Empirical Models of the Time Evolution of SPX Option Prices

Abstract

The key objective of this paper is to develop an empirical model for pricing SPX options that can be simulated over future paths of the SPX. To accomplish this, we formulate and rigorously evaluate several statistical models, including neural network, random forest, and linear regression. These models use the observed characteristics of the options as inputs -- their price, moneyness and time-to-maturity, as well as a small set of external inputs, such as the SPX and its past history, dividend yield, and the risk-free rate. Model evaluation is performed on historical options data, spanning 30 years of daily observations. Significant effort is given to understanding the data and ensuring explainability for the neural network. A neural network model with two hidden layers and four neurons per layer, trained with minimal hyperparameter tuning, performs well against the theoretical Black-Scholes-Merton model for European options, as well as two other empirical models based on the random forest and the linear regression. It delivers arbitrage-free option prices without requiring these conditions to be imposed.