Pricing options and computing implied volatilities using neural networks

TL;DR

The paper tackles the computational bottleneck of pricing and calibrating options in high-dimensional models by training an artificial neural network (ANN) surrogate on data generated from high-fidelity solvers for both Black-Scholes and Heston frameworks. The method comprises an offline data-generation phase and an online prediction phase, enabling fast pricing of $V(t,S)$ and implied volatilities $\sigma^\ast$ (via $BS(\sigma^\ast;\cdot)=V^{mkt}$) with substantial speedups on GPUs while maintaining high accuracy. The authors demonstrate the approach on the analytic Black-Scholes solution, the COS method for the Heston model, and Brent-like root-finding for IV, achieving near machine-precision errors (e.g., $\text{MAE}\approx10^{-4}$ for IV and $\text{RMSE}\sim10^{-4}$–$10^{-5}$ for prices) and up to two orders of magnitude faster performance on GPU batch computations. The work shows the practical viability of data-driven solvers for real-time calibration and risk management, with immediate applicability to European options and potential extensions to Greeks and more complex derivatives. Overall, the study confirms that neural surrogates can accelerate parametric financial models while preserving accuracy, enabling real-time workflows in calibrations and risk assessment.

Abstract

This paper proposes a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options and to calculate implied volatilities with the aim of accelerating the corresponding numerical methods. With ANNs being universal function approximators, this method trains an optimized ANN on a data set generated by a sophisticated financial model, and runs the trained ANN as an agent of the original solver in a fast and efficient way. We test this approach on three different types of solvers, including the analytic solution for the Black-Scholes equation, the COS method for the Heston stochastic volatility model and Brent's iterative root-finding method for the calculation of implied volatilities. The numerical results show that the ANN solver can reduce the computing time significantly.

Figures (12)

• Figure S1: Vega tends to be zero in certain regions of deep ITM or OTM options.
• Figure S2: Illustration of an MLP configuration.
• Figure S3: Average training loss against varying learning rates.
• Figure S4: Different learning rate schedules for training ANNs on Heston model.
• Figure S5: The history of training and validation losses for the Heston model.