Fitting Accumulated Stock Returns with Tempered Skew t-Distribution

Battle Count: 50

The paper provides important insights into the statistical properties of multi-day accumulated returns, particularly the tempering of power-law tails and asymmetry between gains and losses. This is relevant for: (1) tail risk estimation over multi-day holding periods, (2) understanding that extreme losses have heavier tails than gains (negative skew), (3) the 'conservation law' of linear variance scaling with time, and (4) realistic modeling of return distributions for position sizing and VaR/CVaR calculations. However, it is primarily a descriptive/statistical modeling paper rather than a predictive or trading strategy paper. The tempering scale discrepancy limits its direct applicability for extreme event modeling.

Implementation requires: (1) Computing the confluent hypergeometric function 1F1 with potentially negative parameters (requires careful numerical handling), (2) Numerical integration for the normalization constant Z, (3) Multi-parameter optimization (5 parameters: βl, βg, α, κ1, µ) for each τ value, (4) Handling the product distribution of stochastic volatility and Gaussian increments, (5) Computing CCDFs for comparison. The special functions and numerical integration add complexity beyond standard distribution fitting. Python with scipy (for 1F1 and optimization) or Mathematica would be suitable.

The paper provides complete analytical derivations, all fitted parameters in tables, and clear mathematical formulations. However, no code or software implementation is provided. The S&P500 data (1980-2025) is publicly available. Reproduction would require implementing the confluent hypergeometric function-based PDF and performing numerical fitting. The normalization constant Z requires numerical integration.