A modified Black–Scholes equation for option pricing prediction based on physics-informed neural networks

Considering the limitations of the classical Black–Scholes equation in modeling real markets, modifications have been made on the volatility and risk free interest rate parameters and then the modified Black–Scholes equation is proposed for calculating and predicting the option price. For volatility, under the assumption that the stock price follows the Brownian motion, volatility is estimated using the Monte Carlo simulation approach. Estimation is performed through 100,000 simulations with multiple price paths and time steps. For the risk free interest rate, the modified volatility is used in conjunction with the Black–Karasinski (BK) interest rate model, and the risk free rate is predicted using the PINNs model. Additionally, the risk free rate is also modified using the classical Cox–Ingersoll–Ross (CIR) rate model. In this paper, the PINNs model is employed to solve the modified Black–Scholes equation and predict option prices, while we make a simple modification to the classic PINNs model by adjusting the number of hidden layers. The results of both models are compared with the corresponding exact solutions. The findings indicate that the PINNs model used in this study maintains a stable magnitude of loss error when solving the modified Black–Scholes equation, with errors smaller than the classical PINNs model. This demonstrates that the PINNs model adopted in this paper is effective. Finally, comparing with actual option data, it is found that the option prices obtained in this study closely match the real data from the options market, indicating that the modified Black–Scholes equation and the PINNs model have strong explanatory power for the market.