Stochastic Hardness in Model Risk: A Framework for Tangible and Auditable Assessments in Banking

Banking executives need quantitative, auditable tools to assess model reliability that satisfy regulatory requirements while reducing operational overhead. This paper introduces a statistical stress-testing framework for model reliability assessment that provides explicit pass/fail thresholds for regulatory compliance. Our approach transforms subjective model validation into objective, data-driven certification through tail-exponent analysis and summability testing. The framework aims to deliver a polynomial-tail threshold at $p = 1$ as a concrete decision criterion, automated monitoring systems that integrate with existing infrastructure, and comprehensive audit trails that strengthen regulatory confidence. The methodology is designed to align with OCC 2011-12, SR 11-7, and Basel III requirements as a design objective while potentially providing superior risk detection compared to traditional backtesting approaches. We present a theoretical framework with operationalization pathways, without claiming realized financial impacts or supervisory approvals. Our choice of tail-exponent analysis is motivated by its direct link to computational complexity: when error rates decay faster than $1/n$, reliability scales polynomially with input size; slower decay indicates potential NP-hard explosion of unreliability. This makes the tail index a natural diagnostic for distinguishing tractable from intractable model behavior. We motivate this approach via the WC--SP dimension~\cite{DyerStougie}: (SP = distributional Stochastic Polynomial-time; “WC” here is a shorthand for worst-case, NP-hard–like regimes) worst‑case intractability (WC-like regimes) versus distributional polynomial‑time behavior (SP) on operational inputs. The critical issue is whether reliability scales tractably on typical data; tail‑exponent monitoring offers an auditable proxy for staying in the SP-like regime without invoking worst‑case assumptions.