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1
Credit Default Swap with Nonlinear Dependence
Chih-Yung Lin
Shwu-Jane Shieh
2006.12.14
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Abstract
The first-to-default Credit Default Swap (CDS) with three assets is priced when the default barrier is changing over time, which is contrast to the assumption in most of the structural-form models.
We calibrate the nonlinear dependence structure of the joint survival function of these assets by applying the elliptical and the Archimedean copula functions.
3
Abstract
The empirical evidences support that the Archimedean copula functions are the best-fitting model to describe the nonlinear asymmetric dependence among the assets.
We investigate the effects and sensitivities of parameters to the survival probability and the par spread of CDS with three correlated assets by a simulation analysis.
4
Abstract
The simulation analysis demonstrates that the joint survival probability increases as these assets are highly positive correlated.
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Outline
1. Introduction
2. Methodology
3. Empirical and Simulation Results
4. Conclusions
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1. Introduction
A. Two Core Issues B. First-to-Default Credit Default Swap
(CDS)
C. Elliptical Copulas and Archimedean
Copulas
D. Related Literatures
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Two core issues
a. structural-form model :
how to use credit default model in the pricing of first-to-default CDS in the structural-form model .
b. copula function :
how to use copula function in the pricing of first-to-default CDS consisting of three assets.
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Structural-form model :
This part is the extension of “the model of Finkelstein, Lardy, Pan, Ta, and Tierney (2002)”.
That discusses the pricing of one firm’s CDS in the structural-form model.
In this paper, we discuss with the pricing of CDS in two companies, three companies and n companies.
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First-to-Default Credit Default Swap
the buyer : may experience a credit default event in the portfolio of
assets which may cause their loss uncertainly.
the seller: receive the payments that the buyers pay every season
or every year before the credit incident happened.
When the first credit incident happened, the sellers have to pay the loss to the buyer.
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Elliptical Copulas and Archimedean Copulas
Elliptical Copulas
a. Gaussian copulas
b. t copulas Archimedean Copulas
a. Clayton copulas
b. Frank copulas
c. Gumble copulas
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Related Literatures
Credit default swap (CDS) Finkelstein, Lardy, Pan, Ta, and Tierney (2002)
apply the credit risk model of KMV to calculate the prices of CDS.
This is the first paper that finds out the closed form result of the pricing of CDS in literature.
The default barrier in their model is variable, but the default barrier of KMV is regular.
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Related Literatures
Li (2000) has a main contribution in discussing with how to use copula function in the default correlation and first-to-default CDS valuation.
Junk, Szimayer, and Wagner (2006) where they calibrate the nonlinear dependence in term structure by applying a class of copula functions and show that one of the Archimedean copula function is the best-fitting model.
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2. Methodology
A. Valuing Credit Default Swap with single firm
B. Definition, basic properties about Copula
C. Solving survival function with copula function
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Valuing Credit Default Swap with single firm
a. Firm’s assets : We assume that the asset's value is a stochastic
process V and V is evolving as a Geometric Brownian Motion (GBM).
……….. (1)
where W is a standard Brownian motion, σ is the asset's standard deviation, and μ is the asset value average. We also assume that 0=μ.
dtdWV
dVt
t
t
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Valuing Credit Default Swap with single firm
b. Recovery rate : We assume that the recovery rate L follows a lognormal
distribution with mean and percentage standard deviation λ as follows:
…….…….. (2)
…….…….. (3)
…….…….. (4)
)(LEL
)(L
DLLD
2/2 Ze
L
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Valuing Credit Default Swap with single firm
Where L is the global average recovery on the debt and D is the firm’s debt-per-share.
The random variable Z is independent of the Brownian motion W, and Z is a standard normal random variable.
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Valuing Credit Default Swap with single firm
c. Stopping time
The survival probability of company at time t can be defined as the probability of the asset value (1) does not reach the default barrier (4) at time t.
……….. (5) DLeV tWt
2/0
2
2/2 Ze
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Valuing Credit Default Swap with single firm
We can do some conversions in the formula (5). First, we introduce a process Xt in the formula (6); then we convert formula (5) to formula (7).
……….. (6)
……….. (7)
2/2/ 22 tZwX tt
)/log( 20
VDLX t
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Valuing Credit Default Swap with single firm
d. Survival function
Now we want to find the first hitting time of Brownian motion in the formula (7). For the process Yt = at + b Wt with constant a and b, we have (see, for example, Musiela and Rutkowski (1998))
……….. (8) )()(,
2/2
tb
yate
tb
yattsyYP bay
s
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Valuing Credit Default Swap with single firm
Apply to Xt we can obtain a closed form formula for the survival probability up to time t.
……….. (9)d
A
dAtS
t
t ))log(
2()( )
)log(
2(
t
t
A
dA
LD
eVd
2
0
222 tAt
Where
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Valuing Credit Default Swap with single firm
e. The pricing of Credit Default Swap
do some transfer in the calculating of the asset standard deviation
…….. (10)d
A
dAtS
t
t ))log(
2()( )
)log(
2(
t
t
A
dA
Where 20 e
LD
DLSd
22*
**2 )(
t
LDS
SA St
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Valuing Credit Default Swap with single firm
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Valuing Credit Default Swap with single firm
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Valuing Credit Default Swap with single firm
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Gaussian copula function
when n=k, we can get the Gaussian copula function as follows ：
))(),...,(),(()( 12
11
12kR
kR uuuuC
)( )(
11
2
1
2
11 1
...}2
1exp{
||)2(
1...
u u
kT
k
k
dxdxXRX
R
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Gaussian copula function
when n=3, we can calculate the joint survival function of three companies as follows ：
……… (17)
)1,1()1,1(2 31132112321 uuCuuCuuu
),,( 321 uuuS
)1,1,1()1,1( 3211233223 uuuCuuC +
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t-copula function:
Y is the random variance of
distribution
……… (18)
2
))(),...,((),...,( 11
1,1, nvvvRnvR ututtuuT
ii XY
vu Where
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t-copula function:
when n=k, we can get t-copula function as follows:
))(),...,((),...,( 11
1,1, kvvvRk
kvR ututtuuT
)( )(
12
)(1
2
2
1
11 1
...)1
1(
))(2
(
||)2
(...
ut ut
k
kvT
n
v kv
dxdxXRXv
vv
Rnv
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t-copula function
when n=3 ：
… (20)
),,( 321 uuuS
)1,1()1,1(2 31,21,321 uutuutuuu vRvR
)1,1,1()1,1( 321,32, uuutuut vRvR
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Clayton copula
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Clayton copula function
when n=3 ：
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Frank copula
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Frank copula function
when n=3 ：
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Gumbel Copula
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Gumbel Copula function
when n=3 ：
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3. Empirical and Simulation Results
A. Data
B. Empirical Results
C. Simulation Results
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Data ( 2004/3/14 to 2006/3/08)
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Data ( basic data)
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Empirical Results
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Empirical Results
Moreover we can find there is steady positive correlation between T and c*. The empirical results support that the dependence among these assets are asymmetric and can be better described by the Archimedean copula functions.
The results obtained here are consistent with those documented by Junker, Szimayer, and Wagner (2006) where they calibrate the nonlinear dependence in term structure by applying a class of copula functions and show that one of the Archimedean copula function is the best-fitting model.
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Simulation Results
In this part, we want to find the relationship between model parameters and the survival probability (S). On the other hand, we also discuss with the relationship between model parameters and the par spread (c*).
So we use AT&T Inc. and Microsoft Corp. to be our reference companies. Then we use the basic stock price on 2006/3/08 to estimate the par spread of CDS in T=480.
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The effect of the correlation (ρ)
In our intuition, the lower correlation leads to the higher survival probability(S). But we find some interesting evidences in the simulation results. We can find that the survival probability is always the highest when the correlation is 0.9. But we can not get a steady result in the comparing between ρ=0.001 and ρ=-0.9.
Thus we can find that the difference of the survival probability between ρ=0.001 and ρ=-0.9 is always very small. In addition, we can find that the positive correlation among assets has a positive effect to the survival probability.
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4. Concluding Remarks
We derive the pricing of Credit Default Swap (CDS) in three companies under the situation where the default barrier is variable, the survival probability is lognormal distributed and the default-free interest rate is constant.
The joint survival probability of three companies is derived by employing the Elliptical and Archimedean copula functions and we can price the CDS with three assets.
In addition, we also investigate how the model parameters affect the survival probability of the three companies by a simulation analysis.
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