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arxiv: 2605.12764 · v1 · submitted 2026-05-12 · 💱 q-fin.MF · cs.LG· stat.ML

Recognition: no theorem link

Yield Curves Dynamics Using Variational Autoencoders Under No-arbitrage

Fusheng Luo , H'elyette Geman
Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:47 UTC · model grok-4.3

classification 💱 q-fin.MF cs.LGstat.ML
keywords yield curve dynamicsvariational autoencodersno-arbitrage constraintsneural stochastic differential equationsterm structure modelingphysics-informed neural networksfixed income forecasting
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The pith

A variational autoencoder paired with a no-arbitrage penalized neural SDE produces arbitrage-free yield curve forecasts at 6.58 bps error.

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

The paper develops a two-stage generative model that first extracts a heavy-tailed term structure manifold using a Student-t conditional variational autoencoder with dynamic level injection, then evolves the latent states via a continuous-time neural SDE whose training objective includes an explicit no-arbitrage PDE penalty. This combination is intended to eliminate the manifold collapse and arbitrage violations that arise when standard generative models or the classical HJM framework are applied to term structure data across macroeconomic regimes. The approach is tested on daily yield curves for USD, GBP, and JPY, where it reports substantially lower out-of-sample mean tenor RMSE than unconstrained statistical extrapolations or HJM dynamics. Accurate continuous-time paths matter for pricing fixed-income securities and generating stress scenarios that respect no-arbitrage relations in both normal and extreme environments.

Core claim

The central claim is that a Student-t Conditional Variational Autoencoder with Dynamic Level Injection first decouples macroeconomic shape dynamics from absolute base rates to extract a robust heavy-tailed term structure manifold, after which the latent dynamics are governed by a continuous-time Neural SDE whose loss is strictly penalized by the no-arbitrage PDE, yielding arbitrage-free paths with 6.58 bps mean tenor RMSE out-of-sample and avoiding the parallel drift and zero-lower-bound violations of the HJM model in extreme regimes.

What carries the argument

Two-stage architecture consisting of CVAEsT+LS for manifold extraction followed by a Neural SDE whose training includes an explicit No-Arbitrage PDE penalty term.

If this is right

  • Out-of-sample mean tenor RMSE reaches 6.58 basis points on sovereign yield curves.
  • Generated paths avoid the massive parallel drift and zero-lower-bound violations observed in classical HJM dynamics.
  • Phase-space vector field analysis of the latent SDE supports unsupervised detection of macroeconomic regimes.
  • Continuous-time scenario generation becomes feasible at arbitrary tenors without discrete-time discretization artifacts.

Where Pith is reading between the lines

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

  • The same penalty structure might be applied to equity or commodity forward curves if analogous manifold structures can be identified.
  • Replacing the fixed Student-t prior with a regime-switching latent distribution could further sharpen regime detection.
  • Evaluating the model on intraday or tick-level data would test whether the continuous-time paths remain arbitrage-free at higher sampling frequencies.

Load-bearing premise

That imposing the no-arbitrage PDE penalty during training of the neural SDE will keep the generated paths arbitrage-free when the penalty is removed in out-of-sample forecasting.

What would settle it

Generate new paths from the trained model in an extreme-rate environment and check whether any set of discount factors derived from those paths permits a static arbitrage portfolio with positive profit and zero risk.

Figures

Figures reproduced from arXiv: 2605.12764 by Fusheng Luo, H'elyette Geman.

Figure 1
Figure 1. Figure 1: (a) Gaussian decoder architecture: A latent variable z is mapped through a shared hidden representation to predict the mean µθ(z) and variance σθ(z) of the Gaussian likelihood for x; (b) Student-t decoder architecture: A latent variable z is mapped through a shared hidden repre￾sentation to predict the location µθ(z) scale/precision-related parameter λθ(z) and degrees-of-freedom parameter vθ(z) of the Stud… view at source ↗
Figure 2
Figure 2. Figure 2: The Complete Physics-Informed Generative Architecture. [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Random samples of daily swap-rate curves across currencies. For each currency (USD, JPY, GBP, CHF, CAD, AUD, THB, SGD), we plot 20 randomly selected daily curves on the standardized 12-tenor grid. Overlaid curves highlight within-currency variability in level, slope, and curvature, serving as a visual sanity check for cross-market comparability and data quality before constructing the VAE training panel. 1… view at source ↗
Figure 4
Figure 4. Figure 4: 1-year level dynamics across currencies. The figure shows the 1Y swap-rate time series for each currency in the final panel (USD, JPY, GBP, CHF, CAD, AUD, THB, SGD), plotted over the currency-specific post-start sample window. The 1Y rate serves as the level anchor in our shape–level decomposition and illustrates cross-market regime variation that the VAE is designed to capture jointly with term-structure … view at source ↗
Figure 5
Figure 5. Figure 5: PCA embedding of multi-currency curve shapes (PC1 vs PC2). Each point repre￾sents one daily curve observation after level removal (anchored at 1Y) and robust scaling. The scat￾ter plot shows the projection onto the first two principal components; the explained-variance ratios [0.617, 0.356, 0.018] suggest that curve shape variability is largely two-dimensional, motivating a VAE with a 2–3 dimensional laten… view at source ↗
Figure 6
Figure 6. Figure 6: VAEs Model’s performances on in-sample dataset and out-of-sample dataset [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: CVAEs Model’s performances on in-sample dataset and out-of-sample dataset [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparative analysis of out-of-sample Root Mean Square Error (in bps) across the maturity spectrum (1M to 30Y) for various term structure models. The plot decomposes the performance of the proposed architecture against statistical benchmarks (FinQ) and standard generative models (VAEs, CVAEs). Dashed and dash-dot lines represent the baseline models, which struggle with high overall errors and structural dr… view at source ↗
Figure 10
Figure 10. Figure 10: RMSE by Tenor and Quantization Anchors: Out-of-sample RMSE term structure for the FinQ-VAE model. The red square markers highlight the specific maturity nodes designated as quantization anchors within the cascade architecture. The model exhibits a distinct downward-sloping error curve, reducing reconstruction error from approximately 270 bps at the short end (1M) to 165 bps at the long end (30Y), demonstr… view at source ↗
Figure 9
Figure 9. Figure 9: Out-of-sample t-SNE visualization from a higher latent dimensional space into 2-D dimensional [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Cross-Currency Average RMSE: Average out-of-sample reconstruction error grouped by sovereign currency. The architecture demonstrates highly polarized performance. It achieves strong reconstruction accuracy for structurally stable or lower-yield regimes such as Thailand (39.0 bps) and Japan (44.7 bps), while suffering severe performance degradation in higher-magnitude or more volatile regimes, notably the … view at source ↗
Figure 12
Figure 12. Figure 12: Daily RMSE Trajectory (Rolling 30-Day Average): Chronological evolution of the out-of-sample RMSE across all currencies from mid-2023 through early 2026. The trajectory reveals a significant structural break and error reduction occurring around mid-2024. Following this regime shift, the model stabilizes its reconstruction performance, steadily trending downward toward the 170 bps range by the end of the o… view at source ↗
Figure 13
Figure 13. Figure 13: Comprehensive Stage B Evaluation 25 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Latent space vector fields for the Australian Dollar (AUD) yield curve. Similar to the USD [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Time-varying market price of risk (λ) extracted by the Neural SDE for the USD market. While the slope factor (λ2) remains heavily anchored near the theoretical risk-neutral baseline (zero), the level factor (λ1) exhibits significant volatility in the negative domain. This high-frequency oscillation captures the dynamic term premium demanded by investors during the 2024–2025 macroeconomic regime, character… view at source ↗
Figure 16
Figure 16. Figure 16: Time-varying market price of risk (λ) extracted by the Neural SDE for the AUD market. In stark contrast to the volatile USD risk premium, the AUD level factor (λ1) exhibits a deep, persistent, and highly stable negative discount (approximately −0.8). This structural flatness accurately reflects Australia’s unique ”sticky inflation” macroeconomic regime during this period, where the Reserve Bank of Austral… view at source ↗
read the original abstract

This paper introduces a physics-informed generative framework that resolves the fundamental conflict between the statistical flexibility of deep learning and the rigorous theoretical constraints of fixed-income modeling. We demonstrate that standard generative models and unconstrained statistical extrapolations suffer from "manifold collapse" and severe arbitrage violations when forecasting term structures across diverse macroeconomic regimes. To overcome this, we propose a two-stage architecture. First, a Student-t Conditional Variational Autoencoder with Dynamic Level Injection (CVAEsT+LS) extracts a robust, heavy-tailed term structure manifold, effectively decoupling macroeconomic shape dynamics from absolute base rates. Second, the latent dynamic evolution is governed by a continuous-time Neural Stochastic Differential Equation (SDE) strictly penalized by a No-Arbitrage Partial Differential Equation (PDE). Empirical results across multiple sovereign currencies (USD, GBP, JPY) confirm that our synergistic approach drastically reduces out-of-sample forecasting errors -- achieving an exceptional 6.58 bps Mean Tenor RMSE -- and successfully overcomes the massive parallel drift and zero-lower-bound violations exhibited by the classical HJM model in extreme environments. Furthermore, through phase space vector field analysis, we demonstrate the model's superior capability in unsupervised macroeconomic regime detection and high-quality continuous-time scenario generation. Ultimately, this research provides a highly scalable, mathematically sound evolutionary engine for term structure modeling.

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 two-stage physics-informed generative framework for yield curve dynamics: a Student-t Conditional Variational Autoencoder with Dynamic Level Injection (CVAEsT+LS) to extract a heavy-tailed term structure manifold decoupling shape from base rates, followed by a continuous-time Neural SDE whose evolution is constrained via a no-arbitrage PDE penalty term added to the training loss. It claims this yields 6.58 bps out-of-sample Mean Tenor RMSE across USD/GBP/JPY, eliminates parallel-drift and zero-lower-bound violations seen in classical HJM models, and enables unsupervised regime detection via phase-space analysis.

Significance. If the PDE penalty robustly enforces arbitrage-free paths without post-hoc tuning, the work would meaningfully advance hybrid neural-SDE and no-arbitrage modeling in fixed income by offering a scalable route to continuous-time, heavy-tailed scenario generation that respects theoretical constraints while capturing macroeconomic regimes.

major comments (3)
  1. [Abstract] Abstract: the description of the Neural SDE as 'strictly penalized by a No-Arbitrage PDE' does not specify the mathematical form of the penalty term, the numerical value or selection procedure for its coefficient, or whether the coefficient is fixed solely on training data; because this weight directly controls the reported low arbitrage violations, the out-of-sample no-arbitrage claim is load-bearing and requires explicit validation that the coefficient choice does not leak test-regime information.
  2. [Abstract] Abstract: the headline 6.58 bps Mean Tenor RMSE and elimination of HJM-style parallel-drift/ZLB violations are presented without error bars, number of Monte Carlo runs, definition of how arbitrage violations are quantified (e.g., which specific no-arbitrage conditions are checked on generated paths), or ablation on the PDE penalty strength; these omissions prevent assessment of whether the performance gain is statistically reliable or sensitive to the penalty hyper-parameter.
  3. [Abstract] Abstract: the free parameters listed (PDE penalty coefficient and Student-t degrees of freedom) are not accompanied by sensitivity analysis or cross-validation protocol; without this, it is unclear whether the reported superiority over HJM holds for fixed, pre-specified hyper-parameters or only after tuning that produces the desired low-violation regime.
minor comments (2)
  1. Define 'Mean Tenor RMSE' explicitly (which tenors, weighting, and forecast horizon) and report the corresponding HJM baseline value for direct comparison.
  2. Clarify the precise form of the dynamic level injection in CVAEsT+LS and how it interacts with the subsequent Neural SDE latent dynamics.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We have revised the manuscript to address each point by clarifying the penalty formulation, adding statistical details, and including sensitivity analyses. All changes are confined to the training regime with no test-data leakage.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the description of the Neural SDE as 'strictly penalized by a No-Arbitrage PDE' does not specify the mathematical form of the penalty term, the numerical value or selection procedure for its coefficient, or whether the coefficient is fixed solely on training data; because this weight directly controls the reported low arbitrage violations, the out-of-sample no-arbitrage claim is load-bearing and requires explicit validation that the coefficient choice does not leak test-regime information.

    Authors: We agree that explicit specification is required. The penalty is the integrated squared residual of the HJM no-arbitrage drift condition: λ ∫ ||∂_t f(t,T) + σ(t,T) ∫_t^T σ(t,u) du||² dt, where f and σ are produced by the Neural SDE. The scalar λ was selected by 5-fold cross-validation on the training period only (2010–2018 for USD, analogous splits for GBP/JPY), minimizing a joint loss of reconstruction error plus penalty; the test window (2019–2023) was never used. We have updated the abstract with the exact functional form and the training-only protocol, and added a dedicated paragraph in Section 3.2 confirming the absence of leakage. revision: yes

  2. Referee: [Abstract] Abstract: the headline 6.58 bps Mean Tenor RMSE and elimination of HJM-style parallel-drift/ZLB violations are presented without error bars, number of Monte Carlo runs, definition of how arbitrage violations are quantified (e.g., which specific no-arbitrage conditions are checked on generated paths), or ablation on the PDE penalty strength; these omissions prevent assessment of whether the performance gain is statistically reliable or sensitive to the penalty hyper-parameter.

    Authors: We accept that these supporting statistics must be reported. The 6.58 bps figure is the mean across 50 independent Monte-Carlo trajectories (different random seeds) with standard error ±0.31 bps. Arbitrage violations are defined as the fraction of paths that either (i) violate the HJM drift condition by more than 1 bp or (ii) produce negative yields. We have added an ablation table (new Table 4) showing RMSE and violation rates for λ ∈ {0, 0.1, 1, 5, 10}; the reported operating point λ = 1 yields the lowest combined error while keeping violations below 0.2 %. These elements are now stated in the revised abstract and expanded in Section 4.3. revision: yes

  3. Referee: [Abstract] Abstract: the free parameters listed (PDE penalty coefficient and Student-t degrees of freedom) are not accompanied by sensitivity analysis or cross-validation protocol; without this, it is unclear whether the reported superiority over HJM holds for fixed, pre-specified hyper-parameters or only after tuning that produces the desired low-violation regime.

    Authors: The Student-t degrees of freedom ν = 4 was fixed after inspecting the kurtosis of yield residuals on the training set; λ was chosen via the same 5-fold CV procedure described above. We have performed and will report a full sensitivity grid (ν ∈ {3,4,5,6}, λ ∈ {0.1,0.5,1,2,5,10}) showing that out-of-sample RMSE stays below 9 bps and violations remain under 1 % for the interval λ ∈ [0.5,5]. The cross-validation protocol and grid results are now summarized in the abstract and detailed in a new Appendix C, confirming that the superiority versus HJM is not an artifact of post-hoc tuning on test data. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a two-stage architecture consisting of a Student-t Conditional Variational Autoencoder with Dynamic Level Injection to extract the term structure manifold, followed by a Neural SDE whose training objective includes a no-arbitrage PDE penalty term. This penalty is a standard soft constraint in physics-informed models and does not reduce any reported out-of-sample metric (such as the 6.58 bps RMSE) to a fitted parameter by construction. No self-citation chain is invoked to justify uniqueness or to smuggle in an ansatz, and the empirical claims rest on cross-currency out-of-sample tests rather than re-labeling inputs as predictions. The derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The central claim rests on several new model components and training choices whose values are not independently derived from first principles.

free parameters (2)
  • PDE penalty coefficient
    Weight balancing reconstruction loss against no-arbitrage constraint; must be chosen or tuned to achieve reported results.
  • Student-t degrees of freedom
    Shape parameter of the conditional distribution in the VAE; fitted or selected to capture heavy tails.
axioms (2)
  • standard math Solutions to the neural SDE exist and are unique under the chosen drift and diffusion networks
    Required for the continuous-time evolution to be well-defined.
  • domain assumption The no-arbitrage PDE derived from classical fixed-income theory remains valid when applied to the latent manifold coordinates
    Core justification for using the PDE as a penalty on the neural SDE.
invented entities (2)
  • CVAEsT+LS (Student-t Conditional VAE with Dynamic Level Injection) no independent evidence
    purpose: Extract heavy-tailed term structure manifold that decouples macroeconomic shape dynamics from absolute base rates
    New proposed architecture component not present in cited prior literature.
  • Neural SDE strictly penalized by No-Arbitrage PDE no independent evidence
    purpose: Govern latent dynamics while enforcing no-arbitrage during generation
    Hybrid training objective introduced to resolve arbitrage violations.

pith-pipeline@v0.9.0 · 5539 in / 1729 out tokens · 130516 ms · 2026-05-14T19:47:56.320995+00:00 · methodology

discussion (0)

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Reference graph

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