Thèse Backtesting Exact en Échantillon Fini des Mesures de Risque H/F - Doctorat.Gouv.Fr

Établissement : Institut Polytechnique de Paris École nationale de la statistique et de l'administration économique École doctorale : Mathématiques Hadamard Laboratoire de recherche : CREST - Centre de recherche en économie et statistique Direction de la thèse : Christian FRANCQ ORCID 0000000315288652 Début de la thèse : 2026-10-01 Date limite de candidature : 2026-06-01T23:59:59 Ce projet développe des méthodes exactes de backtesting sur échantillons finis pour les mesures de risque dans des contextes où l'hypothèse nulle de référence correcte dépasse le cadre classique des processus de hit de Bernoulli i.i.d. Il s'attaque aux limites des backtests asymptotiques sur des échantillons courts, lorsque les probabilités de queue sont faibles et lorsque les seuils de risque sont sous-estimés. La recherche comporte trois volets : le backtesting exact pour la VaR de simulation historique en cas de sous-estimation du risque ; le backtesting implicite exact pour le shortfall attendu par discrétisation des queues à états finis ; et le backtesting exact pour les mesures de risque systémique telles que la CoVaR et le MES par le biais de processus d'états catégoriels conditionnés par la détresse. Le projet combine la théorie exacte des échantillons finis avec la programmation dynamique sparse afin de fournir des méthodes, des logiciels et des preuves empiriques exploitables pour la gestion et la réglementation des risques financiers. Backtesting is a key statistical tool for evaluating tail-risk forecasts. In the standard
VaR framework, the main implications under correct calibration concern unconditional
coverage, independence, and conditional coverage of the hit sequence (Kupiec, 1995;
Christoffersen, 1998). However, these procedures are often applied in settings where
asymptotic approximations may be unreliable, especially with short evaluation samples,
small tail probabilities, and estimated risk thresholds (Christoffersen and Gonçalves, 2005;
Escanciano and Olmo, 2010; Barendse et al., 2023). This issue is also important in
regulation, because under the Basel market-risk framework expected shortfall is used
for capital measurement, while VaR-based exception backtesting remains operationally
central (Basel Committee on Banking Supervision, 2019).
This project builds on the exact finite-sample backtesting framework implemented in the
ExactVaRTest package (Chen, 2025). Its main objective is to extend exact calibration
to settings in which the correct benchmark null goes beyond the classical i.i.d. Bernoulli
hit-process model, with applications to estimated VaR, expected shortfall, and systemic
risk measures. The project has four main objectives:
1. To derive exact finite-sample benchmark null distributions for backtesting problems in which the standard i.i.d. Bernoulli framework is no longer the appropriate
benchmark because of estimation risk or multilevel tail conditioning.
2. To construct exact tests of unconditional coverage, dependence, and conditional
coverage for VaR, ES-related finite-state proxies, and systemic risk measures.ancial
3. To design computationally tractable algorithms, especially based on sparse dynamic
programming and pathwise recursion, that remain feasible at economically relevant
sample sizes.
4. To provide reproducible software and empirical applications demonstrating the
practical value of exact calibration in regulatory and systemic risk contexts. The project develops exact finite-sample backtesting methods based on analytic enumeration and deterministic recursion, avoiding asymptotic critical values and simulation error.
The main computational tool will be sparse dynamic programming, which makes exact
procedures feasible at practically relevant sample sizes.
The research has three components. First, it studies fixed-scheme historical-simulation
VaR under estimation risk, where the correct finite-sample null is no longer the standard
Bernoulli model (Christoffersen and Gonçalves, 2005; Escanciano and Olmo, 2010; Escanciano and Pei, 2012; Barendse et al., 2023). Second, it develops an exact implicit backtesting approach for expected shortfall by discretizing the tail into a finite-state process
(Fissler and Ziegel, 2016; Kratz et al., 2018). Third, it extends the same exact-calibration
logic to systemic risk measures such as CoVaR and MES through distress-conditioned categorical state processes (Adrian and Brunnermeier, 2016; Banulescu-Radu et al., 2021;
Fissler and Hoga, 2024; Francq and Zakoïan, 2025). Across these components, the project
combines finite-sample statistical theory with implementable computation and software
development.

M2 en finance quantitative