Maximum Entropy Distributions Inferred from Option Portfolios on an Asset

TL;DR

This paper develops a mathematically rigorous method to infer maximum entropy probability distributions for assets from option prices, producing realistic volatility surfaces and validated on S&P 500 options.

Contribution

It provides a new, robust algorithm for maximum entropy distribution inference from option data, with a formal proof of existence and exponential form, improving prior methods.

Findings

Produces realistic volatility surfaces from limited data

Validates approach with S&P 500 options

Offers a rigorous mathematical foundation for maximum entropy inference

Abstract

We obtain the maximum entropy distribution for an asset from call and digital option prices. A rigorous mathematical proof of its existence and exponential form is given, which can also be applied to legitimise a formal derivation by Buchen and Kelly. We give a simple and robust algorithm for our method and compare our results to theirs. We present numerical results which show that our approach implies very realistic volatility surfaces even when calibrating only to at-the-money options. Finally, we apply our approach to options on the S&P 500 index.