Neural Network Learning of Black-Scholes Equation for Option Pricing

ArXiv ID: 2405.05780 “View on arXiv”

Authors: Unknown

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.

Keywords: option pricing, neural networks, Black-Scholes, time series forecasting, deep learning

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper involves advanced mathematics, including solving the Black-Scholes PDE (a parabolic PDE) and neural network training, justifying a high math score. It uses real-world data from specific stocks, compares results with analytical solutions, and discusses implementation for forecasting, giving it solid empirical rigor.

  flowchart TD
    A["Research Goal<br>Develop Neural Network approach to solve Black-Scholes<br>PDE for Option Pricing"] --> B["Data Preparation<br>Time Series: Petrobras & Vale<br>Call Options Prices"]
    B --> C["Model Architecture<br>Feedforward Neural Network<br>Inputs: S, K, T, r, σ<br>Output: Option Price"]
    C --> D["Training Process<br>Learn Black-Scholes Solution<br>from Real Market Data"]
    D --> E["Prediction Output<br>Neural Network Pricing<br>vs<br>Traditional Analytical BS"]
    E --> F["Key Findings<br>NN achieves higher accuracy<br>than Black-Scholes analytical<br>enables short-term forecasting"]