Maximum entropy distribution of stock price fluctuations

TL;DR

This paper derives a maximum entropy distribution for stock price fluctuations based on no-arbitrage principles, providing a practical model that incorporates market-implied volatility for better derivative pricing.

Contribution

It introduces a novel maximum entropy framework constrained by expected volatility, linking option market data to the distribution of stock returns.

Findings

Model accurately matches observed return distributions

Expected volatility can be extracted from option prices

Analytic probability density function derived for practical use

Abstract

The principle of absence of arbitrage opportunities allows obtaining the distribution of stock price fluctuations by maximizing its information entropy. This leads to a physical description of the underlying dynamics as a random walk characterized by a stochastic diffusion coefficient and constrained to a given value of the expected volatility, taking in this way into account the information provided by the existence of an option market. This model is validated by a comprehensive comparison with observed distributions of both price return and diffusion coefficient. Expected volatility is the only parameter in the model and can be obtained by analysing option prices. We give an analytic formulation of the probability density function for price returns which can be used to extract expected volatility from stock option data. This distribution is of high practical interest since it should be preferred to a Gaussian when dealing with the problem of pricing derivative financial contracts.