Paper: arXiv 2405.00576

Authors: Jian He, Asma Khedher, Peter Spreij

Abstract

In this paper we develop Maximum likelihood (ML) based algorithms to calibrate the model parameters in credit rating transition models. Since the credit rating transition models are not Gaussian linear models, the celebrated Kalman filter is not suitable to compute the likelihood of observed migrations. Therefore, we develop a Laplace approximation of the likelihood function and as a result the Kalman filter can be used in the end to compute the likelihood function. This approach is applied to so-called high-default portfolios, in which the number of migrations (defaults) is large enough to obtain high accuracy of the Laplace approximation. By contrast, low-default portfolios have a limited number of observed migrations (defaults). Therefore, in order to calibrate low-default portfolios, we develop a ML algorithm using a particle filter (PF) and Gaussian process regression. Experiments show that both algorithms are efficient and produce accurate approximations of the likelihood function and the ML estimates of the model parameters.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces sophisticated non-Gaussian state-space calibration techniques using Laplace approximations and particle filtering tailored to different credit portfolio default densities. The mathematical framework is robust and directly addresses a classic quantitative credit risk estimation problem, though empirical validation remains primarily synthetic or algorithmic rather than extensively market-tested.

Research Flowchart

  flowchart TD
    A["Research Goal: Calibrate parameters in non-Gaussian Credit Rating Transition Models"] --> B{"Portfolio Type & Migration Data"}
    
    B -- "High-Default Portfolio (Large Migration Data)" --> C["Apply Laplace Approximation + Kalman Filter"]
    B -- "Low-Default Portfolio (Limited Migration Data)" --> D["Apply Particle Filter + Gaussian Process Regression"]
    
    C --> E["Maximum Likelihood Estimation (MLE) Computation"]
    D --> E
    
    E --> F["Outcomes: Efficient algorithms & accurate ML estimates for model parameters"]

Cited References (49)

  1. Modelling default risk (2003)
  2. A dimension reduction approach for loss valuation in credit risk modeling (2023) — link
  3. CreditMetrics - Technical document (1997)
  4. CreditRisk+ - A credit risk management framework (1997)
  5. Bayesian inference for generalized linear mixed models of portfolio credit risk (2007)
  6. Modelling dependent defaults (2001)
  7. Dependent defaults in models of portfolio credit risk (2003)
  8. Estimating default correlations from short panels of credit rating performance data (2002)
and 41 more
  1. Maximum likelihood estimate of default correlations (2007)
  2. Empirical credit cycles and capital buffer formation (2005) — link
  3. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking (2002)
  4. Novel approach to nonlinear/non- G aussian B ayesian state estimation (1993)
  5. Time series analysis by state space methods (2012)
  6. Robust extended K alman filtering (1999)
  7. Dual Extended Kalman Filter Methods (2002) — link
  8. The Unscented Kalman Filter (2002) — link
  9. An overview of existing methods and recent advances in sequential M onte C arlo (2007)
  10. A Kalman particle filter for online parameter estimation with applications to affine models (2021)
  11. Maximum Likelihood from Incomplete Data Via EM Algorithm (1977) — link
  12. A view of the EM algorithm that justifies incremental, sparse, and other variants (1998)
  13. Expectation-Maximization as lower bound maximization (2000)
  14. The EM Algorithm and Extensions (Wiley Series in Probability and Statistics) (2007) — link
  15. Expectation Maximization Algorithm (2010)
  16. Particle methods for optimal filter derivative: application to parameter estimation (2005) — link
  17. Particle Methods for Change Detection, System Identification, and Control (2004) — link
  18. Parameter estimation for discrete-time nonlinear systems using EM (2008) — link
  19. Smooth Particle Filters for Likelihood Evaluation and Maximisation (2002)
  20. On the pricing of corporate debt: The risk structure of interest rates (1974)
  21. Valuing credit default swaps II : Modeling default correlations (2001)
  22. Distance to default (2001)
  23. Credit risk: pricing, measurement, and management (2004)
  24. Credit barrier models (2003)
  25. Credit risk: Drawing the analogy (1992)
  26. Pricing derivatives on financial securities subject to credit risk (1995)
  27. Dependence modelling, model risk and model calibration in models of portfolio credit risk (2002)
  28. The multi-state latent factor intensity model for credit rating transitions (2008)
  29. Dependent credit migrations (2006)
  30. Stability of rating transitions (2000)
  31. Ratings migration and the business cycle, with application to credit portfolio stress testing (2002)
  32. Macroeconomic default modeling and stress testing (2018)
  33. Bank credit cycles (2008)
  34. Business and default cycles for credit risk (2005)
  35. Stochastic migration models with application to corporate risk (2005)
  36. An Overview of Sequential Monte Carlo Methods for Parameter Estimation in General State-Space Models (2009) — link
  37. The distribution of loan portfolio value (2002)
  38. Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning) (2005)
  39. Scalable Inference for Structured Gaussian Process Models (2011)
  40. Fast near-GRID Gaussian process regression (2013)
  41. Infinite Tucker decomposition: Nonparametric Bayesian models for multiway data analysis (2011) — link

0 of 49 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.