Sharp Large Deviations and Gibbs Conditioning for Threshold Models in Portfolio Credit Risk

ArXiv ID: 2509.19151 “View on arXiv”

Authors: Fengnan Deng, Anand N. Vidyashankar, Jeffrey F. Collamore

Abstract

We obtain sharp large deviation estimates for exceedance probabilities in dependent triangular array threshold models with a diverging number of latent factors. The prefactors quantify how latent-factor dependence and tail geometry enter at leading order, yielding three regimes: Gaussian or exponential-power tails produce polylogarithmic refinements of the Bahadur-Rao $n^{"-1/2"}$ law; regularly varying tails yield index-driven polynomial scaling; and bounded-support (endpoint) cases lead to an $n^{"-3/2"}$ prefactor. We derive these results through Laplace-Olver asymptotics for exponential integrals and conditional Bahadur-Rao estimates for the triangular arrays. Using these estimates, we establish a Gibbs conditioning principle in total variation: conditioned on a large exceedance event, the default indicators become asymptotically i.i.d., and the loss-given-default distribution is exponentially tilted (with the boundary case handled by an endpoint analysis). As illustrations, we obtain second-order approximations for Value-at-Risk and Expected Shortfall, clarifying when portfolios operate in the genuine large-deviation regime. The results provide a transferable set of techniques-localization, curvature, and tilt identification-for sharp rare-event analysis in dependent threshold systems.

Keywords: Large Deviation Theory, Triangular Array Threshold Models, Gibbs Conditioning Principle, Laplace-Olver Asymptotics, Dependent Latent Factors, Risk Management / Credit Risk

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 3.2/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, featuring advanced probability theory (large deviations, Laplace-Olver asymptotics, triangular arrays) and dense mathematical derivations without any code, backtesting, or empirical implementation. Its rigor is purely theoretical, aimed at establishing general asymptotic results rather than data-driven validation.

  flowchart TD
    A["Research Goal<br/>Sharp rare-event analysis in<br/>dependent credit threshold models"] --> B["Methodology<br/>Laplace-Olver asymptotics<br/>& Conditional Bahadur-Rao"]
    B --> C["Computational Processes<br/>Localization, curvature,<br/>tilt identification"]
    C --> D{"Data/Input: Tail Geometry?"}
    D --> E["Regime 1: Gaussian/Exp-Power<br/>Polylogarithmic refinement"]
    D --> F["Regime 2: Regularly Varying<br/>Polynomial scaling"]
    D --> G["Regime 3: Bounded-Support<br/>n^{"-3/2"} prefactor"]
    
    E --> H["Findings: Gibbs Conditioning<br/>Defaults asymptotically i.i.d.<br/>LGD exponentially tilted"]
    F --> H
    G --> H
    
    H --> I["Outcomes: Risk Metrics<br/>VaR/ES 2nd-order approx.<br/>Large-deviation regime identification"]