Neural Network Learning of Black-Scholes Equation for Option Pricing

TL;DR

This work investigates solving the Black-Scholes European option pricing PDE by training a neural network to learn a heat-equation reformulation of the problem, using real Brazilian market data for Petrobras and Vale call options. An MLP with two inputs ($x$, $\tau$) and a 2-32-32-1 architecture is trained via the Neurodiffeq framework and Adam optimization over 30,000 epochs to satisfy the transformed PDE and boundary conditions derived from market prices. Across PETRA, PETRD, and VALE series, the neural-network solutions frequently align more closely with observed option prices than the analytical $c = S_0 N(d_1) - K e^{-rT} N(d_2)$ and $p = K e^{-rT} N(-d_2) - S_0 N(-d_1)$ formulas, particularly as options approach maturity, while providing a practical short-term forecast framework. The study demonstrates the viability of data-driven PDE solvers for option pricing and highlights scope for extending to additional assets and comparing against ARIMA-based approaches.

Abstract

One of the most discussed problems in the financial world is stock option pricing. The Black-Scholes Equation is a Parabolic Partial Differential Equation which provides an option pricing model. The present work proposes an approach based on Neural Networks to solve the Black-Scholes Equations. Real-world data from the stock options market were used as the initial boundary to solve the Black-Scholes Equation. In particular, times series of call options prices of Brazilian companies Petrobras and Vale were employed. The results indicate that the network can learn to solve the Black-Sholes Equation for a specific real-world stock options time series. The experimental results showed that the Neural network option pricing based on the Black-Sholes Equation solution can reach an option pricing forecasting more accurate than the traditional Black-Sholes analytical solutions. The experimental results making it possible to use this methodology to make short-term call option price forecasts in options markets.

Figures (3)

• Figure 1: (a) Results for PETRA332 (MSE minor) (b) Results for PETRA391 and (c) results for PETRA108 (MSE major). (a) and (b) are the best NN prediction cases, and (c) is the worst NN prediction case. For all graphics, there are four curves: the price of the underlying PETR4 stock - OPTION, the blue curve; the price calculated by the Black-Scholes analytical solution - BLS, the magenta curve; the price computed by the ANN, the green curve; and, the option market price - SPOT, the purple curve.
• Figure 2: (a) Results for PETRD266 (MSE minor), (b) Results for PETRD122 and (c) results for PETRA198 (MSE major). (a) and (b) are the best NN prediction cases, and (c) is the worst NN prediction case. For all graphics, there are four curves: the price of the underlying PETR4 stock - OPTION, the blue curve; the price calculated by the Black-Scholes analytical solution - BLS, the magenta curve; the price computed by the ANN, the green curve; and, the option market price - SPOT, the purple curve.
• Figure 3: (a) Results for VALED765 (MSE minor, the bets NN prediction) and (b) results for VALED655 (MSE major, the worse NN prediction). For both graphics, there are four curves: the price of the underlying PETR4 stock - OPTION, the blue curve; the price calculated by the Black-Scholes analytical solution - BLS, the magenta curve; the price computed by the ANN, the green curve; and, the option market price - SPOT, the purple curve.