Empirical Models of the Time Evolution of SPX Option Prices

2025-06-20

AI Summary

This paper develops an empirical SPX option pricing model using 30 years of daily data to generate arbitrage-free, interpretable, and generalizable simulations of future S&P 500 index paths. Methodologically, it incorporates financial priors into feature engineering and comparatively evaluates a compact two-layer neural network (4 nodes per layer), random forests, and linear regression—achieving, for the first time without explicit constraints, intrinsic arbitrage-free outputs from the neural network. Contributions include: (1) reconciling high predictive performance of black-box models with economic interpretability; (2) systematically demonstrating the robust outperformance of nonlinear models over decades of market history; and (3) substantially surpassing both the Black–Scholes–Merton model and two empirical baselines across prediction accuracy, robustness, and arbitrage-free consistency—thereby unifying these traditionally competing objectives.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Develop empirical model for SPX option pricing simulation
• Compare neural network, random forest, linear regression models
• Ensure arbitrage-free prices without imposed conditions

Innovation

• Neural network models SPX option prices
• Random forest and linear regression compared
• Arbitrage-free prices without imposed conditions