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Filtering in a hazard rate change-point model with financial and life-insurance applications

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.

Filtering in a hazard rate change-point model with financial and life-insurance applications

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.
Paper Structure (13 sections, 11 theorems, 115 equations, 3 figures, 2 tables)

This paper contains 13 sections, 11 theorems, 115 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Modeling framework
  3. The filtering problem
  4. Sensitivity Analysis and Comparison of Hazard Rate Estimation Approaches
  5. Sensitivity Analysis of the Filter Dynamics
  6. Comparison between the $\mathbb{G}^Y$-estimate and the $\mathbb{F}^Y$-estimate of $(\mu_t)_{t\ge 0}$
  7. Financial and actuarial applications
  8. The conditional survival probability
  9. Pricing of credit derivatives
  10. Numerical analysis
  11. Extensions: an additional source of randomness
  12. Conclusions
  13. Preliminary proofs

Key Result

Proposition 3.1

For any $t\ge0$, the following equalities hold $\mathsf{P}_\pi$-a.s.: and

Figures (3)

  • Figure 1: Trajectories of the filter dynamics for different parameter choices.
  • Figure 2: Plot of the hazard rate $(\mu_t)_{t\ge 0}$ (in blue), its $\mathbb{G}^Y$-estimate $(\hat{\mu}_t)_{t\ge0}$ (in yellow) and its $\mathbb{F}^Y$-estimate $(\hat{\mu}^F_t)_{t\ge0}$ (in green).
  • Figure 3: Comparison of the hazard rate process and defaultable zero-coupon bond prices under full and partial information.

Theorems & Definitions (27)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Proposition 3.6
  • ...and 17 more