際際滷

際際滷Share a Scribd company logo

Criticism on Li's Copula Approach

Oleguer Sagarra
Oleguer Sagarra

This document summarizes a paper on modeling default correlation using copula functions. It discusses: 1) How credit default swaps and collateralized debt obligations securitize and trade default risk. 2) Two approaches for modeling individual and joint default probabilities - the Li model uses hazard rate functions and a Gaussian copula. 3) Simulations test the Gaussian and Student copulas and find small differences, but the hazard rate function has a bigger impact on results than the copula. 4) The Li model is criticized for unrealistic assumptions, inability to model extreme events, and circular reasoning in using CDS-priced data. While simple, it fails to capture important features of default correlation.

1 of 25
On the modelling of default correlation using copula functions Econophysics Final Work. Master in Oleguer Sagarra Pascual Computational Physics, UB-UPC 2011. June 2011
Quick Review of Contents Introduction: The Risk-Credit based Trading Securizing the Default Risk: CDS CDOs Modelling the Default Risk: Assumptions Individual Default: Credit Curves Correlated Default: Copula Approach Pricing the Risk: Lis Model Simulating the model: Results Criticism
Risk-Credit Based Trading I Before... (Traditional Banking) Investor puts Money on Bank Borrower ask money to the Bank Bank evaluates the Borrower, lends money (takes a risk, or not!) and charges him a penalty, that is returned to the investors. Key Point: Good Credit Risk assessment. If Borrower defaults (fails to pay), Bank loses money.
Risk-Credit Based Trading II Irruption of Derivatives: We can trade with everything! Why not trade with risk? Securization Now the Bank sells the risk from the Borrower to an Insurer. Borrower defaults : Insurer pays a penalty Borrower pays: Insurer gets payed a periodic premium for assuming the risk. Advantages: SPV: Outside the books. No taxes. Capital freed. Allows more Leverage. Macro-Economic mainstream: Good: It diffuses risk on the system (?多!) Magic: Bank is risk safe ? No, because the it is doubly exposed to default: By Insurer and/or by Borrower! Please Remember: More Risk = More Premium = More Business! (Or at least until something goes wrong...). And the banks no longer care about risk... they are insured!
Securizing the Default Risk I Individual assets subject to Credit Default events: Mortgages, Student Debts, Credit Card... CDS (Credit Default Swaps) Key Point: Probability S(t) of an asset to survive to time t.
Securizing the Default Risk II Please note: One can generate many CDS contracts from the same asset! = More volume As in all derivatives: Cheaper than assets! Some 鍖gures... Starts in the 90s: 100 billion* $ by the end of 1998. Booms on the new millennia**: 1 trillion $ in 2000, 60 trillion $ by 2008. * 1 American Billion=1000 Million 1 American Trillion= 10 000 Million **Lis 鍖rst paper appears on 1999
Securizing the Default Risk III One step beyond: Collateralized Debt Obligations (CDOs) Take N default-susceptible assets and pool them together in a portfolio. Tranche the pool and sell the risk: Senior: (Low risk: 80%) AAA Mezzanine: (Med Risk: 15%) BBB Equity: (High Risk: 5 %) Unrated
Securizing the Default Risk IV Rating becomes independent of the subjacent assets Key Point: Joint Probability S(t1,t2,t3...) of survival to k-th default of correlated assets.
Modelling the Default Risk I Assumptions: Market is fair : The prices are correct. Market is ef鍖cient: Information is accessible to determine evolution of market. Procedure: Model individual default probabilities (Marginals) Model joint default probabilities Problem: Solution is not unique, if the assets are correlated!
Modelling the Default Risk II Individual Default Modelling: 3 approaches: Rating agencies + Historical data Merton approach (stochastic random walk) Current Market Data approach De鍖nitions: S(t)= 1- F(t) : Survival Function to time t. h(t) : Hazard Rate Function. Proba of defaulting in the interval [t,t +dt].
Modelling the Default Risk III We can easily solve this using B.C: (S(0)=1, S(inf)=0) Assuming h(t) piecewise constant function*, And the problem is solved (assuming we are able to construct h(t)). *h(t): Stochastic nature. But in Lis model is piecewise constant
Modelling the Default Risk IV Joint Default Modelling: Copula Approach : Characterise correlation of variables with the copula (independently of marginals) Problem not unique: Many families of copulas exist
Modelling the Default Risk V Important feature: Tail Coef鍖cient (extreme events*) Two Examples: Gaussian (Lis Model), T-Student * Such as crisis
Modelling the Default Risk IV
Pricing the Default Risk I Suppose we have a set of hazard rate functions {hi(t)}... We generate a set of correlated {Ui=Ti(Ti)} using a copula. We obtain joint default times via the transform {Ti=F-1 (Ui)}. Once we have that, it is simple to derive the fair price of the CDO/CDS contract using no-arbitrage arguments.
Pricing the Default Risk II Lis procedure: Infer h(t) piecewise constant from the market for each price, based on the price of the CDS contracts at different maturities T (expiring times). Determine 1-Factor from market data using ML methods. Use 1-Factor Gaussian Copula* to generate default times via MC simulation and obtain prices for CDO averaging. * Extreme additional assumption: Pairwise correlation is constant between assets.
Pricing the Default Risk III Weaknesses: Unrealistic assumption for h(t), . Bad characterisation of extreme events Massive presence of Bias: Relied on data from CDS, priced from other CDS! Strengths: Simple, computationally easy. Few parameters to estimate. So... (almost) everybody used it!
Simulating the model: Results and Criticism I We apply two tests to both the Student and Gaussian Copula: Error spread: We apply 5% random errors to both and h(t). We simulate a crisis, with a h(t) non piece-wise function.
Simulating the model: Results and Criticism II Relevant Magnitudes: Mean default time Mean survival rate Extreme events: Probability of k-assets defaulting Times to k-th default
Simulating the model: Results and Criticism III h(t) functions used: Piece-wise constant function Continuos function with random normal noise
Simulating the model: Results and Criticism IV Error check: Good convergence as N grows. Small differences between two copulas. key factor on convergence.
Simulating the model: Results and Criticism VI Number of k-th defaults Increasing clusters events Small differences between two copulas.
Simulating the model: Results and Criticism VI k-th time default h(t) effect is more important than the copula. In fact, correlation might be included twice in the model. Mean k-th defaults are more clustered in the Student copula.
Criticism: Theory strongly dependent on h(t). Is it possible to estimate h(t) from market data? Reduces correlation to a single factor. Modelling: Inability to do stress testing. Inadequate usage of Mathematical/Econophysics formulas. Very quantitative results. Inconclusive results. Feedback: Bubble Effect. Complete fail to reproduce fat tails (extreme events)
A question arises: Could all this have been avoided ? All Models are Wrong but some are useful (George P. Box 1987)

More Related Content

Criticism on Li's Copula Approach

  • 1. On the modelling of default correlation using copula functions Econophysics Final Work. Master in Oleguer Sagarra Pascual Computational Physics, UB-UPC 2011. June 2011
  • 2. Quick Review of Contents Introduction: The Risk-Credit based Trading Securizing the Default Risk: CDS CDOs Modelling the Default Risk: Assumptions Individual Default: Credit Curves Correlated Default: Copula Approach Pricing the Risk: Lis Model Simulating the model: Results Criticism
  • 3. Risk-Credit Based Trading I Before... (Traditional Banking) Investor puts Money on Bank Borrower ask money to the Bank Bank evaluates the Borrower, lends money (takes a risk, or not!) and charges him a penalty, that is returned to the investors. Key Point: Good Credit Risk assessment. If Borrower defaults (fails to pay), Bank loses money.
  • 4. Risk-Credit Based Trading II Irruption of Derivatives: We can trade with everything! Why not trade with risk? Securization Now the Bank sells the risk from the Borrower to an Insurer. Borrower defaults : Insurer pays a penalty Borrower pays: Insurer gets payed a periodic premium for assuming the risk. Advantages: SPV: Outside the books. No taxes. Capital freed. Allows more Leverage. Macro-Economic mainstream: Good: It diffuses risk on the system (?多!) Magic: Bank is risk safe ? No, because the it is doubly exposed to default: By Insurer and/or by Borrower! Please Remember: More Risk = More Premium = More Business! (Or at least until something goes wrong...). And the banks no longer care about risk... they are insured!
  • 5. Securizing the Default Risk I Individual assets subject to Credit Default events: Mortgages, Student Debts, Credit Card... CDS (Credit Default Swaps) Key Point: Probability S(t) of an asset to survive to time t.
  • 6. Securizing the Default Risk II Please note: One can generate many CDS contracts from the same asset! = More volume As in all derivatives: Cheaper than assets! Some 鍖gures... Starts in the 90s: 100 billion* $ by the end of 1998. Booms on the new millennia**: 1 trillion $ in 2000, 60 trillion $ by 2008. * 1 American Billion=1000 Million 1 American Trillion= 10 000 Million **Lis 鍖rst paper appears on 1999
  • 7. Securizing the Default Risk III One step beyond: Collateralized Debt Obligations (CDOs) Take N default-susceptible assets and pool them together in a portfolio. Tranche the pool and sell the risk: Senior: (Low risk: 80%) AAA Mezzanine: (Med Risk: 15%) BBB Equity: (High Risk: 5 %) Unrated
  • 8. Securizing the Default Risk IV Rating becomes independent of the subjacent assets Key Point: Joint Probability S(t1,t2,t3...) of survival to k-th default of correlated assets.
  • 9. Modelling the Default Risk I Assumptions: Market is fair : The prices are correct. Market is ef鍖cient: Information is accessible to determine evolution of market. Procedure: Model individual default probabilities (Marginals) Model joint default probabilities Problem: Solution is not unique, if the assets are correlated!
  • 10. Modelling the Default Risk II Individual Default Modelling: 3 approaches: Rating agencies + Historical data Merton approach (stochastic random walk) Current Market Data approach De鍖nitions: S(t)= 1- F(t) : Survival Function to time t. h(t) : Hazard Rate Function. Proba of defaulting in the interval [t,t +dt].
  • 11. Modelling the Default Risk III We can easily solve this using B.C: (S(0)=1, S(inf)=0) Assuming h(t) piecewise constant function*, And the problem is solved (assuming we are able to construct h(t)). *h(t): Stochastic nature. But in Lis model is piecewise constant
  • 12. Modelling the Default Risk IV Joint Default Modelling: Copula Approach : Characterise correlation of variables with the copula (independently of marginals) Problem not unique: Many families of copulas exist
  • 13. Modelling the Default Risk V Important feature: Tail Coef鍖cient (extreme events*) Two Examples: Gaussian (Lis Model), T-Student * Such as crisis
  • 15. Pricing the Default Risk I Suppose we have a set of hazard rate functions {hi(t)}... We generate a set of correlated {Ui=Ti(Ti)} using a copula. We obtain joint default times via the transform {Ti=F-1 (Ui)}. Once we have that, it is simple to derive the fair price of the CDO/CDS contract using no-arbitrage arguments.
  • 16. Pricing the Default Risk II Lis procedure: Infer h(t) piecewise constant from the market for each price, based on the price of the CDS contracts at different maturities T (expiring times). Determine 1-Factor from market data using ML methods. Use 1-Factor Gaussian Copula* to generate default times via MC simulation and obtain prices for CDO averaging. * Extreme additional assumption: Pairwise correlation is constant between assets.
  • 17. Pricing the Default Risk III Weaknesses: Unrealistic assumption for h(t), . Bad characterisation of extreme events Massive presence of Bias: Relied on data from CDS, priced from other CDS! Strengths: Simple, computationally easy. Few parameters to estimate. So... (almost) everybody used it!
  • 18. Simulating the model: Results and Criticism I We apply two tests to both the Student and Gaussian Copula: Error spread: We apply 5% random errors to both and h(t). We simulate a crisis, with a h(t) non piece-wise function.
  • 19. Simulating the model: Results and Criticism II Relevant Magnitudes: Mean default time Mean survival rate Extreme events: Probability of k-assets defaulting Times to k-th default
  • 20. Simulating the model: Results and Criticism III h(t) functions used: Piece-wise constant function Continuos function with random normal noise
  • 21. Simulating the model: Results and Criticism IV Error check: Good convergence as N grows. Small differences between two copulas. key factor on convergence.
  • 22. Simulating the model: Results and Criticism VI Number of k-th defaults Increasing clusters events Small differences between two copulas.
  • 23. Simulating the model: Results and Criticism VI k-th time default h(t) effect is more important than the copula. In fact, correlation might be included twice in the model. Mean k-th defaults are more clustered in the Student copula.
  • 24. Criticism: Theory strongly dependent on h(t). Is it possible to estimate h(t) from market data? Reduces correlation to a single factor. Modelling: Inability to do stress testing. Inadequate usage of Mathematical/Econophysics formulas. Very quantitative results. Inconclusive results. Feedback: Bubble Effect. Complete fail to reproduce fat tails (extreme events)
  • 25. A question arises: Could all this have been avoided ? All Models are Wrong but some are useful (George P. Box 1987)