An uncertainty-aware physics-informed neural network solution for the Black-Scholes equation: a novel framework for option pricing

ArXiv ID: 2511.05519 “View on arXiv”

Authors: Sina Kazemian, Ghazal Farhani, Amirhessam Yazdi

Abstract

We present an uncertainty-aware, physics-informed neural network (PINN) for option pricing that solves the Black–Scholes (BS) partial differential equation (PDE) as a mesh-free, global surrogate over $(S,t)$. The model embeds the BS operator and boundary/terminal conditions in a residual-based objective and requires no labeled prices. For American options, early exercise is handled via an obstacle-style relaxation while retaining the BS residual in the continuation region. To quantify \emph{“epistemic”} uncertainty, we introduce an anchored-ensemble fine-tuning stage (AT–PINN) that regularizes each model toward a sampled anchor and yields prediction bands alongside point estimates. On European calls/puts, the approach attains low errors (e.g., MAE $\sim 5\times10^{"-2"}$, RMSE $\sim 7\times10^{"-2"}$, explained variance $\approx 0.999$ in representative settings) and tracks ground truth closely across strikes and maturities. For American puts, the method remains accurate (MAE/RMSE on the order of $10^{"-1"}$ with EV $\approx 0.999$) and does not exhibit the error accumulation associated with time-marching schemes. Against data-driven baselines (ANN, RNN) and a Kolmogorov–Arnold FINN variant (KAN), our PINN matches or outperforms on accuracy while training more stably; anchored ensembles provide uncertainty bands that align with observed error scales. We discuss design choices (loss balancing, sampling near the payoff kink), limitations, and extensions to higher-dimensional BS settings and alternative dynamics.

Keywords:

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper is highly mathematical, incorporating advanced PDE theory, residual-based neural network training, and anchored ensembles for uncertainty quantification; while it reports strong empirical results (low MAE/RMSE, explained variance) and benchmarks against ANN/RNN/KAN baselines, it lacks real market data and focuses on synthetic/simulated settings, limiting its practical backtest readiness.

  flowchart TD
    A["Research Goal<br>Mesh-free solution to Black-Scholes PDE<br>with epistemic uncertainty quantification"] --> B["Methodology: Physics-Informed PINN"]
    
    B --> C["Input: Unsupervised data<br>Random sample from domain S,t"]
    B --> D["Core Computational Process<br>Residual minimization of BS PDE + boundary conditions"]
    
    D --> E["Adaptation for American Options<br>Obstacle-style relaxation"]
    D --> F["UQ: Anchored-Ensemble Fine-tuning<br>AT-PINN for prediction bands"]
    
    C & E & F --> G["Outcomes"]
    
    subgraph G ["Results"]
        G1["High Accuracy<br>MAE ~5e-2, EV ~0.999"]
        G2["Stable Performance<br>No error accumulation vs time-marching"]
        G3["Reliable Uncertainty<br>Bands align with error scales"]
    end