The Evolution of Option Pricing Models: From Black-Scholes and Heston Models to Differentiable Deep Neural Networks

Abstract

Option pricing has moved away from the previous analytical and stochastic models to machine-learning-based methods. Therefore, a good model of the market and an efficient one for computation is required. This paper reviews the development of the Black-Scholes and Heston models to DNN- and DDN-based surrogate models, as well as their applications in quantitative finance. Give some attention to computational efficiency in the repeated pricing and calibration. Another issue in the paper is the trade-off between the accuracy of prices and the cost of computation; high-speed calculation may not be entirely precise in finance. Differentiability in the context of risk management and hedging, as well as Delta and Vega sensitivity measures, will be introduced. The problems with the current ones are: dependence on training data, lack of interpretability, and issues of robustness and generalization. In the future, one will aim to build a more stable and flexible surrogate model, enhance its interpretability, address financial constraints, and integrate traditional financial models with machine learning in a hybrid manner.