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

TL;DR

This work develops an uncertainty-aware physics-informed neural network (PINN) to price European and American options by solving the Black-Scholes PDE as a mesh-free surrogate over $(S,t)$. A two-stage anchor-transfer training with anchored ensembling provides both point estimates and calibrated epistemic uncertainty, yielding prediction bands that respect no-arbitrage bounds. The AT–PINN demonstrates strong accuracy on European calls/puts and American puts, without the error accumulation typical of time-marching schemes, and compares favorably to data-driven baselines and a Kolmogorov–Arnold FINN variant. The approach offers a practical framework for risk-aware option pricing and outlines directions for higher-dimensional extensions, adaptive collocation, and formal uncertainty calibration.

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.

Figures (4)

• Figure 1: Schematic of the physics-informed neural network used to approximate $V(S,t)$: inputs $S,t$, four tanh hidden layers (50 units each), and a single scalar output $\hat{V}_\theta$.
• Figure 2: Left Panel: Call price vs stock price shown for ground truth in blue and the prediction in orange curve, with the shaded light blue representing uncertainty at $pm2$ for better visualization. Right Panel: Absolute error $(\hat{y} - y)$ at each stock price between the ground truth and the prediction is plotted.
• Figure 3: Top Panel: Orange dashed line is the prediction, and the solid blue line is the groundtruth with a shaded model uncertainty shown at 2 standard deviations. Bottom Panel: The error between the ground truth and the prediction and the shaded region is the uncertainty of the prediction at 1 standard deviation. In both panels, the left figure shows $t = 0$, middle panel shows midway to the expiry date, and the right panel is the expiry date.
• Figure 4: Ground truth (pink), KAN (dashed dark green), PINN-anchor (dotted purple), RNN (cyan stars), and ANN (solid blue). Left Panel: European, Right Panel: American