Copula approach to Default Correlation and the Credit Crisis of 2008/2009
1. Modeling
Correla-
tion in
Credit
Modeling Correlation in Credit Risk
Risk
“Copula Functions”
Robbin
Tops
“The
Crisis” Robbin Tops
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
Advisor: Dr. Bas Kleijn
KdV Instituut voor wiskunde
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Universiteit van Amsterdam
6. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
7. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
The Credit Crisis brought two groups together
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
8. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
Which represent
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
9. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
10. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
Investors wanted to turn their money into...
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
11. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
MORE MONEY!
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
12. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
Normally, investors go to the
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
13. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
and get so called T-BONDS or Treasury Bond
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
14. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
BUT
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
15. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
Federal Reserve Chairman Alan Greenspan lowered interest rates.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
And investors said: “Thank, but no thanks”.
16. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
On the flipside U.S. Bank could borrow for almost nothing.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
17. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
18. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
27. Modeling
Correla-
tion in
Credit
Example: Leverage
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
Thus Leverage make good deals into GREAT deals.
28. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
We still have our investors sitting on a lot of money, wanting to make
“The more!
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
29. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
This gives Wallstreet an idea!
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
30. Modeling
Correla-
tion in
Credit
What is the Credit Crisis?
Risk
Robbin
Tops
Connecting investors to homeowners through mortgages.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
31. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
“The
A family wants to buy a house
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
32. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
They go to a mortgage broker.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
33. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
The mortgage broker links the homeowners to a mortgage lender.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
34. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
An investment banker from Wallstreet calls the mortgage lender.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
35. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
The mortgage lender sells the mortgage to the investment banker.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
36. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops The investment banker buys many of these mortgages to make a
“The
deal with a lot of leverage!
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
Thus borrowing a lot of money from the federal reserve!
37. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
All these mortgages are now in a box and the investment banker
“The receives all the mortgage payments from the homeowners.
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
38. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
“The
Crisis” This box is called a CDO.
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
39. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
“The
The math-wizards from Wallstreet cut this box in three slices.
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
40. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
If some homeowners default the bottom tray may not get filled.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
41. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
Other banks or insurance companies (e.g. AIG) will insure the “safe”
“The
Crisis”
slice with a CDS.
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
42. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
Thus the rating agencies will give a rating according to the slices in
“The the CDO.
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
43. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
INTERESTING: Mortgages alone are almost never rated AAA but the
“The top slice receives AAA ratings.
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
44. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops Now the investment banker sells the slices individually:
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
The “Safe” slice is sold to investors only wanting safe investments.
The “Okay” slice is sold to other investment bankers.
The “Risky” slice is sold to hedge funds and other risk takers.
45. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
“The
MORE!!!
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
46. Modeling
Correla-
tion in
Credit
Connection: homeowners and investors
Risk
Robbin
Tops
The whole process repeats itself, but no more families.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
47. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
This gives the investment banker another idea!!!
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
48. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
“The
If a homeowner defaults on his mortgage...
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
49. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
...the investment banker owns the house.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
But housing prices have been rising practically forever!
50. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
Thus the investment banker adds more risk to the mortgages.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
55. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
Again the investment banker makes a CDO, now with the sub-prime
“The mortgages!
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
56. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops Now some of the sub-prime mortgages default.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
No big deal!?
57. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
BUT more sub-prime mortgages defaulted!
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
58. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
This changed the relation between supply and demand.
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
59. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
This created an interesting situation for homeowners who did not
“The default.
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
60. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
“The
Thus they walked away from their mortgages!
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
61. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
Now the investment banker has a box full of worthless houses and no
“The one wants to buy them!
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
62. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
But he was not the only one!
“The
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
63. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops Because the investment banker used a lot of leverage to amplify his
“The
deal.
Crisis”
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
He was in a lot of trouble!
64. Modeling
Correla-
tion in
Credit
Crisis
Risk
Robbin
Tops
Consequently the whole financial system freezes, creating a frozen
“The
Crisis”
credit market!
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
68. Modeling
Correla-
tion in
Credit
Credit Default Swap
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products
Pricing
periodic payments
Pricing
?? bps above LIBOR (CDS spread)
model: −→
Credit-
Metrics
Protection Buyer Protection Seller
Critical
←−
View Specified payment in case
Conclusion of default (Loss given Default)
69. Modeling
Correla-
tion in
Credit
‘Cash’ Collateralized Debt Obligation
Risk
Robbin
Tops
“The
Crisis”
Credit
Portfolio
Derivative
Company 1 → Bond 1
Products
Company 2 → Bond 2
Periodic coupon Super Senior Tranche
Pricing Periodic payments Lowest return/Residual loss
. .
payments ?? bps above LIBOR
Pricing . .
−→ −→ Senior Tranche
model: . .
Credit- 2nd lowest return/3rd ..% of loss
Metrics SPV
Mezzanine Tranche
Critical 2nd highest return/2nd ..% of loss
View ←− ←−
↓ ↓
Sp. payment Sp. payment
Conclusion (in case of default) (in case of default) Equity Tranche
Highest return/1st ..% of loss
Company n → Bond n
71. Modeling
Correla-
tion in
Credit
Usual scenario
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products
Individual Bonds/Loans
Pricing
Bond issuer may default on the bond/loan (Credit Risk),
Pricing
Money that is loaned to bond issuer is illiquid (Market Risk).
model:
Credit-
Metrics
Critical
View
Conclusion
72. Modeling
Correla-
tion in
Credit
Usual scenario
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products
Individual Bonds/Loans
Pricing
Bond issuer may default on the bond/loan (Credit Risk),
Pricing
Money that is loaned to bond issuer is illiquid (Market Risk).
model:
Credit-
Metrics CDSs
Critical Protection buyer may default (Credit Risk),
View
Protection seller may default (Credit Risk).
Conclusion
Default dependence of protection buyer and seller!
73. Modeling
Correla-
tion in
Credit
CDO scenario
Risk
Robbin
Tops
“The
Crisis”
Cash CDOs
Credit
Derivative Any amount of bond issuers may default (Credit Risk),
Products
Money that is loaned is illiquid (Market Risk).
Pricing
Pricing
model:
Credit-
Metrics
Default dependence of all bond issuers!
Critical
View
Conclusion
74. Modeling
Correla-
tion in
Credit
CDO scenario
Risk
Robbin
Tops
“The
Crisis”
Cash CDOs
Credit
Derivative Any amount of bond issuers may default (Credit Risk),
Products
Money that is loaned is illiquid (Market Risk).
Pricing
Pricing
model:
Credit-
Metrics
Default dependence of all bond issuers!
Critical
View
Synthetic CDOs
Conclusion
Any amount of protection buyer may default (Credit Risk),
Any amount of protection seller may default (Credit Risk).
Default dependence between buyers and seller!
75. Modeling
Correla-
tion in
Credit
Introduction
Risk
Robbin
Tops What will follow
“The
Li’s approach to default correlation:
Crisis” Model survival time of credit entities,
Credit Model asset correlation,
Derivative
Products Use copula and correlation to create dependence structure,
Pricing Rescale marginals of joint distribution to survival time distributions,
Pricing Generate from copula to calculate default correlation.
model:
Credit-
Metrics
Critical
View
Conclusion
76. Modeling
Correla-
tion in
Credit
Introduction
Risk
Robbin
Tops What will follow
“The
Li’s approach to default correlation:
Crisis” Model survival time of credit entities,
Credit Model asset correlation,
Derivative
Products Use copula and correlation to create dependence structure,
Pricing Rescale marginals of joint distribution to survival time distributions,
Pricing Generate from copula to calculate default correlation.
model:
Credit-
Metrics
Diagram
Critical
View
Survival Time
Conclusion
Distributions
Joint Survival
Times &
Default
Correlation
Asset Value
Gaussian
Processes &
Copula
Asset −→
Function
Correlation
77. Modeling
Correla-
tion in
Credit
Survival time distribution
Risk
Robbin
Tops Definition (Time-to-default)
Let Ti be the random variable time-to-default of financial entity i and
“The
Crisis” STi (t ) := P (Ti > t ) is the survival function of i.
Credit
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
Cox, D. R. , Oake, D. , Analysis of Survival Data, London: Chapman and Hall, 1984.
78. Modeling
Correla-
tion in
Credit
Survival time distribution
Risk
Robbin
Tops Definition (Time-to-default)
Let Ti be the random variable time-to-default of financial entity i and
“The
Crisis” STi (t ) := P (Ti > t ) is the survival function of i.
Credit
Derivative
Products
Definition (Hazard Rate Function)
Pricing If T is absolutely continuous and define fT (t ) as the density function
Pricing of T , then
model:
f (t ) −ST (t )
Credit-
Metrics h (t ) : = T =
Critical
ST (t ) ST (t )
View
is the hazard rate function.
Conclusion
Cox, D. R. , Oake, D. , Analysis of Survival Data, London: Chapman and Hall, 1984.
79. Modeling
Correla-
tion in
Credit
Survival time distribution
Risk
Robbin
Tops Definition (Time-to-default)
Let Ti be the random variable time-to-default of financial entity i and
“The
Crisis” STi (t ) := P (Ti > t ) is the survival function of i.
Credit
Derivative
Products
Definition (Hazard Rate Function)
Pricing If T is absolutely continuous and define fT (t ) as the density function
Pricing of T , then
model:
f (t ) −ST (t )
Credit-
Metrics h (t ) : = T =
Critical
ST (t ) ST (t )
View
is the hazard rate function.
Conclusion
Thus:
t
h(s )ds
ST (t ) = e 0
Cox, D. R. , Oake, D. , Analysis of Survival Data, London: Chapman and Hall, 1984.
80. Modeling
Correla-
tion in
Credit
Hazard Rate h(s )
Risk
Robbin
Estimation
Tops Historical default information,
“The
Merton option theoretical approach,
Crisis” Implied approach using market price of defaultable bonds or
Credit asset swap spreads.
Derivative
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
Duffie, D. and Singleton, J. , Modeling Term Structure of Defaultable Bonds. In: Rev. Financ. Stud. 12, pp. 687–720, 1999.
Li, D. X. , On Default Correlation: A Copula Function Approach. In: Journal of Fixed Income 9(4), pp. 43–54, 2000.
81. Modeling
Correla-
tion in
Credit
Hazard Rate h(s )
Risk
Robbin
Estimation
Tops Historical default information,
“The
Merton option theoretical approach,
Crisis” Implied approach using market price of defaultable bonds or
Credit asset swap spreads.
Derivative
Products
Pricing Example (Hazard rate function for B rating)
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
Duffie, D. and Singleton, J. , Modeling Term Structure of Defaultable Bonds. In: Rev. Financ. Stud. 12, pp. 687–720, 1999.
Li, D. X. , On Default Correlation: A Copula Function Approach. In: Journal of Fixed Income 9(4), pp. 43–54, 2000.
82. Modeling
Correla-
tion in
Credit
Asset correlation
Risk
Robbin
Definition (Asset value process)
Tops The asset valuation of a financial entity i is assumed to follow a
“The
standard geometric Brownian motion, i.e.
Crisis”
Credit 1 2 √
Derivative
i
Vti = V0 exp (µi − σi )t + σi tZt
Products 2
Pricing
Pricing where µi represents the mean of the rate of return, σi the volatilities
model:
Credit- of returns on assets and Zt ∼ N (0, 1).
Metrics
Critical
View
Conclusion
Merton, R. , On the pricing of corporate debt: The risk structure of interest rates. In: Journal of Finance 28, pp. 449–470, 1974.
83. Modeling
Correla-
tion in
Credit
Asset correlation
Risk
Robbin
Definition (Asset value process)
Tops The asset valuation of a financial entity i is assumed to follow a
“The
standard geometric Brownian motion, i.e.
Crisis”
Credit 1 2 √
Derivative
i
Vti = V0 exp (µi − σi )t + σi tZt
Products 2
Pricing
Pricing where µi represents the mean of the rate of return, σi the volatilities
model:
Credit- of returns on assets and Zt ∼ N (0, 1).
Metrics
Critical
View Thus
Conclusion
qi := P (Vti ≤ vdef )
is assumed to be the individual default probability.
j
ρasset := ρ(Rti , Rt )
is the (linear) correlation coefficient between the normalized asset
returns of financial entities i and j.
Merton, R. , On the pricing of corporate debt: The risk structure of interest rates. In: Journal of Finance 28, pp. 449–470, 1974.
84. Modeling
Correla-
tion in
Credit
Copulas
Risk
Robbin
Definition (Copula)
Tops
If F is a n-dimensional joint distribution function with marginals
“The F1 , F2 , . . . , Fn then a copula C is a function
Crisis”
Credit C : [0, 1]n −→ [0, 1]
Derivative
Products
Pricing
such that
Pricing
C (F1 (x1 ), . . . , Fn (xn )) := F (x1 , . . . , xn ).
model:
Credit-
Metrics
Critical
View
Conclusion
Nelson, R. B. , An Introduction to Copulas. In: Journal of Finance 28, New York: Springer, 1999.
85. Modeling
Correla-
tion in
Credit
Copulas
Risk
Robbin
Definition (Copula)
Tops
If F is a n-dimensional joint distribution function with marginals
“The F1 , F2 , . . . , Fn then a copula C is a function
Crisis”
Credit C : [0, 1]n −→ [0, 1]
Derivative
Products
Pricing
such that
Pricing
C (F1 (x1 ), . . . , Fn (xn )) := F (x1 , . . . , xn ).
model:
Credit-
Metrics
Critical
Theorem (Sklar’s Theorem (modified))
View Let F be a n-dimensional distribution function with continuous
Conclusion marginals F1 , . . . , Fn . Then there exists a unique n-dimensional
copula C such that for all x ∈ Rn , ¯
F (x1 , . . . , xn ) = C (F1 (x1 ), . . . , Fn (xn )). (1)
Conversely, if C is a copula and F1 , . . . , Fn are univariate continuous
distribution functions, then the function F defined in (1) is a
multivariate distribution function with marginals F1 , . . . , Fn .
Nelson, R. B. , An Introduction to Copulas. In: Journal of Finance 28, New York: Springer, 1999.
86. Modeling
Correla-
tion in
Credit
Gaussian Copula
Risk
Robbin
Tops
“The
Bivariate Gaussian Copula
Crisis” Let F1 and F2 be standard normal distribution functions, then
Credit
Derivative z1 z2
Ga
Products
Cρ (F1 (z1 ), F2 (z2 )) = φ2 (x, y |ρ)dxdy = Φ2 (z1 , z2 , ρ)
Pricing −∞ −∞
Pricing
model:
Credit-
with ρ the correlation parameter between F1 and F2 .
Metrics
Critical
View
Conclusion
87. Modeling
Correla-
tion in
Credit
Gaussian Copula
Risk
Robbin
Tops
“The
Bivariate Gaussian Copula
Crisis” Let F1 and F2 be standard normal distribution functions, then
Credit
Derivative z1 z2
Ga
Products
Cρ (F1 (z1 ), F2 (z2 )) = φ2 (x, y |ρ)dxdy = Φ2 (z1 , z2 , ρ)
Pricing −∞ −∞
Pricing
model:
Credit-
with ρ the correlation parameter between F1 and F2 .
Metrics
Critical Thus:
View
The joint default probability is assumed to be
Conclusion
P (Vti ≤ vdef , Vti ≤ vdef ) = Cρasset (Φ(ri ), Φ(rj )) = Φ2 (ri , rj , ρasset )
i i Ga
vk
ln def −(µk − 1 σk )t
2
Vk 2
where k
vdef are default thresholds and rk := 0
√ for
σk t
k = i, j are normalized thresholds.
88. Modeling
Correla-
tion in
Credit
Rescale Marginals to Survival Time and
Risk
Default correlation
Robbin
Tops
Joint Survival Function
“The If Sk (t ) = 1 − Gk (t ) with k = i, j are the survival functions for Ti and
Crisis”
Tj , then the joint survival function can be defined as
Credit
Derivative
Products
P (Ti ≤ ti , Tj ≤ tj ) Ga
= Cρasset (Gi (ti ), Gj (tj ))
Pricing
(2)
Pricing
model:
Credit-
= Φ2 (Φ−1 (Gi (ti )), Φ−1 (Gj (tj )), ρasset ).
Metrics
Critical
View
Conclusion
Li, D. X. , On Default Correlation: A Copula Function Approach. In: Journal of Fixed Income 9(4), pp. 43–54, 2000.
89. Modeling
Correla-
tion in
Credit
Rescale Marginals to Survival Time and
Risk
Default correlation
Robbin
Tops
Joint Survival Function
“The If Sk (t ) = 1 − Gk (t ) with k = i, j are the survival functions for Ti and
Crisis”
Tj , then the joint survival function can be defined as
Credit
Derivative
Products
P (Ti ≤ ti , Tj ≤ tj ) Ga
= Cρasset (Gi (ti ), Gj (tj ))
Pricing
(2)
Pricing
model:
Credit-
= Φ2 (Φ−1 (Gi (ti )), Φ−1 (Gj (tj )), ρasset ).
Metrics
Critical
View
Default correlation
Conclusion
We generate from Equation (2) to calculate (linear) default correlation
as follows,
E (Ti Tj ) − E (Ti )E (Tj )
ρdef = ρ(Ti , Tj ) = .
Var (Ti )Var (Tj )
Li, D. X. , On Default Correlation: A Copula Function Approach. In: Journal of Fixed Income 9(4), pp. 43–54, 2000.
90. Modeling
Correla-
tion in
Credit
Asset values
Risk
Robbin
Tops
Theorem
“The
Crisis” Let Di (t ) and Dj (t ) be the comprehensive default events of the
Credit
Derivative
financial entities i and j, respectively. If asset value processes Vti and
Products j
Vt for financial entities i and j, respectively, then
Pricing
Pricing j j
model: Vti < vti , Vt < vt ⊂ Di ( t ) ∩ Dj ( t ) .
Credit-
Metrics
Critical
View
Conclusion
91. Modeling
Correla-
tion in
Credit
Asset values
Risk
Robbin
Tops
Theorem
“The
Crisis” Let Di (t ) and Dj (t ) be the comprehensive default events of the
Credit
Derivative
financial entities i and j, respectively. If asset value processes Vti and
Products j
Vt for financial entities i and j, respectively, then
Pricing
Pricing j j
model: Vti < vti , Vt < vt ⊂ Di ( t ) ∩ Dj ( t ) .
Credit-
Metrics
Critical
View
Conclusion
Example
Insurance
Company XYZ
insurance insurance
Zero asset correlation
Financial Financial
Entity i
←→ Entity j
92. Modeling
Correla-
tion in
Credit
Tail dependence of Gaussian Copula
Risk
Robbin
Tops
“The
Definition (Tail Dependence)
Crisis” The upper and lower tail dependence coefficients are defined by
Credit
Derivative
− −
Products λu = lim P X2 > F2 1 (q )|X1 > F1 1 (q )
Pricing q ↑1
Pricing
− −
model:
Credit- λl = lim P X2 < F2 1 (q )|X1 < F1 1 (q ) ,
Metrics q ↑1
Critical
View respectively, and measure the probability of joint extreme events.
Conclusion
93. Modeling
Correla-
tion in
Credit
Tail dependence of Gaussian Copula
Risk
Robbin
Tops
“The
Definition (Tail Dependence)
Crisis” The upper and lower tail dependence coefficients are defined by
Credit
Derivative
− −
Products λu = lim P X2 > F2 1 (q )|X1 > F1 1 (q )
Pricing q ↑1
Pricing
− −
model:
Credit- λl = lim P X2 < F2 1 (q )|X1 < F1 1 (q ) ,
Metrics q ↑1
Critical
View respectively, and measure the probability of joint extreme events.
Conclusion
Theorem
The upper and lower tail dependence coefficients of the Gaussian
Ga
copula Cρ are zero, that is,
λu = λl = 0
for ρ < 1.
94. Modeling
Correla-
tion in
Credit
Tail dependence of Gaussian Copula
Risk
Robbin
Tops
“The
Crisis”
Example
Credit
Derivative Tails of bivariate Gaussian versus bivariate t-distribution.
Products
Pricing
Pricing
model:
Credit-
Metrics
Critical
View
Conclusion
95. Modeling
Correla-
tion in
Credit
Dependence Structure in Rescaling
Risk
Marginals
Robbin
Tops Theorem
“The
Let H1 and H2 be strictly monotonic continuous functions defined on
Crisis” the range of random variables X1 and X2 , respectively, then
Credit
Derivative
Products
|ρ(X1 , X2 )| > |ρ (H1 (X1 ), H2 (X2 )) |,
Pricing
where ρ is the linear correlation coefficient.
Pricing
model: ⇒ Dependence structure always reduces if marginals are
Credit-
Metrics transformed!
Critical
View
Conclusion
96. Modeling
Correla-
tion in
Credit
Dependence Structure in Rescaling
Risk
Marginals
Robbin
Tops Theorem
“The
Let H1 and H2 be strictly monotonic continuous functions defined on
Crisis” the range of random variables X1 and X2 , respectively, then
Credit
Derivative
Products
|ρ(X1 , X2 )| > |ρ (H1 (X1 ), H2 (X2 )) |,
Pricing
where ρ is the linear correlation coefficient.
Pricing
model: ⇒ Dependence structure always reduces if marginals are
Credit-
Metrics transformed!
Critical
View Example
Conclusion
97. Modeling
Correla-
tion in
Credit
Summary
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products
Pricing Which flaws have we seen
Pricing Defaultable bond prices or asset swap spreads to estimate survival
model:
Credit- functions,
Metrics Asset value as underlying information for dependence structure,
Critical
View
Gaussian copula and simultaneous extreme events,
Conclusion
Rescaling results in weaker correlation structure,
Linear correlation has many undesirable properties.
98. Modeling
Correla-
tion in
Credit
Conclusion
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products CreditMetric Approach
Pricing Pricing of CDOs is complicated, due to complexity,
Pricing
model:
Assumptions should not be indiscriminately excepted.
Credit-
Metrics
Critical
View
Conclusion
99. Modeling
Correla-
tion in
Credit
Conclusion
Risk
Robbin
Tops
“The
Crisis”
Credit
Derivative
Products CreditMetric Approach
Pricing Pricing of CDOs is complicated, due to complexity,
Pricing
model:
Assumptions should not be indiscriminately excepted.
Credit-
Metrics
Critical
Possible Solutions
View Jump Levy component in asset value process,
Conclusion Factor models for default,
t-copula model,
Non-parametric model.