Fitting Accumulated Stock Returns with Tempered Skew t-Distribution

We analyze the distributions of historic S&P500 multi-day returns for the number of days of accumulation from 20 to 120. With the increase in the number of days of accumulation, we observe clear tempering of power-law tails toward a seemingly finite value. To explain this phenomenon, we employ a model that produces a “capped inverse gamma” stationary (steady-state) distribution for stochastic volatility which, in turn, produces a “tempered Student-t” distribution for returns. We then employ Jones–Faddy-like symmetry-breaking mechanism that produces a “tempered Skew-t” distribution. This distribution provides rather good fits to the distributions of accumulated multi-day S&P500 returns, which exhibit symmetry breaking between gains and losses—as reflected by a positive mean and negative skew. Tempered Skew-t fits are also consistent with near-perfect linear dependence on the number of days of accumulation of the mean values and, even more so, the variances (mean squared realized volatility) of the distributions.