Probability-Weighting Functions for Normal Tempered-Stable Laws: Orthogonal Channels and Information-Theoretic Indices
Akash Deep, Bhathiya Divelgama, A. Alexandre Trindade, W. Brent Lindquist, Svetlozar T. Rachev, Frank J. Fabozzi
Abstract
A probability-weighting function (PWF) maps objective probabilities into decision weights. We construct PWFs from changes in the distribution of returns rather than from a parametric family: with prior and posterior return distributions $F_{prior}$ and $F_{post}$, the PWF is $w(p) = F_{post}(F_{prior}^{-1}(p))$. Both distributions are normal tempered-stable (NTS), and the posterior differs from the prior along one of three channels, each isolated with the variance held fixed: scale, skewness and tail thickness. Around a symmetric benchmark with excess kurtosis 3, a 40% change in scale moves cumulative probabilities by up to 7 to 8 percentage points, a skewness of $\pm 0.5$ by 2 percentage points, and a change of $\pm 0.3$ in the tail index by about 2 percentage points. The three channels leave distinct signatures: the deviation $w(p) - p$ changes sign once, twice and three times, respectively. Three information-theoretic indices (a Fisher-normalized logit shift, a signed Jensen-Shannon divergence and a log-odds elasticity) summarize the distortions and avoid the endpoint singularity of the classical ratio $(w(p)-p)/[p(1-p)]$. Applied to NTS parameters calibrated to S&P 500 ETF daily returns over 2015-2025, the same channels produce distortions of 2 to 7 percentage points. Code that reproduces every figure and number in the paper is provided.
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