Credit Default Swap with
Nonlinear Dependence
Chih-Yung Lin
Shwu-Jane Shieh
2006.12.14
0
Abstract
1

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.
Abstract
2

The empirical evidences support that the
Archimedean copula functions are the bestfitting 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.
Abstract

3
The simulation analysis demonstrates that
the joint survival probability increases as
these assets are highly positive correlated.
Outline
4

1. Introduction

2. Methodology

3. Empirical and Simulation Results

4. Conclusions
1. Introduction

A. Two Core Issues

B. First-to-Default Credit Default Swap
(CDS)

C. Elliptical Copulas and Archimedean
Copulas

5
D. Related Literatures
Two core issues
6

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.
Structural-form model :
7

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.
First-to-Default Credit Default Swap




8

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.
Elliptical Copulas and Archimedean
Copulas


9
Elliptical Copulas
a. Gaussian copulas
b. t copulas
Archimedean Copulas
a. Clayton copulas
b. Frank copulas
c. Gumble copulas
Related Literatures
10

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.
Related Literatures
11

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.
2. Methodology
12

A. Valuing Credit Default Swap with single
firm

B. Definition, basic properties about Copula

C. Solving survival function with copula
function
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).
dVt
   dWt    dt ……….. (1)
Vt

13
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.
Valuing Credit Default Swap
with single firm

b. Recovery rate :

We assume that the recovery rate L follows a
lognormal distribution with mean L and percentage
standard deviation λ as follows:


L  E (L)
…….…….. (2)
   (L)
…….…….. (3)

14
LD  L D  e
Z 2 / 2
…….…….. (4)
Valuing Credit Default Swap
with single firm
15

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.
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.
V0 e
16
Wt  t / 2
2

 L D e
Z 2 / 2
……….. (5)
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).
X t  wt  Z   2 t / 2  2 / 2
……….. (6)

X t  log( L D / V0  2 )
17
……….. (7)
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))
PYs  y, s  t  (
18
at  y
b t
)e
2 ay / b 2
(
at  y
b t
)
……….. (8)
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.
At log( d )
A log( d )
)
S (t )  ( 
)  d (  t 
……….. (9)
2
At
2
At
Where
d 
2
V0 e
LD
At2   2 t  2
19
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
At log( d )
At log( d )

(


)
S (t )  ( 
)d
2
At
2
At

Where
d 
S*
2
2
A  (
)
t


S *  LD
2
t
20
S 0  L D 2
e
LD
*
S
…….. (10)
Valuing Credit Default Swap
with single firm
21
Valuing Credit Default Swap
with single firm
22
Valuing Credit Default Swap
with single firm
23
Gaussian copula function
when n=k, we can get the Gaussian copula
function as follows：

1
1
1
C (u )   ( (u1 ),  (u 2 ),...,  (u k ))
k
R

2
R
 1 ( u1 )


24
 1 ( uk )
...


1
k
2
(2 ) | R |
1
2
1
exp{ X T R 1 X } dx1...dxk
2
Gaussian copula function

when n=3, we can calculate the joint survival
function of three companies as follows：
S (u1 , u 2 , u3 )
 u1  u2  u3  2  C12 (1  u1 ,1  u2 )  C13 (1  u1 ,1  u3 )
+
25
C23 (1  u 2 ,1  u3 )  C123 (1  u1 ,1  u 2 ,1  u3 )
……… (17)
t-copula function:

Y is the random variance of
distribution

2
TR,v (u1 ,..., un )  t R,v (tv1 (u1 ),..., tv1 (un ))
……… (18)
Where
26
v
ui 
Xi
Y
t-copula function:
when n=k, we can get t-copula function as
follows:

1
v
1
v
T (u1 ,..., uk )  t R,v (t (u1 ),..., t (uk ))
k
R,v

vn
(
)| R| 2
( vk )

1
T
1
2
2
(
1

X
R
X
)
dx1...dxk
n
v
v
2
( )(v )
2
1

tv1 ( u1 )


27
tv1 ( uk )
...


t-copula function

when n=3 ：
S (u1 , u 2 , u3 )
 u1  u2  u3  2  t R ,v (1  u1 ,1  u2 )  t R ,v (1  u1 ,1  u3 )
 t R ,v (1  u2 ,1  u3 )  t R ,v (1  u1 ,1  u2 ,1  u3 )
… (20)
28
Clayton copula
29
Clayton copula function
when
30
n=3 ：
Frank copula
31
Frank copula function
when
32
n=3 ：
Gumbel Copula
33
Gumbel Copula function
when
34
n=3 ：
3. Empirical and Simulation
Results
35

A. Data

B. Empirical Results

C. Simulation Results
Data
( 2004/3/14 to 2006/3/08)
36
Data
( basic data)
37
Empirical Results
38
39
40
41
Empirical Results
42

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.
Simulation Results
43

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.
The effect of the correlation (ρ)
44

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.
45
4. Concluding Remarks



46
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.