Paper: arXiv 2610.26746

Authors: 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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a mathematically sophisticated approach to constructing probability-weighting functions using Normal Tempered-Stable distributions. It demonstrates strong empirical rigor through calibration to S&P 500 ETF data and provides code for full reproducibility. The novelty lies in deriving PWFs from distributional changes rather than parametric families and introducing new information-theoretic indices.

Research Flowchart

  flowchart TD
    A[Research Goal: Construct Probability-Weighting Functions for NTS Laws] --> B{Methodology: Define PWF from Distribution Changes};
    B --> C[Inputs: Prior & Posterior NTS Return Distributions (F_prior, F_post)];
    C --> D{Computational Process: Isolate Channels (Scale, Skewness, Tail Thickness)};
    D --> E[Outcomes: Quantify Impact on Cumulative Probabilities];
    E --> F{Outcomes: Characterize Channel Signatures (w(p)-p changes sign)};
    F --> G[Outcomes: Develop Information-Theoretic Indices (Fisher-normalized logit shift, Signed Jensen-Shannon divergence, Log-odds elasticity)];
    G --> H[Application: Calibrated NTS parameters to S&P 500 ETF daily returns (2015-2025)];
    H --> I[Key Findings: Distinct distortions (2-8 percentage points) for each channel];