Credit Migration and Derivatives Pricing Using Copulas
Bibliographic record
Abstract
The multivariate modelling of default risk is a crucial aspect of the pricing of credit derivative products referencing a portfolio of underlying assets, and the evaluation of Value at Risk of such portfolios. This paper proposes a model for the joint dynamic behavior of credit ratings for several rms. Namely, individual credit ratings are modelled by univariate continuous time Markov chain, while their joint dynamics is modelled using copulas. A by-product of the method is the joint laws of the default times of all the rms in the portfolio. The use of copulas allows us to incorporate our knowledge of the modelling of univariate processes, into a multivariate framework. The Normal and Student copulas commonly used in the literature as well as by practitioners do not produce very di¤erent estimates of default risk prices. We show that this result is restricted to these two two basic copulas. That is, for any other family of copula, the choice of the copula greatly a¤ects the pricing of default risk. Key Words: Copula, Markov chain, credit risk, credit rating migration J.E.L. classi cation: G10, G20, G28, C16 Send correspondence to Nicolas Papageorgiou,Finance Department, HEC Montreal, 3000 Cote Sainte-Catherine, Montreal QC H3T 2A7, Canada. or at nicolas.papageorgiou@hec.ca . All the authors are at HEC Montreal can be reached at www.hec.ca/pages/ rstname.lastname. Funding in partial support of this work was provided by the Natural Sciences and Engineering Research Council of Canada, the Fonds quebecois de la recherche sur la nature et les technologies, and the Institut de nance mathematique de Montreal. We thank Hyung-Seob Kim for his help in creating the database.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".