Paper: arXiv 2505.13185

Authors: Matteo Buttarazzi, Claudia Ceci

Abstract

This paper develops a continuous-time filtering framework for estimating a hazard rate subject to an unobservable change-point. This framework naturally arises in both financial and insurance applications, where the default intensity of a firm or the mortality rate of an individual may experience a sudden jump at an unobservable time, representing, for instance, a shift in the firm’s risk profile or a deterioration in an individual’s health status. By employing a progressive enlargement of filtration, we integrate noisy observations of the hazard rate with default-related information. We characterise the filter, i.e. the conditional probability of the change-point given the information flow, as the unique strong solution to a stochastic differential equation driven by the innovation process enriched with the discontinuous component. A sensitivity analysis and a comparison of the filter’s behaviour under various information structures are provided. Our framework further allows for the derivation of an explicit formula for the survival probability conditional on partial information. This result applies to the pricing of credit-sensitive financial instruments such as defaultable bonds, credit default swaps, and life insurance contracts. Finally, a numerical analysis illustrates how partial information leads to delayed adjustments in the estimation of the hazard rate and consequently to mispricing of credit-sensitive instruments when compared to a full-information setting.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using progressive enlargement of filtration, change-of-measure techniques, and stochastic calculus for nonlinear filtering in a change-point model. However, it is largely theoretical and lacks real-world empirical validation or extensive data analysis, relying instead on synthetic numerical examples.

Research Flowchart

  flowchart TD
    A[Research Goal: Estimate unobservable hazard rate change-point & price credit/insurance contracts] --> B[Inputs: Noisy hazard rate observations + Default/Survival timing information]
    B --> C[Methodology: Progressive enlargement of filtration & SDE filtering framework]
    C --> D[Computation: Solve SDE for conditional change-point probability & derive explicit survival formula]
    D --> E[Sensitivity Analysis: Compare filter behavior across different information structures]
    E --> F[Outcomes: Precise pricing formulas for bonds/CDS/life insurance & identification of partial-info mispricing delays]