[Paper Review] Non-Gaussianity of the Intraday Returns Distribution: its evolution in time

M. A. Virasoro|arXiv (Cornell University)|Dec 4, 2011

Complex Systems and Time Series Analysis7 references3 citations

TL;DR

This paper introduces a non-parametric measure of intraday return non-Gaussianity—specifically, a time-varying proxy for kurtosis (p-kurtosis)—to capture market stress beyond volatility. It finds that p-kurtosis exhibits long-term persistence and evolves independently of volatility, with a sharp rise preceding the 1987 crash and a sustained decline through the dot-com bubble, challenging the assumption of a universal return distribution shape.

ABSTRACT

We find a remarkable time persistence of various proxies for the kurtosis (p-kurtosis) of the intraday returns distribution for the S&P500 index and this permits a significant measure of their evolution from 1983 to 2004. There appears a long time scale dramatic variation of the p-kurtosis uncorrelated with the variation of the volatility thus falsifying any hypothesis of a universal shape for the probability distribution of the returns. A large increase in the kurtosis anticipates the October 87 crash. During the years 1991-2003 it continuously decreases even when the volatility grows during the dot-com bubble. We propose some speculative interpretations of these results.

Motivation & Objective

• To identify and measure a new collective market state variable beyond price and volatility, specifically the non-Gaussianity of intraday returns.
• To investigate whether the kurtosis of intraday return distributions evolves persistently over time, independent of volatility dynamics.
• To challenge the assumption of a universal return distribution shape by demonstrating time-dependent, non-Gaussian features in market returns.
• To explore the implications of such non-Gaussianity for market behavior, particularly in relation to investor uncertainty and model trust.
• To provide a data-driven narrative of market evolution using a non-parametric approach that avoids overfitting and model dependence.

Proposed method

• The study uses high-frequency S&P500 data to compute intraday returns over a fixed time interval δt, forming a time series of returns.
• A temporal window Δt is applied to group returns into time-dependent samples, enabling the measurement of time-varying statistical properties.
• A proxy for kurtosis (p-kurtosis) is computed for each Δt window, normalized by subtracting the Gaussian baseline to isolate non-Gaussianity.
• The p-kurtosis is corrected for systematic intraday volatility patterns (e.g., opening/closing effects) to reduce bias.
• The method is non-parametric, avoiding assumptions of specific dynamic models, and generalizes the concept of realized volatility to realized kurtosis.
• The analysis compares p-kurtosis evolution with realized volatility and identifies periods of divergence, particularly around market events.

Experimental results

Research questions

• RQ1 Does the non-Gaussianity of intraday returns, as measured by p-kurtosis, evolve persistently over time in a way uncorrelated with volatility?
• RQ2 Can p-kurtosis anticipate major market events such as the 1987 crash, and if so, how?
• RQ3 How does the non-Gaussianity of returns change during prolonged market trends like the dot-com bubble, despite rising volatility?
• RQ4 To what extent does the observed time evolution of p-kurtosis challenge the assumption of a universal return distribution shape?
• RQ5 What behavioral or structural market mechanisms might underlie the observed long-term evolution of non-Gaussianity?

Key findings

• The p-kurtosis of S&P500 intraday returns shows remarkable time persistence from 1983 to 2004, indicating a measurable and stable time-varying feature of market dynamics.
• A significant and sustained increase in p-kurtosis preceded the October 1987 crash, suggesting a detectable signal of rising market stress.
• During the dot-com bubble (1991–2003), p-kurtosis continuously decreased even as volatility increased, indicating a decoupling of non-Gaussianity from volatility.
• The evolution of p-kurtosis is largely uncorrelated with realized volatility, falsifying the hypothesis of a universal return distribution shape.
• The observed patterns are compatible with behavioral interpretations involving investor uncertainty, model distrust, and delayed reactions to market anomalies.
• The results suggest that non-Gaussianity, as captured by p-kurtosis, is a distinct and measurable market state variable with implications for risk assessment and market microstructure.