analysis of copula functions and applications to credit risk management · 2005-09-26 · u2...
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Analysis of Copula Functions and
Applications to Credit Risk
Management
Philipp KoziolMichael Kunisch
Financial Modelling Workshop Ulm, 2005
Philipp Koziol WHU - Otto Beisheim School of Management
Analysis of Copula Functions and Applications to Credit Risk Management
Motivation
Modelling dependent defaults is crucial in credit risk
Copula Functions are a very useful toolto model joint default distributions
But:
• Impacts of copulas on credit derivative prices?
• Which copula should be used?
Literature:
• Standard: Gauss Copula
• Li (2000) and Schonbucher/Schubert (2001):
=⇒ No Analysis of copulas
Goal: Influence of copulas on creditderivative prices
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Analysis of Copula Functions and Applications to Credit Risk Management
Idea of Copulas
joint distribution function:F (x1, . . . , xn)
↙ ↘
dependence betweenthe random variables:
C(u1, . . . , un)
MarginalDistributions:
F1(x1), ..., Fn(xn)
↘ ↙
F (x1, . . . , xn) = C(F1(x1), . . . , Fn(xn))
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Analysis of Copula Functions and Applications to Credit Risk Management
Basics in Copulas and Dependence (1)
• Global Dependence Measure:
◦ Rank Correlation: Kendalls tau
+ capture nonlinearities+ independent of marginal distr.+ only dependent of copula parameters
=⇒ sensitivities of τ
• Local Dependence Measure:
– Tail Dependence:
−4 −2 0 2 4−4
−3
−2
−1
0
1
2
3
4
u1
u2
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Analysis of Copula Functions and Applications to Credit Risk Management
Basics in Copulas and Dependence (2)
Important Examples of Copulas
•Elliptical C.: Combination of elliptical distr.
−4 −2 0 2 4−4
−3
−2
−1
0
1
2
3
4
u1
u2
(a) Gauss Copula
−4 −2 0 2 4−4
−3
−2
−1
0
1
2
3
4
u1
u2
(b) t4-Copula
=⇒ Symmetrical Structure
•Archimedean C.: artificially generated by ϕ(t)
−4 −2 0 2 4−4
−3
−2
−1
0
1
2
3
4
u1
u2
(c) Gumbel C.
−4 −2 0 2 4−4
−3
−2
−1
0
1
2
3
4
u1
u2
(d) Clayton C.
−4 −2 0 2 4−4
−3
−2
−1
0
1
2
3
4
u1
u2
(e) Frank C.
−4 −2 0 2 4−4
−3
−2
−1
0
1
2
3
4
u1
u2
(f) Nelsen C.
=⇒ ”Non” Symmetrical Structure
•Farlie-Gumbel-Morgenstern C.: easy expression
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Analysis of Copula Functions and Applications to Credit Risk Management
Models
1. Li Copula Model:
• Marginal default distribution Fi(ti) = P(τi ≤ ti)=⇒ Poisson or Cox Processes
• Dependence Structure through a Copula Function:
P(τ1 ≤ t1, ..., τn ≤ tn) = CLi(F1(t1), ..., Fn(tn))
• Valuation: Monte-Carlo Simulation of (τ1, . . . , τn)
2. Schonbucher/Schubert Model:
• Extension of the Li Model:
=⇒ Dynamics of the default intensities
• Simulation of the default times is exactly the same
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Analysis of Copula Functions and Applications to Credit Risk Management
Data
Goal: Comparison of credit derivative prices=⇒ Influence of copulas
• Credit Portfolio: n = 2 identical credits
=⇒ 2 Turkey Zerobonds (B1 Rating), Maturity=8.6a
• Marginal distribution: Cox Process
dλi(t) = µi · λi(t) · dt+ σi · λi(t) · dWi(t), i = 1, 2,
• where µ1 = µ2 = 0,
• σ1 = σ2 ≈ 0.11,
• E(dW1(t) · dW2(t)) = 0.
• Default Dependency: Copula Functions
• Simulation of default times (τ1, τ2)
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Analysis of Copula Functions and Applications to Credit Risk Management
Applications in Credit Risk Manag. (1)
1. Valuation of nth-to-Default Swaps:
• First-to-Default Swap FtD(t,T):
FtD(t, T ) = E
(
e−∫ min(τ1,τ2)t rsds · 1{min(τ1,τ2)≤T}
∣
∣ Ht
)
=⇒ considered time horizon: T = 1year at t = 0
Distribution of τmin = min{τ1, τ2} crucial
Strong dependence in [0, T ] =⇒ Low FtD premia
⇓
Simulation Resultsin the Schonbucher/Schubert model
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Analysis of Copula Functions and Applications to Credit Risk Management
Archimedean Copulas
• Archimedean copulas have variable price behavior
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Analysis of Copula Functions and Applications to Credit Risk Management
Elliptical Copulas
• Elliptical copulas have equal price characteristics
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Analysis of Copula Functions and Applications to Credit Risk Management
Numbers
Fix Kendall’s tau
Kendall’s tau ∆FtD=max-min % of upper bound
0.2 0.0208 11.2%
0.5 0.034 20%
Fix FtD premium
Fix FtD premium τ -range ∆τ
0.18 [0.08; 0.4] 0.32
0.17 [0.13; 0.61] 0.48
⇓
Choice of the copula is important for FtD premia
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Analysis of Copula Functions and Applications to Credit Risk Management
Results for the FtD Swap
• Tail Dependence is crucial for FtD premia
• Archimedean copulas are more flexible
• t-copula differs strongly for low dependencies
⇓
Choice of the copula is important for FtD premia
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Analysis of Copula Functions and Applications to Credit Risk Management
Applications in Credit Risk Manag. (2)
2. Valuation of defaultable Zerobonds:
B1(t, T ) = E
(
e−∫ Tt rsds · 1{τ1>T}
∣
∣ Ht
)
• Defaultable Zerobond at t = 1• maturity T = 8.6 years• firm 2 has already defaulted
Distribution of τ1 important
⇓
Simulation Resultsin the Schonbucher/Schubert model
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Analysis of Copula Functions and Applications to Credit Risk Management
Archimedean Copulas
• Archimedean copulas have variable price behavior
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Analysis of Copula Functions and Applications to Credit Risk Management
Elliptical Copulas
• Elliptical copulas have equal price characteristics
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Analysis of Copula Functions and Applications to Credit Risk Management
Numbers
Fix Kendall’s tau
Kendall’s tau ∆CS=max-min % of upper bound
0.1 503bp 41.9%
0.4 1300bp 44.8%
Fix Credit Spread
Fix credit spread τ -range ∆τ
1250bp [0.075; 0.28] 0.205
1600bp [0.18; 0.4] 0.22
⇓
Choice of the copula is important for credit spreads
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Analysis of Copula Functions and Applications to Credit Risk Management
Results for defaultable Zerobonds
• probability mass in the center is crucial
=⇒ Tail Dependence is less important
• copulas without T. D. =⇒ highest spreads
• Archimedean copulas are more flexible
• t-copula differs strongly for low dependencies
Choice of the copula is important for credit spreads
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Analysis of Copula Functions and Applications to Credit Risk Management
Conclusion
• Choice of the copula has a high impact on prices
• Copula behavior varies for different creditderivatives
• Simultaneous Consideration of global and localDependencies is necessary
• Reduction of model risk by identification ofnecessary copula properties
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