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

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.

Figures (19)

• Figure 1: S&P500; $r_t=\log(S_t/S_0)$, $S_t$ is price on day $t$, $t$ changes in daily increments ($\tau =1$ in text).
• Figure 2: S&P500; $x_t = r_t - \mu_1 t$ where index in $\mu_1$ reflects daily increments of $t$ ($\tau =1$ in text).
• Figure 3: left: $x_t$ (\ref{['xt']}) for S&P daily returns ($\tau=1$) -- centered on $m_1$; right: $x_t-m_1$($\tau=1$) -- centered on $0$.
• Figure 4: PDF of stocks returns and fits with distributions (\ref{['fSt']}), (\ref{['fhSt']}), (\ref{['fmJF1']}), and (\ref{['fmJF2']}).
• Figure 5: Expanded view of the area around the mode of the distribution in Fig. \ref{['fplot']}.