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

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
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. 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” (?¿!)
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 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 Million
1 American Trillion= 10 000 Million
**Li’s first paper appears on 1999

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. 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 efficient: 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
Definitions:
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 Li’s 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 Coefficient (extreme events*)
Two Examples: Gaussian (Li’s Model), T-Student
* Such as crisis

14. Modelling the Default Risk IV

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
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. Pricing the Default Risk III
Weaknesses:
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. 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. 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.
ρ key factor on convergence.

22. Simulating the model: Results and Criticism VI
Number of k-th defaults
Increasing ρ 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)