Market-Based Probability of Stock Returns

TL;DR

This paper explores how market-based statistical moments of stock returns depend on trade value moments and correlations, deriving formulas for volatility and higher moments, and emphasizing the importance of these moments for financial modeling and forecasting.

Contribution

It introduces a new framework linking trade value moments to return moments, providing approximations for return distributions and insights for improving macroeconomic and market models.

Findings

Derived dependence of return moments on trade value moments and correlations.

Provided formulas for approximating return distribution using finite moments.

Highlighted the importance of market-based moments for financial forecasting.

Abstract

This paper describes the dependence of market-based statistical moments of returns on statistical moments and correlations of the current and past trade values. We use Markowitz's definition of value weighted return of a portfolio as the definition of market-based average return of trades during the averaging period. Then we derive the dependence of market-based volatility and higher statistical moments of returns on statistical moments, volatilities, and correlations of the current and past trade values. We derive the approximations of the characteristic function and the probability of returns by a finite number q of market-based statistical moments. To forecast market-based average and volatility of returns at horizon T, one should predict the first two statistical moments and correlation of current and past trade values at the same horizon. We discuss the economic reasons that limit the number of predicted statistical moments of returns by the first two. That limits the accuracy of the forecasts of probability of returns by the accuracy of the Gaussian approximations. To improve the reliability of large macroeconomic and market models like BlackRock's Aladdin, JP Morgan, and the U.S. Fed., the developers should use market-based statistical moments of returns.