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Forward-Time Black-Scholes Reconstruction via Regularized Legendre Reduction

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Pith's one-line read Projecting the forward Black-Scholes PDE onto a finite shifted Legendre basis yields a stable ODE system that Tikhonov regularization solves for the terminal price profile.

desk verdict The paper stabilizes forward Black-Scholes reconstruction with Legendre projection and Tikhonov, proving well-posedness for fixed N and showing decent numerics. read the letter →

arxiv 2606.12450 v1 pith:YDTT46O6 submitted 2026-05-30 q-fin.CP cs.NAmath.NA

Phuong M. Nguyen , Matt Nguyen , Loc H. Nguyen This is my paper · ORCID
classification q-fin.CPcs.NAmath.NA
keywords Black-ScholesequationforwardproblemLegendrepolynomialsTikhonovregularizationill-posedoptionpricingspectralmethodsdimensionreduction
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 addresses the ill-posed forward-time Black-Scholes problem, in which the current option-price profile is given and the profile at maturity must be recovered, by first reducing the spatial dimension via projection onto shifted Legendre polynomials. This produces a finite system of ODEs in time whose solution approximates the PDE while avoiding the S-squared degeneracy at the asset-price boundary. The authors then apply Tikhonov regularization to this reduced system and prove existence, uniqueness, stability with respect to noisy data, and convergence for each fixed truncation level. Numerical tests on smooth, butterfly-spread, and European-put payoffs show that the resulting Legendre-Tikhonov reconstructions recover the terminal profile accurately even when the input data contain noise, while also outperforming conventional quasi-reversibility in stability.

What carries the argument

The finite shifted Legendre expansion in the asset-price variable, which projects the PDE onto a stable ODE system and relaxes the S-squared boundary degeneracy before Tikhonov regularization is applied.

What would settle it

A test in which, for successively smaller noise levels in the initial price profile, the recovered terminal profile does not converge to the known terminal payoff or in which the reconstruction error increases rather than decreases with higher truncation levels.

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Extended reading notes

Core claim

The dimension-reduced Legendre-Tikhonov method, formed by projecting the forward Black-Scholes equation with state-dependent volatility onto a finite shifted Legendre basis in the asset-price variable and then regularizing the resulting ODE system, admits a unique solution for each truncation level that is stable to perturbations in the initial data and converges to the true terminal profile as the noise level tends to zero.

Load-bearing premise

The finite shifted Legendre expansion in the asset-price variable acts as an effective spectral cutoff that relaxes the S-squared degeneracy at the zero-price boundary and yields a stable reduced ODE system whose solution approximates the original PDE solution sufficiently well for reconstruction purposes.

Editorial extensions

If this is right

  • For any fixed truncation level the reduced problem possesses a unique solution that depends continuously on the initial data.
  • The reconstruction converges to the true terminal profile as the noise amplitude in the current price profile tends to zero.
  • The Legendre reduction demonstrably stabilizes the problem relative to the conventional physical-space quasi-reversibility method.
  • The same reduced system admits a PINN solver that serves as a consistent numerical benchmark for the Tikhonov reconstructions.

Reading between the lines

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

  • The same Legendre projection could be inserted into other forward parabolic pricing equations that suffer from similar boundary degeneracy.
  • Choosing the truncation level adaptively from the noise amplitude might further improve practical accuracy without altering the proved stability properties.
  • Because the reduced system is low-dimensional, the method may allow repeated reconstructions inside calibration loops that would be prohibitive with full PDE solvers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper studies the ill-posed forward-time Black-Scholes equation with state-dependent volatility, where the current option-price profile is given and the terminal profile at T must be recovered. It projects the PDE onto a finite shifted Legendre basis in the asset-price variable to obtain a reduced ODE system, applies Tikhonov regularization to this finite-dimensional inverse problem, and claims to prove existence, uniqueness, data stability, and convergence for each fixed truncation level N. Numerical experiments recover terminal profiles for smooth, butterfly-spread, and European put payoffs from noisy data, with comparisons to a reduced PINN solver and the physical-space quasi-reversibility method.

Significance. If the well-posedness and convergence results hold, the Legendre reduction provides a spectral cutoff that relaxes the S^2 degeneracy at S=0 and yields a stable reconstruction method for an important class of inverse option-pricing problems. The combination of dimension reduction, regularization, and secondary PINN benchmark is technically coherent and offers a concrete alternative to existing stabilization techniques.

minor comments (3)
  1. The abstract and introduction state that proofs of existence, uniqueness, stability, and convergence are given for fixed N, but the manuscript should explicitly reference the relevant theorem numbers and indicate where the spectral-cutoff argument for the degeneracy relaxation is formalized.
  2. Numerical section: the reported error metrics and noise levels should be tabulated with explicit values of N and the Tikhonov parameter for each payoff to allow direct reproducibility.
  3. The comparison with quasi-reversibility would benefit from a brief statement of the precise discretization and regularization parameters used in that baseline.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. The recommendation for minor revision is noted. Since no specific major comments were raised, we interpret the request as pertaining to minor editorial or presentational adjustments that can be addressed in the revised version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained via standard spectral and regularization theory

full rationale

The paper projects the forward Black-Scholes PDE onto a finite shifted Legendre basis in the asset-price variable to obtain a reduced ODE system, applies Tikhonov regularization to the resulting finite-dimensional inverse problem, and invokes standard existence/uniqueness/stability arguments for fixed truncation level N. These steps rely on classical spectral cutoff properties and Tikhonov theory applied to the explicitly constructed reduced system; no quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation. The claims remain independent of the target reconstruction result.

Assumptions & free parameters 2 free parameters · 1 assumptions · 0 invented entities

The central claim depends on the completeness and orthogonality properties of shifted Legendre polynomials on the price interval and on the well-posedness of the resulting finite ODE system after projection. The truncation level and Tikhonov parameter are introduced as fixed or tunable quantities without independent determination from first principles.

free parameters (2)
  • truncation level N
    Finite number of Legendre modes selected to balance approximation quality and stability; directly controls the spectral cutoff.
  • Tikhonov regularization parameter
    Scalar chosen to trade off data fidelity against solution smoothness in the reduced system.
assumptions (1)
  • standard math Shifted Legendre polynomials form a complete orthogonal basis suitable for expansion of functions on the asset-price domain.
    Invoked to justify the projection that converts the PDE into a finite ODE system.

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Cite this review

Pith. "Pith review of Forward-Time Black-Scholes Reconstruction via Regularized Legendre Reduction." pith.science (2026). https://pith.science/paper/YDTT46O6

@misc{pith2026260612450,
  author       = {Pith},
  title        = {Pith review of: Forward-Time Black-Scholes Reconstruction via Regularized Legendre Reduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDTT46O6}},
  note         = {Machine review of arXiv:2606.12450}
}
read the original abstract

We study a forward-time formulation of the Black-Scholes equation with state-dependent volatility. In contrast to the classical terminal-value pricing problem, where the option payoff is prescribed at maturity and the price is computed backward in time, the present problem prescribes the current option-price profile and seeks to recover the option-price profile at the expiration date T. This formulation is ill-posed, since the equation evolves in the unstable direction of the parabolic operator and high-frequency perturbations in the initial data may be strongly amplified. To address this difficulty, we introduce a price-dimensional reduction based on shifted Legendre polynomials. The original Black-Scholes equation is projected onto a finite-dimensional Legendre basis in the asset-price variable, leading to a system of ordinary differential equations in time for the expansion coefficients. This reduction acts as a spectral cutoff and also relaxes the degeneracy caused by the factor S^2 at the zero-price boundary. The main reconstruction method is a dimension-reduced Legendre--Tikhonov method. We prove existence, uniqueness, data stability, and convergence for each fixed truncation level. We also include a reduced PINN solver as a secondary computational comparison after the Legendre reduction. Numerical experiments with smooth, butterfly-spread, and European put payoffs show that the Legendre--Tikhonov method recovers the terminal option-price profile from noisy initial data, while the reduced PINN solver provides a useful additional benchmark. Comparisons with the conventional physical-space quasi-reversibility method demonstrate the stabilizing effect of the Legendre reduction.

Figures

Figures reproduced from arXiv: 2606.12450 by the authors.

Figure 1
Figure 1. Numerical results for Test 1 with final time [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Numerical results for Test 2 with final time [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Numerical results for Test 3 with final time [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Parameter selection for Test 1 with 10% noise. The L-curve in Figure 4a is used to guide [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: Effect of incorrectly chosen truncation numbers for Test 1 with 10% noise. The choice [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Comparison between the proposed Legendre-reduction method and the conventional [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]

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Cited by 1 Pith paper

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