Sharp Large Deviations and Gibbs Conditioning for Threshold Models in Portfolio Credit Risk
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| Format: | Preprint |
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2025
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| author | Deng, Fengnan Vidyashankar, Anand N. Collamore, Jeffrey F. |
| author_facet | Deng, Fengnan Vidyashankar, Anand N. Collamore, Jeffrey F. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_19151 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Large Deviations and Gibbs Conditioning for Threshold Models in Portfolio Credit Risk Deng, Fengnan Vidyashankar, Anand N. Collamore, Jeffrey F. Probability Statistics Theory Mathematical Finance Portfolio Management Risk Management 60F10, 60G70, 62P05, 91G70 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. |
| title | Sharp Large Deviations and Gibbs Conditioning for Threshold Models in Portfolio Credit Risk |
| topic | Probability Statistics Theory Mathematical Finance Portfolio Management Risk Management 60F10, 60G70, 62P05, 91G70 |
| url | https://arxiv.org/abs/2509.19151 |