Towards Fast Option Pricing PDE Solvers Powered by PIELM

TL;DR

The paper addresses the computational bottleneck of PINNs in financial PDEs for option pricing and model calibration by introducing Physics-Informed Extreme Learning Machines (PIELMs). PI-ELMs use a single-hidden-layer network with randomly initialized hidden weights and analytic training of output weights, producing a fast, deterministic surrogate that enforces PDE and boundary conditions via a least-squares residual. Applied to Black–Scholes and HHW forward pricing and to inverse calibration through Bayesian optimization, the approach achieves accuracy on par with PINNs but with substantial speedups (up to $30\times$). This enables near real-time pricing and parameter estimation from noisy data, illustrating strong potential for real-time financial modeling and calibration tasks. The work also outlines a Bayesian inverse framework and discusses avenues for future enhancements such as adaptive bases and extensions to American options.

Abstract

Partial differential equation (PDE) solvers underpin modern quantitative finance, governing option pricing and risk evaluation. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving the forward and inverse problems of partial differential equations (PDEs) using deep learning. However they remain computationally expensive due to their iterative gradient descent based optimization and scale poorly with increasing model size. This paper introduces Physics-Informed Extreme Learning Machines (PIELMs) as fast alternative to PINNs for solving both forward and inverse problems in financial PDEs. PIELMs replace iterative optimization with a single least-squares solve, enabling deterministic and efficient training. We benchmark PIELM on the Black-Scholes and Heston-Hull-White models for forward pricing and demonstrate its capability in inverse model calibration to recover volatility and interest rate parameters from noisy data. From experiments we observe that PIELM achieve accuracy comparable to PINNs while being up to $30\times$ faster, highlighting their potential for real-time financial modeling.

Figures (5)

• Figure 1: Solution Obtained for European Call Option using PIELM.
• Figure 2: Difference between Heston-Hull-White (calculated using PIELM) and Black-Scholes call prices as a function of volatility. Mean reversion in the Heston model produces the characteristic bell shape.
• Figure 3: Difference between Heston-Hull-White and Black-Scholes call prices as a function of the short rate. Stochastic mean reversion of rates leads to systematically higher prices than under constant rate discounting, particularly when initial rates are low.
• Figure 4: European Call Option with stochastic volatility and stochastic rates PIELM solution.
• Figure 5: An instance of Best-so-far trajectories during Bayesian optimization for the Black–Scholes inverse problem. Panels show (top) volatility $\sigma$ and (middle) risk-free rate $r$ best-so-far estimates (solid blue) alongside reference values (dashed red), and (bottom) best-so-far objective (green, log scale) versus BO iteration. The true parameters were $\sigma_{\text{true}} = 0.62$ and $r_{\text{true}} = 0.035$, while the Bayesian optimization recovered $\hat{\sigma} = 0.6216$ and $\hat{r} = 0.03691$. The objective is the mean-squared error between model predictions and observations at selected $(S,t)$ points.