Forward-Time Black–Scholes Reconstruction via Regularized Legendre Reduction

Battle Count: 79

The paper addresses option price reconstruction from current market observations, which is relevant to derivatives pricing and risk management. However, it is primarily a mathematical/methodological contribution focused on solving an ill-posed PDE problem rather than a direct trading strategy. The forward-time formulation (predicting terminal prices from current prices) has potential applications in derivatives valuation and hedging, but practical implementation would require significant adaptation for real market conditions.

Implementation requires: (1) computing shifted Legendre polynomial integrals for coefficient matrices, (2) assembling and solving large augmented least-squares systems for Tikhonov regularization, (3) parameter selection via L-curve criterion, (4) optional PINN training with automatic differentiation. The mathematical framework is sophisticated, involving spectral methods, functional analysis, and regularization theory. MATLAB implementation is described but no code is provided.

Algorithms are clearly described (Algorithm 1 and 2), parameter selection procedures are detailed, and numerical setup is specified. However, no code repository is provided, and data is only available upon request. The synthetic data generation process is fully described, enabling reproduction of experiments.