How Volatility Distributions Shape Financial Return Tails

Summary

This paper studies return distributions under an assumption that returns are independent once their time-varying variance is specified. Because volatility itself changes randomly, the distribution of aggregate returns reflects the distribution of those volatility levels. The authors derive a scaling form connecting return-distribution shape directly to volatility-distribution shape.

The framework explains power-law tails in returns as arising from power-law tails in volatility, and it relates Bitcoin's stretched-exponential return distribution to a similarly shaped volatility distribution. The stated tests use S&P 500 index data, Apple and Paramount shares, and Bitcoin. The approach offers a statistical explanation for observed distribution shapes, but it relies on conditional independence and on the assumed volatility behavior; the abstract does not establish that these assumptions capture all sources of dependence or tail risk in markets.

Key ideas

• Returns are modeled as independent conditional on their changing variance.
• The distribution of volatility determines the scaling form and shape of returns.
• Power-law volatility tails can produce power-law tails in returns.
• The framework links Bitcoin's stretched-exponential returns to stretched-exponential volatility.
• The predictions are tested on an index, individual stocks, and Bitcoin, subject to the model assumptions.

Tags

# Scaling and shape of financial returns distributions modeled as conditionally independent random variables


# Scaling and shape of financial returns distributions modeled as conditionally independent random variables









We show that assuming that the returns are independent when conditioned on the value of their variance (volatility), which itself varies in time randomly, then the distribution of returns is well described by the statistics of the sum of conditionally independent random variables. In particular, we show that the distribution of returns can be cast in a simple scaling form, and that its functional form is directly related to the distribution of the volatilities. This approach explains the presence of power-law tails in the returns as a direct consequence of the presence of a power law tail in the distribution of volatilities. It also provides the form of the distribution of Bitcoin returns, which behaves as a stretched exponential, as a consequence of the fact that the Bitcoin volatilities distribution is also closely described by a stretched exponential. We test our predictions with data from the S\&P 500 index, Apple and Paramount stocks; and Bitcoin.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.