Comparative analysis of CDO pricing models
Credit Correlation Modelling Comparative analysis of CDO pricing models
Purpose of the presentation
Global Derivatives 2005
24 May 2005
Some insights about current issues in CDO modelling Gaussian copula
Jean-Paul Laurent Professor, ISFA Actuarial School, University of Lyon Scientific consultant, BNP-Paribas
[email protected], http://laurent.jeanpaul.free.fr
Model dependence/Choice of copula
Joint work with X. Burtschell and J. Gregory (BNP-Paribas)
A comparative analysis of CDO pricing models available on www.defaultrisk.com
One factor Gaussian copula Ordering of risks Correlation sensitivities and Gaussian extensions Student t, double t, Clayton, Marshall-Olkin, Stochastic correlation Calibration issues Distribution of conditional default probabilities
Matching the correlation skew Further issues
1
2
Comparative analysis of CDO pricing models
Agenda
Semi explicit pricing, conditional default probabilities
Factor approaches to joint default times distributions:
Conditional default probabilities and pricing of CDOs One factor Gaussian copula
Dependence to the correlation parameter
Gaussian extensions, correlation sensitivities
V: low dimensional factor
Conditionally on V, default times are independent.
Conditional default and survival probabilities:
Model dependence/Choice of copula
Student t, double t, Clayton, Marshall-Olkin, Stochastic correlation
Calibration
Empirical results
Why factor models ?
Need tractable dependence between defaults:
Matching the correlation skew
Tackle with large dimensions (125 names in I-TRAXX) Parsimonious modelling Semi-explicit computations for CDO tranches
3
4
Semi explicit pricing, conditional default probabilities
Semi explicit pricing, conditional default probabilities
Semi-explicit pricing for CDO tranches
Characteristic function:
Laurent & Gregory [2003]
Default payments are based on the accumulated losses on the pool of credits:
By conditioning upon V and using conditional independence:
n
L(t ) = ∑ LGDi 1{τ i ≤t} , LGDi = N i (1 − δ i ) i =1
Tranche premiums only involves call options on the accumulated losses
Distribution of L(t) can be obtained by FFT
+ E ⎡( L(t ) − K ) ⎤ ⎣ ⎦
Or other inversion technique iV
5
Only need of conditional (on factor) probabilities pt 6
1
One factor Gaussian copula
Semi explicit pricing, conditional default probabilities
One factor Gaussian copula:
independent Gaussian,
Default times:
Fi marginal distribution function of default times
Conditional default probabilities:
mezzanine
senior
0%
5341
560
0.03
With respect to correlation Gaussian copula
equity
CDO margins (bps pa)
Attachment points: 3%, 10%
10%
3779
632
4.6
100 names
30%
2298
612
20
Unit nominal Credit spreads 100 bp
50%
1491
539
36
5 years maturity
70%
937
443
52
100%
167
167
91
7
8
One factor Gaussian copula
One factor Gaussian copula
Equity tranche premiums are decreasing wrt ρ
n
∀x ∈
→ n
, ∀ε , δ > 0
Let X and Y be Gaussian vectors with zero mean
⎛ 1 σ 12X ⎜ X ⎜ σ 21 1 ⎜ X Σ =⎜ ⎜ ⎜ ⎜⎜ ⎝
∆i f ( x ) = f ( x + ε ei ) − f ( x ) ε
∆εi ∆δj f ( x ) ≥ 0
Supermodular order (increase in dependence) X = ( X 1 ,… , X n )
« Supermodular » order of Gaussian vectors
General result ? Supermodular function f is such that:
f :
Y = (Y1 ,…, Yn )
σ Yji
1
Müller & Scarsini (2000), Müller (2001)
10
One factor Gaussian copula
« Stop-Loss » order
Accumulated losses: L(t ), L '(t )
L(t ) ≤sl L '(t ) ⇔ E ⎡( L(t ) − K ) ⎤ ≤ E ⎡( L '(t ) − K ) ⎤ , ∀K ≥ 0 ⎣ ⎦ ⎣ ⎦ +
1
1
Σ X ≤ ΣY ⇔ X ≤sm Y
One factor Gaussian copula
def
σ jiX
σ ijY
1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 1 ⎟⎠
def
9
1
⎛ 1 σ 12Y ⎜ Y ⎜ σ 21 1 ⎜ Y Σ =⎜ ⎜ ⎜ ⎜⎜ ⎝
Σ X ≤ ΣY ⇔ σ ijX ≤ σ ijY , ∀i, j
X ≤ sm Y ⇔ E ⎡⎣ f ( X ) ⎤⎦ ≤ E ⎡⎣ f (Y ) ⎤⎦ , ∀f supermodular
σ ijX
1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 1 ⎟⎠
+
Supermodular order of latent variables implies stop-loss order of accumulated losses
Thus,equity tranche premium is always decreasing with correlation
Guarantees uniqueness of « base correlation »
Monotonicity properties extend to Student t, Clayton (Wei & Hu [2002]) and Marshall-Olkin copulas 11
Second issue
Equity tranche premium decrease with correlation
Does ρ = 100% correspond to some lower bound?
ρ = 100% corresponds to « comonotonic » default dates:
(τ1 ,…,τ n ) comonotonic
where U is uniform
Tchen (1980)
⇔ (τ 1 ,…,τ n ) = ( F1−1 (U ),…, Fn−1 (U ) ) d
(τ 1,…,τ n ) ≤sm ( F1−1 (U ),…, Fn−1 (U ) )
ρ = 100% is a model free lower bound for the equity tranche premium 12
2
One factor Gaussian copula
One factor Gaussian copula
Third issue
Does ρ = 0% corresponds to the higher bound on the equity tranche premium? ρ = 0% corresponds to the independence case between default dates The answer is no, negative dependence can occur Base correlation does not always exists
Even in Gaussian copula models
Factor models are usually associated with positive dependence
Gaussian extensions
i.e. independent default dates are smaller with respect to supermodular order
Pairwise correlation sensitivities
Intra and intersector correlations In the core of correlation, Gregory & Laurent, Risk october 2004
ρ12
⎛ 1 ⎜ ⎜ ρ 21 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ 1 ⎜ ⎜ β1 ⎜β ⎜ 1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎟ ρij + δ ⎟ ⎟ ⎟ . ⎟ 1 ⎟ 1 .⎟ ⎟ . 1⎟⎠
1 1 .
ρij + δ
β1 1
β1 β1
β1
1
γ 1 . . 1
γ
1
βm
βm βm
βm
1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ βm ⎟ ⎟ βm ⎟ ⎟ 1 ⎠
13
14
Model dependence / choice of copula
Model dependence / choice of copula
Stochastic corrrelation copula
independent Gaussian variables Bi = 1 correlation ρ , Bi = 0 correlation β
)
(
(
Vi = Bi ρV + 1 − ρ 2 Vi + (1 − Bi ) β V + 1 − β 2 Vi
Student t copula
Embrechts, Lindskog & McNeil, Greenberg et al, Mashal et al, O’Kane & Schloegl, Gilkes & Jobst
⎧ X = ρV + 1 − ρ 2 V i ⎪⎪ i ⎨ Vi = W × X i ⎪ τ = F −1 ( t (V ) ) i i ν ⎪⎩ i
)
Vi = ( Bi ρ + (1 − Bi ) β ) V + 1 − ( Bi ρ + (1 − Bi ) β ) Vi 2
τ i = Fi pti|V
−1
(Φ(Vi ))
⎛ − ρV + Φ −1 ( Fi (t ) ) ⎞ ⎛ − β V + Φ −1 ( Fi (t ) ) ⎞ ⎟ + (1 − p )Φ ⎜ ⎟ = pΦ ⎜ 2 ⎜ ⎟ ⎜ ⎟ − 1 ρ 1− β 2 ⎝ ⎠ ⎝ ⎠
V , Vi independent Gaussian variables 2 ν follows a χν distribution W
Conditional default probabilities (two factor model)
⎛ − ρV + W −1/ 2 tν−1 ( Fi (t ) ) ⎞ ⎟ pti|V ,W = Φ ⎜ ⎜ ⎟ 1− ρ2 ⎝ ⎠
15
Model dependence / choice of copula
Model dependence / choice of copula
Clayton copula
Double t model (Hull & White) ⎛ν − 2 ⎞ Vi = ρi ⎜ ⎟ ⎝ ν ⎠
1/ 2
Schönbucher & Schubert, Rogge & Schönbucher, Friend & Rogge, Madan et al
⎛ ln U i ⎞ Vi = ψ ⎜ − ⎟ ⎝ V ⎠
τ i = Fi
−1
(Vi ) ψ ( s) = (1 + s )
−1/ θ
V: Gamma distribution with parameter θ
U1,…, Un independent uniform variables
Conditional default probabilities (one factor model) iV
(
)
⎛ν − 2 ⎞ V + 1 − ρi2 ⎜ ⎟ ⎝ ν ⎠
1/ 2
Vi
V , Vi are independent Student t variables
Marshall-Olkin construction of archimedean copulas
pt = exp V (1 − Fi (t ) −θ )
16
with ν and ν degrees of freedom
τ i = Fi −1 ( H i (Vi ) )
where Hi is the distribution function of Vi pti|V
17
1/ 2 ⎛ ⎛ν − 2 ⎞ −1 ⎜ 1/ 2 H i ( Fi (t ) ) − ρ i ⎜ ⎟ V ν ν ⎛ ⎞ ⎝ ⎠ = tν ⎜ ⎜ ⎜ ⎝ ν − 2 ⎟⎠ 1 − ρ i2 ⎜⎜ ⎝
⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠
18
3
Model dependence / choice of copula
Shock models (multivariate exponential copulas)
Duffie & Singleton, Giesecke, Elouerkhaoui, Lindskog & McNeil, Wong
Modelling of default dates: Vi = min (V ,Vi )
V ,Vi exponential with parameters α ,1 − α
(
Default dates τ i = Si−1 exp− min (V ,Vi )
Model dependence / choice of copula
)
Well suited for homogeneous portfolios
Dependence is « monotonic » in the parameter
Calibration procedure
Conditionally on V ,τ i are independent.
Conditional default probabilities
senior
0%
5341
560
0.03
3779
632
4.6
2298
612
20
With respect to correlation
100 names
30%
Unit nominal
Credit spreads 100 bp
5 years maturity
50%
Given the previous parameters
equity
mezzanine
senior
0%
5341
560
0.03
Attachment points: 3%, 10%
10%
3779
632
4.6
100 names
30%
2298
612
20
Unit nominal Credit spreads 100 bp
50%
1491
539
36
5 years maturity
70%
937
443
52
100%
167
167
91
CDO margins (bps pa)
Attachment points: 3%, 10%
Or given market quotes on equity trances
Reprice mezzanine and senior CDO tranches
Model dependence / choice of copula
mezzanine
Computed under one factor Gaussian model
20
equity
10%
CDO margins (bps pa) Gaussian copula
Fit Clayton, Student t, double t, Marshall Olkin parameters onto CDO equity tranches
19
Model dependence / choice of copula
One parameter copulas
iV
Si marginal survival function
qt = 1V >− ln Si ( t ) Si (t )1−α
Calibration issues
1491
539
36
70%
937
443
52
100%
167
167
91
With respect to correlation Gaussian copula
21
Model dependence / choice of copula ρ θ ρ62 ρ122
ρ ρ ρ ρ ρ
0% 0
10% 0.05
30% 0.18 14%
50% 0.36 39%
70% 0.66 63%
100% ∞ 100%
22%
45%
67%
100%
22
Model dependence / choice of copula ρ 0% 10% 30% 50% 70% Gaussian 560 633 612 539 443 Clayton 560 637 628 560 464 Student (6) 637 550 447 Student (12) 621 543 445 t(4)-t(4) 560 527 435 369 313 t(5)-t(4) 560 545 454 385 323 t(4)-t(5) 560 538 451 385 326 t(3)-t(4) 560 495 397 339 316 t(4)-t(3) 560 508 406 342 291 MO 560 284 144 125 134 Table 6: mezzanine tranche (bps pa)
0% 12% 34% 55% 73% 100% 0% 13% 36% 56% 74% 100% 0% 12% 34% 54% 73% 100% 0% 10% 32% 53% 75% 100% 0% 11% 33% 54% 73% 100% α 0 28% 53% 69% 80% 100% Table 5: correspondence between parameters
t(4)-t(4) t(5)-t(4) t(4)-t(5) t(3)-t(4) t(4)-t(3)
23
100% 167 167 167 167 167 167 167 167 167 167
24
4
Model dependence / choice of copula ρ Gaussian Clayton Student (6) Student (12) t(4)-t(4) t(5)-t(4) t(4)-t(5) t(3)-t(4) t(4)-t(3) MO
0% 0.03 0.03
10% 4.6 4.0
30% 50% 70% 20 36 52 18 33 50 17 34 51 19 35 52 0.03 11 30 45 60 0.03 10 29 45 59 0.03 10 29 44 59 0.03 12 32 47 71 0.03 12 32 47 61 0.03 25 49 62 73 Table 7: senior tranche (bps pa)
Model dependence / choice of copula ρ 0% 10% 30% 50% 70% 100% Gaussian 0% 0% 0% 0% 0% 100% Clayton 0% 0% 2% 15% 35% 100% Student (6) 5% 12% 25% 100% Student (12) 1% 4% 13% 100% t(4)-t(4) 0% 0% 1% 10% 48% 100% t(5)-t(4) 0% 0% 0% 0% 0% 100% t(4)-t(5) 0% 100% 100% 100% 100% 100% t(3)-t(4) 0% 100% 100% 100% 100% 100% t(4)-t(3) 0% 0% 0% 0% 0% 100% MO 0% 28% 53% 69% 80% 100% Table 8: coefficient of lower tail dependence (%)
100% 91 91 91 91 91 91 91 91 91 91
Gaussian, Clayton and Student t CDO premiums are close
Tail dependence is poorly related to CDO tranche premiums
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26
Model dependence / choice of copula
Model dependence / choice of copula ρ Gaussian Clayton Student (6) Student (12) MO
0% 0% 0%
10% 1% 3%
30% 6% 8% 9% 14% 0% 16% 36% Table 9: Kendall’s
50% 16% 15% 25% 30% 53% τ (%)
70% 33% 25% 44% 47% 67%
ρ2
100% 100% 100% 100% 100% 100%
0%
10%
30%
50%
70%
100%
Gaussian 0.66% 0.91% 1.54% 2.41% 3.59% 8.1% Clayton 0.66% 0.88% 1.45% 2.24% 3.31% 8.1% Student (6) 1.41% 2.31% 3.52% 8.1% Student (12) 1.49% 2.36% 3.56% 8.1% t(4)-t(4) 0.66% 1.22% 2.38% 3.49% 4.67% 8.1% t(5)-t(4) 0.66% 1.16% 2.27% 3.38% 4.57% 8.1% t(4)-t(5) 0.66% 1.18% 2.28% 3.37% 4.54% 8.1% t(3)-t(4) 0.66% 1.34% 2.57% 3.69% 5.02% 8.1% t(4)-t(3) 0.66% 1.31% 2.55% 3.70% 4.87% 8.1% MO 0.66% 2.63% 4.53% 5.65% 6.53% 8.1% Table 13: bivariate default probabilities (5 year time horizon)
Kendall’s tau is poorly related to CDO tranche premiums
Bivariate default probabilities are well related to tranche premiums 27
28
Model dependence / choice of copula
Distribution of conditional default probabilities
Why Clayton and Gaussian copulas provide same SL premiums?
Loss distributions only depend on the distribution of conditional default probabilities iV t
p
(
= exp V (1 − Fi (t )
−θ
))
iV t
p
⎛ − ρV + Φ −1 ( Fi (t ) ) ⎞ = Φ⎜ ⎟ ⎜ ⎟ 1− ρ 2 ⎝ ⎠
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5 1
0,9
0,9
0,85 0,8
0,8
0,75 0,7
0,7
0,65
Distribution functions of conditional default probabilities 1
0 1 0,95
0,6
0,6
0,55
0,5 1
0,5
0,95 0,9
0,9
0,45
0,85 0,8
0,5
0,4
0,8
0,4
Clayton Gaussian MO independence comonotonic stoch.
0,75 0,7
0,35
0,7
0,65 0,6
0,6
0,55 0,5
0,5
0,45 0,4
0,4
0,3 Clayton Gaussian MO independence comonotonic stoch.
0,2
0,2
0,15
0,35 0,3
0,3
0,25
0,3
0,1
0,2
0,05
0,1
0,25 0,2 0,15 0,1
0,1 0,05 0 0,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
0,40
0,45
0 0,50
29
0 0,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
0,40
0,45
0 0,50
30
5
Matching the correlation skew Tranches Market [0-3%] 916 [3-6%] 101 [6-9%] 33 [9-12%] 16 [12-22%] 9
Matching the correlation skew
Gaussian Clayton Student (12) t(4)-t(4) Stoch. MO 916 916 916 916 916 916 163 163 164 82 122 14 48 47 47 34 53 11 17 16 15 22 29 11 3 2 2 13 8 11
implied compound correlation 40% 35% 30%
M ar ket
Table 17: CDO tranche premiums iTraxx (bps pa)
Gaussi an
25%
doubl e t 4/ 4 20%
Tranches Market [0-3%] 916 [0-6%] 466 [0-9%] 311 [0-12%] 233 [0-22%] 128
Gaussian Clayton Student (12) t(4)-t(4) Stoch. MO 916 916 916 916 916 916 503 504 504 456 479 418 339 339 340 305 327 272 253 253 254 230 248 203 135 135 135 128 135 113
cl ayton exponenti al
15%
t-Student 12 10%
Stoch.
5% 0% 0-3
Table 18: “equity tranche” CDO tranche premiums iTraxx (bps pa) 31
3-6
6-9
9-12
12-22
32
Conclusion
Matching the skew with second generation models
Pricing bespoke portfolios, CDO squared with a consistent model Not yet fully satisfactory
RFL, stochastic correlation, double t Conditional default probability distributions are the drivers
Matching 5Y/10Y with same set of parameters Stability of parameters through time Dynamics of the correlation skew / risk management Calibration of multiple parameters, possibly name dependent
Still more work on the quant agenda. 33
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