Criticism on Li's Copula Approach


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Slides accompaining the work on Li's Formula on CDS pricing.

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Criticism on Li's Copula Approach

  1. 1. On the modelling of default correlation using copula functions Econophysics Final Work. Master in Oleguer Sagarra PascualComputational Physics, UB-UPC 2011. June 2011
  2. 2. Quick Review of Contents Introduction: The Risk-Credit based Trading Securizing the Default Risk: CDS CDO’s Modelling the Default Risk: Assumptions Individual Default: Credit Curves Correlated Default: Copula Approach Pricing the Risk: Li’s Model Simulating the model: Results Criticism
  3. 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. 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” (?¿!) 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. 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. 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 figures... Starts in the 90’s: 100 billion* $ by the end of 1998. Booms on the new millennia**: 1 trillion $ in 2000, 60 trillion $ by 2008. * 1 American Billion=1000 Million1 American Trillion= 10 000 Million**Li’s first paper appears on 1999
  7. 7. Securizing the Default Risk III One step beyond: Collateralized Debt Obligations (CDO’s) 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. 8. Securizing the Default Risk IVRating becomes independent of the subjacent assets Key Point: Joint Probability S(t1,t2,t3...) of survival to k-th default of correlated assets.
  9. 9. Modelling the Default Risk IAssumptions: Market is fair : The prices are “correct”. Market is efficient: Information is accessible to determine evolution of market.Procedure: Model individual default probabilities (Marginals) Model joint default probabilitiesProblem: Solution is not unique, if the assets arecorrelated!
  10. 10. Modelling the Default Risk IIIndividual Default Modelling: 3 approaches: Rating agencies + Historical data Merton approach (stochastic random walk) Current Market Data approachDefinitions: 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. 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 Li’s model is piecewise constant
  12. 12. Modelling the Default Risk IVJoint Default Modelling: Copula Approach : Characterise correlation of variables with the copula (independently of marginals)Problem not unique: Many families of copulas exist
  13. 13. Modelling the Default Risk VImportant feature: Tail Coefficient (extreme events*)Two Examples: Gaussian (Li’s Model), T-Student * Such as crisis
  14. 14. Modelling the Default Risk IV
  15. 15. Pricing the Default Risk ISuppose 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 ofthe CDO/CDS contract using no-arbitrage arguments.
  16. 16. Pricing the Default Risk II Li’s 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 ρ 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. 17. Pricing the Default Risk IIIWeaknesses: Unrealistic assumption for h(t), ρ. 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. 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 ρ and h(t). We simulate a crisis, with a h(t) non piece-wise function.
  19. 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. 20. Simulating the model: Results and Criticism IIIh(t) functions used: Piece-wise constant function Continuos function with random normal noise
  21. 21. Simulating the model: Results and Criticism IV Error check: Good convergence as N grows. Small differences between two copulas. ρ key factor on convergence.
  22. 22. Simulating the model: Results and Criticism VI Number of k-th defaults Increasing ρ clusters events Small differences between two copulas.
  23. 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. 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. 25. A question arises: Could all this have been avoided ? “All Models are Wrong but some are useful” (George P. Box 1987)