Forward-Time Black-Scholes Reconstruction via Regularized Legendre Reduction
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.
Versions
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BibTeX
@misc{nguyen2026forwardtime,
title = {{Forward-Time Black-Scholes Reconstruction via Regularized Legendre Reduction}},
author = {Phuong M. Nguyen and Matt Nguyen and Loc H. Nguyen},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.12450},
eprint = {2606.12450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12450},
}
RIS
TY - UNPB
TI - Forward-Time Black-Scholes Reconstruction via Regularized Legendre Reduction
AU - Phuong M. Nguyen
AU - Matt Nguyen
AU - Loc H. Nguyen
PY - 2026
DA - 2026/06/01
PB - arXiv
DO - 10.48550/arxiv.2606.12450
UR - https://arxiv.org/abs/2606.12450
AB - 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.
ER -