Optimization of Continuous-Time Financial Models Driven by Machine Learning: A Core Scenario of Option Pricing

The exact price of financial derivatives, especially options, is still a core of quantitative finance and risk management. Traditional continuous-time models, such as the Black-Scholes-Merton (BSM) framework, provide a theoretical foundation but they make simplifying assumptions, mainly constant volatility, vastly different from those in markets. This gives rise to systematic pricing errors in the price model, and ruins the usefulness of the hedge. This paper explores novel approaches that integrate machine learning with traditional quantitative finance methods to improve pricing accuracy. Focusing on the key case of European option pricing, it demonstrates that a Feedforward Neural Network (FNN) can learn the complex, non-linear implied volatility surface directly from market data without the strong assumptions of traditional models. We train our network with a large set of historical S&P 500 index options, taking moneyness and time to maturity as inputs and predicting implied volatility as outputs. The performance of this machine learning-based approach is compared to that of the BSM framework. The model is also compared to more sophisticated alternatives, such as the Heston Stochastic Volatility Model. Empirically, our approach significantly reduces pricing errors, as evidenced by low Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) values on out-of-sample test data. It also offers excellent computational speed, it can be calibrated quickly and deployed in real time for high-frequency trading, risk management and more. This research demonstrates the transformative potential of machine learning in quantitative finance: it enables the development of more robust, accurate, and computationally efficient models, which will drive the development of a new generation of more powerful quantitative tools over time.