatility model. The method simultaneously predicts the option price and the time-dependent exercise boundary, claiming to provide a robust, efficient alternative to traditional numerical methods with the advantage of rapid inference after training.

Significance. If the numerical results can be shown to match or exceed the accuracy of established methods (e.g., finite-difference schemes or Monte Carlo with regression) while delivering fast inference, the work would add a useful data-driven tool for pricing American options with stochastic volatility. The curriculum-learning and adaptive-resampling strategies address known training difficulties in free-boundary PINNs and therefore represent a plausible incremental advance, but the significance hinges on quantitative validation that is currently absent.

major comments (3)
  1. [Abstract and §4] Abstract and §4 (Numerical Results): the claim that 'the numerical results demonstrate the effectiveness of the proposed learning framework' is unsupported because no quantitative metrics (L² or L^∞ errors, convergence rates, or direct comparisons against finite-difference or other PINN baselines) are reported, nor is any detail given on how the free-boundary complementarity condition is enforced in the loss.
  2. [§3] §3 (Methodology): the coupled-PINN loss formulation is described only at a high level; the precise weighting of the PDE residual, boundary losses, and free-boundary penalty terms, together with the mechanism that couples the price and boundary networks, must be stated explicitly so that readers can verify that the American early-exercise condition is correctly imposed.
  3. [§4 and §5] §4 and §5: no sensitivity study or ablation on hyper-parameter choices (network depth/width, loss weights, curriculum schedule, resampling frequency) is provided, leaving the weakest assumption—that the method converges reliably without significant initialization or hyper-parameter dependence—unexamined.
minor comments (2)
  1. [§2] Notation for the Heston parameters (κ, θ, σ, ρ) and the free-boundary function B(t) should be introduced once in §2 and used consistently thereafter.
  2. [§4] A brief comparison table listing training time, inference time, and error against at least one standard benchmark (e.g., the Heston finite-difference solver) would greatly improve readability of the results section.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript accordingly to strengthen the presentation of results and methodology.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (Numerical Results): the claim that 'the numerical results demonstrate the effectiveness of the proposed learning framework' is unsupported because no quantitative metrics (L² or L^∞ errors, convergence rates, or direct comparisons against finite-difference or other PINN baselines) are reported, nor is any detail given on how the free-boundary complementarity condition is enforced in the loss.

    Authors: We agree that the current manuscript would benefit from explicit quantitative validation. In the revised version we will add tables reporting L² and L^∞ errors for both the option price and the free boundary, computed against a high-resolution finite-difference reference solution. We will also include convergence plots with respect to training iterations and network capacity. Regarding the complementarity condition, we will explicitly state that it is enforced by augmenting the loss with a penalty term ∫ max(V - payoff, 0) dx dt over the continuation region, whose weight is chosen to balance the PDE residual. revision: yes

  2. Referee: [§3] §3 (Methodology): the coupled-PINN loss formulation is described only at a high level; the precise weighting of the PDE residual, boundary losses, and free-boundary penalty terms, together with the mechanism that couples the price and boundary networks, must be stated explicitly so that readers can verify that the American early-exercise condition is correctly imposed.

    Authors: We will expand Section 3 with the exact loss expression L = λ_PDE L_PDE + λ_BC L_BC + λ_FB L_FB + λ_comp L_comp, providing the numerical values used for each λ. The coupling mechanism will be described as follows: the boundary network outputs the time-dependent exercise boundary b(t), which is then used to partition the spatial domain for the price network; the price network is trained only in the continuation region while the complementarity penalty is applied globally. This formulation ensures the early-exercise condition is imposed both by domain restriction and by the penalty term. revision: yes

  3. Referee: [§4 and §5] §4 and §5: no sensitivity study or ablation on hyper-parameter choices (network depth/width, loss weights, curriculum schedule, resampling frequency) is provided, leaving the weakest assumption—that the method converges reliably without significant initialization or hyper-parameter dependence—unexamined.

    Authors: We acknowledge the absence of systematic hyper-parameter studies. In the revised manuscript we will add a dedicated subsection containing ablation experiments that vary network depth and width, the relative loss weights, the number of curriculum stages, and the frequency of adaptive resampling. For each variation we will report both final accuracy and training stability metrics, thereby addressing concerns about initialization and hyper-parameter sensitivity. revision: yes

Circularity Check

0 steps flagged

No circularity: standard PINN residual minimization for free-boundary PDE

full rationale

The paper applies coupled PINNs to minimize the Heston PDE residual plus boundary and free-boundary losses, augmented by curriculum learning and adaptive resampling. This is a direct numerical solver construction with no self-definitional loops, no fitted parameters renamed as predictions, and no load-bearing self-citations that reduce the central claim to prior author work. Numerical results are presented as empirical validation of the training strategy rather than a closed derivation that collapses to its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim depends on the standard Heston PDE formulation and the assumption that PINNs can solve it effectively with the proposed training strategy.

free parameters (1)
  • PINN hyperparameters (network depth, width, loss weights)
    Typical in neural network training, chosen to stabilize the coupled optimization for price and boundary.
axioms (1)
  • domain assumption The Heston stochastic volatility model PDE with free boundary condition for American options
    Standard stochastic volatility model used in finance.

pith-pipeline@v0.9.0 · 5494 in / 1245 out tokens · 48979 ms · 2026-05-11T01:14:34.600344+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Reference graph

Works this paper leans on

29 extracted references · 2 canonical work pages · 1 internal anchor

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