Broken Symmetry of Stock Returns -- a Modified Jones-Faddy Skew t-Distribution

TL;DR

This paper introduces a modified skew t-distribution to model stock returns, capturing the asymmetry caused by broken symmetry in stochastic volatility for gains and losses, improving tail and distribution fitting.

Contribution

It proposes a novel modified Jones-Faddy skew t-distribution that accounts for asymmetry in stock return distributions due to differing volatility parameters for gains and losses.

Findings

The distribution effectively captures asymmetry in S&P 500 returns.

It provides better tail modeling compared to traditional distributions.

Application demonstrates improved fit to real market data.

Abstract

We argue that negative skew and positive mean of the distribution of stock returns are largely due to the broken symmetry of stochastic volatility governing gains and losses. Starting with stochastic differential equations for stock returns and for stochastic volatility we argue that the distribution of stock returns can be effectively split in two -- for gains and losses -- assuming difference in parameters of their respective stochastic volatilities. A modified Jones-Faddy skew t-distribution utilized here allows to reflect this in a single organic distribution which tends to meaningfully capture this asymmetry. We illustrate its application on distribution of daily S&P500 returns, including analysis of its tails.