Copulas: A Tool For Modelling Dependence In Finance

Transcription

Copulas: A Tool For Modelling
Dependence In Finance
Statistical Methods in Integrated Risk Management
Frontières en Finance, Paris, 01/26/2001
Thierry Roncalli
Groupe de Recherche Opérationnelle
Crédit Lyonnais
Joint work with a lot of people (see ref. further)
The Working Paper “Copulas For Finance” is available on the web
site: http://www.gloriamundi.org/var/wps.html
1
Introduction
Definition 1 A copula function C is a multivariate uniform
distribution (a multivariate distribution with uniform margins).
Theorem 1 Let F1, . . . , FN be N univariate distributions. It comes
that C (F1 (x1) , . . . , Fn (xn) , . . . , FN (xN )) defines a multivariate
distributions F with margins F1, . . . , FN (because the integral
transforms are uniform distributions).
⇒ F belongs to the Fréchet class F (F1, . . . , FN ) — F is a
distribution with given marginals.
⇒ Copulas are also a general tool to construct multivariate
distributions, and so multivariate statistical models — see for
example Song [2000].
Copulas in Finance
Introduction
1-1
2
The dependence function
• Canonical representation
• Concordance order
• Measure of dependence
From 1958 to 1976, virtually all the results concerning
copulas were obtained in connection with the study and
development of the theory of probabilistic metric spaces
(Schweizer [1991]).
⇒ Schweizer and Wolff [1976] = connection with rank statistics (see
also Deheuvels [1979b]).
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2.1 Canonical representation
Theorem 2 (Sklar’s theorem) Let F be a N -dimensional
distribution function with continuous margins F1, . . . , FN . Then F
has a unique copula representation
F (x1, . . . , xN ) = C (F1 (x1) , . . . , FN (xN ))
⇒ Copulas are also a powerful tool, because the modelling problem
could be decomposed into two steps:
• Identification of the marginal distributions;
• Defining the appropriate copula function.
In terms of the density, we have the following canonical
representation f (x1, . . . , xN ) = c (F1 (x1) , . . . , FN (xN )) ×
Copulas in Finance
N
Y
fn (xn).
n=1
2-2
The copula function of random variables (X1, . . . , XN ) is invariant
under strictly increasing transformations (∂xhn (x) > 0):
CX1,... ,XN = Ch1(X1),... ,hN (XN )
... the copula is invariant while the margins may be changed
at will, it follows that is precisely the copula which captures
those properties of the joint distribution which are invariant
under a.s. strickly increasing transformations (Schweizer and
Wolff [1981]).
⇒ Copula = dependence function of random variables.
This property was already etablished by Deheuvels [1978,1979a].
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2.2 Examples
For the Normal copula, We have

C (u1, . . . , uN ; ρ) = Φρ Φ−1 (u1) , . . . , Φ−1 (uN )
and

1 > −1
c (u 1 , . . . , u N ; ρ) =
ρ −I ς
1 exp − ς
2
|ρ| 2
1

For the Gumbel copula, We have

C (u1, u2) = exp − (− ln u1)δ + (− ln u2)δ
1
δ
!
Other copulas: Archimedean, Plackett, Frank, Student, Clayton, etc.
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2.3 Concordance order
The copula C1 is smaller than the copula C2 (C1 ≺ C2) if
∀ (u 1 , . . . , u N ) ∈ I N ,
C1 (u1, . . . , uN ) ≤ C2 (u1, . . . , uN )
⇒ The lower and upper Fréchet bounds C− and C+ are

C− (u1, . . . , uN ) = max 
N
X
n=1

un − N + 1, 0
C+ (u1, . . . , uN ) = min (u1, . . . , uN )
We can show that the following order holds for any copula C:
C− ≺ C ≺ C+
⇒ The minimal and maximal distributions of the Fréchet class
F (F1, F2) are then C− (F1 (x1) , F2 (x2)) and C+ (F1 (x1) , F2 (x2)).
Example of the bivariate Normal copula (C⊥ (u1, u2) = u1u2):
C− = C−1 ≺ Cρ<0 ≺ C0 = C⊥ ≺ Cρ>0 ≺ C1 = C+
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Mikusiński, Sherwood and Taylor [1991] give the following
interpretation of the three copulas C−, C⊥ and C+:
• Two random variables X1 and X2 are countermonotonic — or
C = C− — if there exists a r.v. X such that X1 = f1 (X ) and
X2 = f2 (X ) with f1 non-increasing and f2 non-decreasing;
• Two random variables X1 and X2 are independent if the
dependence structure is the product copula C⊥;
• Two random variables X1 and X2 are comonotonic — or
C = C+ — if there exists a random variable X such that
X1 = f1 (X ) and X2 = f2 (X ) where the functions f1 and f2 are
non-decreasing;
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2.4 Measures of association or dependence
If κ is a measure of concordance, it satisfies the properties:
−1 ≤ κC ≤ 1; C1 ≺ C2 ⇒ κC1 ≤ κC2 ; etc.
Schweizer and Wolff [1981] show that Kendall’s tau and Spearman’s
rho can be (re)formulated in terms of copulas
τ = 4
ZZ
% = 12
2
ZZI
C (u1, u2) dC (u1, u2) − 1
I2
u1u2 dC (u1, u2) − 3
⇒ The linear (or Pearson) correlation is not a measure of
dependence.
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2.5 Some misinterpretations of the correlation
The following statements are false:
1. X1 and X2 are independent if and only if ρ (X1, X2) = 0;
2. For given margins, the permissible range of ρ (X1, X2) is [−1, 1];
3. ρ (X1, X2) = 0 means that there are no relationship between X1
and X2.
• We consider the cubic copula of Durrleman, Nikeghbali and
Roncalli [2000b]
C (u1, u2) = u1u2 + α [u1(u1 − 1)(2u1 − 1)] [u2(u2 − 1)(2u2 − 1)]
with α ∈ [−1, 2]. If the margins F1 and F2 are continous and
symmetric, the Pearson correlation is zero. Moreover, if α 6= 0,
the random variables X1 and X2 are not independent.
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• Wang [1997] shows that the min. and max. correlations of
X1 ∼ LN (µ1, σ1) and X2 ∼ LN (µ2, σ2) are
e−σ1σ2 − 1
ρ− =
1
1 ≤ 0
2
2
2
2
eσ2 − 1
eσ1 − 1
eσ1σ2 − 1
ρ+ =
1
1 ≥ 0
2
2
σ22
σ12
e −1
e −1
ρ− and ρ+ are not necessarily equal to −1 and 1. Example with
σ1 = 1 and σ2 = 3:
Copula
C−
ρ = −0.7
C⊥
ρ = 0.7
C+
Copulas in Finance
ρ (X1, X2)
−0.008
'0
0
' 0.10
0.16
τ (X1, X2)
−1
−0.49
0
0.49
1
% ( X1 , X 2 )
−1
−0.68
0
0.68
1
2-9
• Using an idea of Ferguson [1994], Nelsen [1999] defines the
following copula

 u

 1
C (u1, u2) = 12 u2


 u +u −1
1
2
1u ≤ 1
0 ≤ u1 ≤ 2
2
2
1
0 ≤ 2 u 2 ≤ u1 ≤ 1 − 1
2 u2
1 ≤ 1 − 1u ≤ u ≤ 1
1
2
2 2
We have cov (U1, U2) = 0, but Pr {U2 = 1 − |2U1 − 1|} = 1, i.e.
“the two random variables can be uncorrelated although one can
be predicted perfectly from the other ”.
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3
Understanding the dependence: three examples
• Mispecifications of marginals and dependence
• Dependence in multi-assets options
• Temporal dependence in Markov processes
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3.1 Mispecifications of marginals and dependence
Example of Costinot, Roncalli and Teı̈letche [2000]
• Normal copula + Gaussian marginals = Gaussian distribution.

It means that

1 0.158 0.175


ρ̂ = 
1
0.0589 
1
C(CAC40,NIKKEI) C(CAC40,DowJones) C(NIKKEI,DowJones)
• Normal copula + Empirical marginals.


1 0.207 0.157


ρ̂ = 
1
0.0962 
1
In this case, we have
C(CAC40,DowJones) C(CAC40,NIKKEI) C(NIKKEI,DowJones)
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3-2
3.2 Dependence in multi-assets options
Ref.: Bikos [2000], Cherubini and Luciano [2000], Durrleman [2001],
Rosenberg [2000].
Vanilla options contain information on the future distribution of
S (T ). This information (RND) is actually used in monetary policy
(see BIS [1999]) (option prices = “forward-looking” indicators).
Bikos [2000]: options on multi-assets contain information on the
future distribution of S (T ) =

S 1 (T ) · · · S N ( T )
>
.
Let Qn and Q be the risk-neutral probability distributions of Sn (T )
and S (T ). With arbitrage theory, we can show that
Q (+∞, . . . , +∞, Sn (T ) , +∞, . . . , +∞) = Qn (Sn (T ))
⇒ The margins of Q are the RND Qn of Vanilla options.
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How to build ‘forward-looking” indicators for the dependence ?
1. estimate the univariate RND Q̂n using Vanilla options;
2. estimate the copula Ĉ using multi-assets options by imposing
that Qn = Q̂n;
3. derive “forward-looking” indicators directly from Ĉ.
Breeden et Litzenberger [1978] remark that European option prices
permit to caracterize the probability distribution of Sn (T )
∂C (T, K )
∂K
= Pr {Sn (T ) ≤ K}
φ (T, K ) := 1 + er(T −t0)
= Q n (K )
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3-4
Durrleman [2000] extends this result in the bivariate case:
1. for a call max option, φ (T, K ) is the diagonal section of the
copula
φ (T, K ) = C (Q1 (K ) , Q2 (K ))
2. for a spread option, we have
φ (T, K ) =
Z +∞
0
∂1C (Q1 (x) , Q2 (x + K )) dQ1 (x)
⇒ Other results are derived in Durrleman [2001] (bounds, general
kernel pricing, etc.)
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Computation of the implied parameter ρ̂ in a spread option:
• BS model: LN distribution calibrated with ATM options; Kernel
pricing = LN distributions + Normal copula
ρ̂1 = −0.341
• Bahra model: mixture of LN distributions calibrated with eight
European prices; Kernel pricing = MLN distributions + Normal
copula
ρ̂2 = 0.767
Remark 1 ρ̂1 and ρ̂2 are parameters of the Normal Copula. ρ̂1 is a
Pearson correlation, not ρ̂2.
⇒ BS model: negative dependence / Bahra model: positive
dependence.
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3.3 Temporal dependence in Markov processes
• Markov operators and 2-copulas
• Markov operators and stochastic kernels
• Markov processes and ∗ product of 2-copulas
• The Brownian copula
• Understanding the temporal dependence structure of diffusion
processes
• New interpretation of properties of Markov processes
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3-7
3.3.1 Markov operators and 2-copulas
Definition 2 (Brown [1966, p. 15]) Let (Ω, F , P ) be a
probabilistic space. A linear operator T : L∞ (Ω) → L∞ (Ω) is a
Markov operator if
(a) T is positive i.e. T [f ] ≥ 0 whenever f ≥ 0;
(b) 1 is a fixed point of T.
(c) E [T [f ]] = E [f ] for every f ∈ L∞ (Ω).
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The relationship between Markov operator T and 2-copula functions
C is given by the following two lemmas:
Lemma 1 (Olsen, Darsow and Nguyen [1996, lemma 2.1, p. 249])
For a copula C, the operator defined by
Z
d 1
T [f ] (x) =
∂2C (x, y ) f (y ) dy
dx 0
is a Markov operator on L∞ ([0, 1]).
Lemma 2 (Olsen, Darsow and Nguyen [1996, lemma 2.2, p. 251])
Let T be a Markov operator on L∞ ([0, 1]). The two place function
defined by
C (x, y ) =
is a 2-copula.
Z x
0
Copulas in Finance
h
i
T 1[0,y] (s) ds
3-9
The multiplication product of copulas have been defined by Darsow,
Nguyen and Olsen [1992] in the following manner
I2 −→ I
R1
(x, y ) 7−→ (C1 ∗ C2) (x, y ) = 0 ∂2C1 (x, s) ∂1C2 (s, y ) ds
The transposition of copula corresponds to the mapping function
C> (x, y ) = C (y, x).
The adjoint T† of the Markov operator T is the operator such that
we verify that (Brown [1966])
Z
[0,1]
f1 (x) T [f2] (x) m (dx) =
Z
[0,1]
f2 (x) T† [f1] (x) m (dx)
In terms of copulas, we have
d
T† [f ] (x) =
dy
Z 1
Copulas in Finance
0
∂1C (x, y ) f (x) dx
3-10
3.3.2 Markov operators and stochastic kernels
Using the Kantorovitch-Vulich-Ryff representation, Foguel [1969]
shows that T hK1i ◦ T hK2i is a Markov operator with kernel
K = K1 ∗ K2 defined by
K (x, y ) := (K1 ∗ K2) (x, y ) =
Z
[0,1]
K1 (x, s) K2 (s, y ) ds
Let P be the stochastic transition function of a Markov process, we
can deduce that
D
E
E
D
D
E
D
T Ps,t = T Ps,θ ∗ Pθ,t = T Ps,θ ◦ T Pθ,t
with

Ps,t (x, A) := Ps,θ ∗ Pθ,t (x, A) =
Z
E
Ps,θ (x, dy ) Pθ,t (y, A)
Ω
= P ? (x, A)
⇒ this is the Chapman-Kolmogorov equation.
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3.3.3 Markov processes and ∗ product of 2-copulas
Darsow, Nguyen and Olsen [1992] prove the following theorem:
Theorem 3 Let X = {Xt, Ft; t ≥ 0} be a stochastic process and let
Cs,t denote the copula of the random variables Xs and Xt. Then the
following are equivalent
(i) The transition probabilities Ps,t (x, A) = Pr {Xt ∈ A | Xs = x}
satisfy the Chapman-Kolmogorov equations
Ps,t (x, A) =
Z ∞
−∞
Ps,θ (x, dy ) Pθ,t (y, A)
for all s < θ < t and almost all x ∈ R.
(ii) For all s < θ < t,
Cs,t = Cs,θ ∗ Cθ,t
Copulas in Finance
(1)
3-12
In the conventional approach, one specifies a Markov process
by giving the initial distribution µ and a family of transition
probabilities Ps,t (x, A) satisfying the Chapman-Kolmogorov
equations. In our approach, one specifies a Markov process by
giving all of the marginal distributions and a family of
2-copulas satisfying (1). Ours is accordingly an alternative
approach to the study of Markov processes which is different
in principle from the conventional one. Holding the transition
probabilities of a Markov process fixed and varying the initial
distribution necessarily varies all of the marginal distributions,
but holding the copulas of the process fixed and varying the
initial distribution does not affect any other marginal
distribution (Darsow, Nguyen and Olsen [1992]).
The Brownian copula
Cs,t (u1, u2) =
Z u
1
0
Φ
√
Copulas in Finance
√ −1
−1
tΦ (u2) − sΦ (u)
√
t−s
!
du
3-13
Understanding the dependence structure of diffusion processes
The copula of a Geometric Brownian motion is the Brownian copula.
The Ornstein-Uhlenbeck copula is
Cs,t (u1, u2) =
Z u
1
0
Φ
~ (t0, s, t) Φ−1 (u2) − ~ (t0, s, s) Φ−1 (u)
with
~ (t0, s, t) =
~ (s, s, t)
q
!
du
e2a(t−s) − e−2a(s−t0)
Remark 2 A new interpretation of the parameter a follows. For
physicists, a is the mean-reverting coefficient. From a copula point of
view, this parameter measures the dependence between the random
variables of the diffusion process. The bigger this parameter, the less
dependent the random variables.
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3-14
New interpretation of properties of Markov processes
Theorem 4 (Darsow, Nguyen and Olsen [1992, theorem 5.1, p. 622])
The set C is a symmetric Markov algebra under ∗ and > as previously
defined. The unit and null elements are C⊥ and C+.
• Cst is a left invertible copula ⇔ the markov process is
deterministic.
• Cst is idempotent ⇔ the Markov process is conditionally
independent.
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3-15
4
An open field for risk management
• Economic capital adequacy
• Market risk
• Credit risk
• Operational risk
⇒ see BDNRR [2000] for a more detailed presentation (in particular
for stress-testing, multivariate extreme value theory and operational
risk) and DNRa [2000] for the problem of quantile aggregation.
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4-1
4.1 Economic capital adequacy
With copulas, it appears that the risk can be splitted into two parts:
the individual risks and the dependence structure between them.
• Coherent multivariate statistical model = Coherent model
for individual risks + coherent dependence function
• Coherent model for individual risks = taking into account
fat-tailed distributions, etc.
• coherent dependence function = understanding the
aggregation of quantiles of the individual risks.
Copulas in Finance
4-2
⇒ The influence of margins
Rating
α
Return time
Φ−1 (α)
Φ−1(0.99)
t−1
4 (α)
t−1
4 (0.99)
VaR
99%
100 days
BBB
99.75%
400 days
A
99.9%
4 years
AA
99.95%
8 years
AAA
99.97%
13 years
1
1.20
1.33
1.41
1.48
1
1.49
1.91
2.30
2.62
⇒ The influence of the dependence function: If a bivariate copula C
is such that∗
C̄ (u, u)
=λ
lim
u→1 1 − u
exists, then C has upper tail dependence for λ ∈ (0, 1] and no upper
tail dependence for λ = 0.
∗ C̄
is the joint survival function, that is
C̄ (u1, u2) = 1 − u1 − u2 + C (u1, u2)
Copulas in Finance
4-3
Remark 3 The measure λ is the probability that one variable is
extreme given that the other is extreme.
⇒ Coles, Currie and Tawn [1999] define the quantile-dependent
measure of dependence as follows
C̄ (u, u)
λ (u) = Pr {U1 > u|U2 > u} =
1−u
1. Normal copula ⇒ extremes are asymptotically independent for
ρ 6= 1, i.e λ = 0 for ρ < 1.
2. Student copula ⇒ extremes are asymptotically dependent for
ρ 6= −1.
Copulas in Finance
4-4
4.2 Market risk
Copulas = a powerful tool for market risk measurement.
Copulas have been already incorporated in some software solutions:
• SAS Risk Dimensions
• Palisade @Risk
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4-5
LME example:
P1
P2
P3
AL
1
-1
2
AL-15
1
-1
1
CU
1
-1
-3
NI
1
1
4
PB
1
1
5
• Gaussian margins and Normal copula
P1
P2
P3
90%
7.26
4.04
13.90
95%
9.33
5.17
17.82
99%
13.14
7.32
25.14
99.5%
14.55
8.09
27.83
99.9%
17.45
9.81
33.43
• Student margins (ν = 4) and Normal copula
P1
P2
P3
90%
9.20
5.33
18.04
Copulas in Finance
95%
12.48
7.08
24.11
99%
20.16
11.16
38.90
99.5%
23.95
13.17
46.45
99.9%
34.07
19.17
69.51
4-6
• Gaussian margins and Student copula (ν = 1)
P1
P2
P3
90%
6.49
3.45
11.99
95%
8.94
5.08
17.53
99%
14.48
9.17
31.88
99.5%
16.67
11.03
37.94
99.9%
21.11
15.77
51.66
Value-at-risk based on Student margins and a Normal copula (Gauss
software, Pentium III 550 Mhz, 100000 simulations)
Number of assets
2
10
100
500
1000
Copulas in Finance
Computational time
0.1 sc
24.5 sc
4 mn 7 sc
33 mn 22 sc
1 hr 44 mn 45 sc
4-7
5
Modelling credit risk
Some articles are available, see for example Coutant, Martineu,
Messines, Riboulet and Roncalli [2001], Li [2000], Lindskog [2000],
Lindskog and McNeil [2000] and Nyfeler [2000].
Other articles are still in progress ⇒ credit risk = one of the most
important topic for financial applications of copulas.
• Dependence in credit risk models
• The credit migration approach (CreditMetrics)
• The actuarial approach (CreditRisk+)
• The survival approach
• Pricing of credit derivatives
Copulas in Finance
5-1
5.1 Dependence in credit risk models
Portfolio with liquids credits 6= Portfolio with not liquids credits.
⇒ downgrading risk =
6 default risk.
What is the influence of introducing a dependence function?
1. impact on the joint migration probability distribution;
2. impact on the joint survival distribution.
Copulas in Finance
5-2
5.1.1 The credit migration approach (CreditMetrics)
Notations
N = number of credits in the portfolio.

πi,j = rating transition matrix from one state to another.
S = set of the eight states {AAA, AA, A, BBB, BB, B, CCC, D}.
S ? = {1, . . . , 8} with 1 → D, 2 → CCC, 3 → B, etc. (8 → AAA).
πi = distribution of the initial rating i (Ri is the corresponding
random variable).
πi = discrete probability distribution defined by {j, πi (j )} with
Pj
πi (j ) = Pr {Ri ≤ j} = k=1 πi,k and j ∈ S ?.
P (i1, . . . , iN ; j1, . . . , jN ) = joint migration probability distribution
n
P (i1, . . . , iN ; j1, . . . , jN ) = Pr Ri1 ≤ j1, . . . , RiN ≤ jN
o
p (i1, . . . , iN ; j1, . . . , jN ) = joint migration probability mass function
n
p (i1, . . . , iN ; j1, . . . , jN ) = Pr Ri1 = j1, . . . , RiN = jN
Copulas in Finance
o
5-3
The bivariate migration probability distribution
CreditMetrics uses a gaussian random variable Zi

(i)
πi (j ) = Pr Zi ≤ zj

Gupton, Finger and Bhatia [1997] define then the joint migration
probability distribution by
P (i1, i2; j1, j2) = Pr

(i )
(i )
Zi1 ≤ zj 1 , Zi2 ≤ zj 2
2
1

(i ) (i )
= Φ zj 1 , zj 2 ; ρ
1
2

where ρ = ρ (C1, C2) is the ‘asset return correlation’ of C1 and C2.
Remark that πi (Ri) = Φ (Zi). It comes that

−1
−1
P (i1, i2; j1, j2) = Φ Φ
πi1 (j1) , Φ
πi2 (j2) ; ρ
We can then write P (i1, i2; j1, j2) as a function of the bivariate
Normal copula C

P (i1, i2; j1, j2) = C πi1 (j1) , πi2 (j2) ; ρ
Copulas in Finance

5-4
The multivariate migration probability distribution
Let C be a copula function. We have

P (i1, . . . , iN ; j1, . . . , jN ) = C πi1 (j1) , . . . , πiN (jN )
The probability mass function p (i1, . . . , iN ; j1, . . . , jN ) is given by the
Radon-Nikodym density of the copula∗
1
X
k1 =0
···
1
X
kN =0
(−1)k1+···+kN

C πi1 (j1 − k1) , . . . , πiN (jN − kN )

Because a copula is a grounded function, we deduce that the joint
default probability is

p (i1, . . . , iN ; D, . . . , D) = C πi1 (1) , . . . , πiN (1)
It comes that

−
C πi1 (1) , . . . , πiN (1) ≤ p (i1, . . . , iN ; D, . . . , D) ≤
∗ We

+
C πi1 (1) , . . . , πiN (1)
use the convention (−1)0 = 1.
Copulas in Finance
5-5
Some illustrations
• migration probabilities p (i1, i2; j1, j2)
• joint default probability p (i1, i2; D, D)
• discrete default correlation
p (i1, i2; D, D) − πi1,1πi2,1
ρD (C 1 , C 2 ) = r

πi1,1 1 − πi1,1 πi2,1 1 − πi2,1
Remark 4 The discrete default correlation is not a good
measure of the dependence between defaults. We note that
+
−1 < ρ−
C
,
C
≤
ρ
C
,
C
≤
ρ
(
(
)
)
1
2
1
2
D
D
D (C 1 , C 2 ) < 1
Even if the dependence is maximal (C = C+), ρD (C1, C2) can be very
small (less than 10%).
⇒ see CMMRR [2001] for the link between ρD (C1, C2) and a 2 × 2
contingency table.
Copulas in Finance
5-6
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Computing the risk of a portfolio
Credit risk measure can be obtained by Monte Carlo methods. To
simulate the random variables Ri1 , . . . , RiN , we consider the following
two step algorithm:
1. we simulate the random numbers u1, . . . , uN from the copula C;
2. the random variates r1, . . . , rN are obtained by the inverse
distribution function method:
(−1)
rn = πi
(un)
⇒ Credit risk measure = credit value-at-risk with two components:
downgrading risk and default risk.
Some problems = credit modelling + recovery rates modelling.
Copulas in Finance
5-7
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5.1.2 The actuarial approach (CreditRisk+)
More results are in CMMRR [2001]. We just give here some ideas
about the copula used by CreditRisk+.
Notations
T = time horizon.
Bn = Bernoulli random variable which indicates if the credit n has
defaulted.
pn = stochastic parameter of Bn with
pn = Pn
X
θn,mXm
Pn is estimated for each obligator via any credit rating system. Xm
are M independent Gamma distributed factors.
Given those factors, the Bn are assumed to be conditionally
independent.
Copulas in Finance
5-8
One common risk factor
Let us denote X ∼ Γ (α, β ), g (x) the pdf of X, F (t1, . . . , tN ) the joint
distribution of defaults, Fn (tn) the marginal distributions and Fxn (tn)
the conditional distribution. We have
F (t1, . . . , tN ) =
It comes that

Z ∞ Y
N
0
x t g x dx
Fn
( n) ( )
n=1

1
1
F (t1, t2) = 1 +
F1 (t1) F2 (t2) − (F1 (t1) + F2 (t2) − 1)+
α
α
The copula is then

1
1
u1u2 − (u1 + u2 − 1)+
C (u1, u2; α) = 1 +
α
α
Remark 5 The previous analysis is not exact. Only the subcopula
can be defined for the four continuity points (see CMMRR [2001] for
more details).
Copulas in Finance
5-9
Approximation case
We suppose that Pn are small. In this case, we use a Poisson
approximation
1 − Pn
X
X
= exp −Pn
αβ
αβ
We can show that

F (t1, . . . , tN ) = φ 
N
X
n=1
with
F0n (tn) =
(

Pn
exp − αβ
1

!

0 t 
− ln Fn
( n)
if 0 ≤ tn < 1
if tn > 1
φ is the Laplace transform associated with Γ (α, β ). By remarking that
Pn
0
− ln Fn (tn) =
' φ−1 (Fn (tn))
αβ
Copulas in Finance
5-10
we have

F (t1, . . . , tN ) = φ 
=

N
X
n=1

φ−1 (Fn (tn))
1
−α
F 1 (t 1 )
+ . . . + FN
(2)
1
−α
(tN )
−N +1
−α
The dependence function is then the Cook-Johnson copula. Note
that equation (2) corresponds to Archimedean copulas. Moreover,
the generator of the Archimedean copulas is the inverse of a Laplace
transform. In this case, we can give a new probabilistic
interpretation, because this model is a frailty model∗ (see CMMRR
[2001] for a discussion on the dependence in CreditRisk+).
⇒ Extensions when the distribution of X is not Gamma can be found
in CMMRR [2001].
∗ This
result comes from the fact that the random variables Bn are assumed to be
conditionally independent.
Copulas in Finance
5-11
5.1.3 The survival approach
see Coutant, Martineu, Messines, Riboulet and Roncalli [2001].
Copulas in Finance
5-12
5.2 Pricing of credit derivatives
A default is generally described by a survival function
S (t) = Pr {T > t}. Let C̆ be a survival copula. A multivariate survival
distributions S can be defined as follows
S (t1, . . . , tN ) = C̆ (S1 (t1) , . . . , SN (tN ))
where (S1, . . . , SN ) are the marginal survival functions. Nelsen [1999]
notices that “C̆ couples the joint survival function to its univariate
margins in a manner completely analogous to the way in which a
copula connects the joint distribution function to its margins”.
⇒ Introducing correlation between defaultable securities can then be
done using the copula framework (see Li [2000] and Maccarinelli and
Maggiolini [2000]).
Copulas in Finance
5-13
5.2.1 First-to-Default valuation
Let us define the first-to-default τ as follows
τ = min (T1, . . . , TN )
Nelsen [1999] shows that the survival function of τ is given by the
diagonal section of the survival copula.
Let C be a copula. Its survival copula is given by the following
formula
C̆ (u1, . . . , uN ) = C̄ (1 − u1, . . . , 1 − un, . . . , 1 − uN )
with


N
X
X

n
C (u)
C̄ (u1, . . . , un, . . . , uN ) =
(−1)

n=0
u∈Z(N −n,N )
o
n
PN
N
where Z (M, N ) denotes the set u ∈ [0, 1] | n=1 X{1} (un) = M .
Copulas in Finance
5-14
When the copula is radially symmetric, we have
C̆ = C
The survival distribution S of τ is
S (t) = C (S1 (t) , . . . , SN (t))
It comes that the density of τ is given by
f (t) = −∂tS (t)
=
N
X
n=1
Copulas in Finance
∂nC (S1 (t) , . . . , SN (t)) × fn (t)
5-15
5.2.2 Example
N credit events, default of each credit event given by a Weibull
survival function (the baseline hazard is constant and equal to 3%
per year and the Weibull parameter is 2).
C is a Normal copula of dimension N = very tractable (N can be
very large) and ∂nC is a Normal copula of dimension N − 1.
Copulas in Finance
5-16
5.2.3 Other credit derivatives
see Coutant, Martineu, Messines, Riboulet and Roncalli [2001].
Copulas in Finance
5-17
6
Conclusion
The study of copulas and the role they play in probability,
statistics, and stochastic processes is a subject still in its
infancy. There are many open problems and much work to be
done (Nelsen [1999], page 4).
In finance, the use of copulas is very recent (Embrechts, McNeil and
Straumann [1999]). In one year, great progress have been made.
Nevertheless, the finance industry needs more works on copulas and
their applications. And there are many research directions to explore.
Moreover, many pedagogical works have to be done in order to
familiarize the finance industry with copulas.
Copulas in Finance
Conclusion
6-1
7
References
[1] Beneš, V. and J. Štĕpán [1997], Distributions with Given Marginals and Moment Problems,
Kluwer Academic Publishers, Dordrecht
[2] Bikos, A. [2000], Bivariate FX PDFs: A Sterling ERI Application, Bank of England, Working
Paper
[3] BIS [1999], Estimating and Interpreting Probability Density Functions, Proceedings of the
workshop held at the BIS on 14 June 1999
[BDNRR] Bouyé, E., V. Durrleman, A. Nikeghbali, G. Riboulet and T. Roncalli [2000], Copulas for
finance — A reading guide and some applications, Groupe de Recherche Opérationnelle, Crédit
Lyonnais, Working Paper
[4] Bouyé, E., Durrleman, V., A. Nikeghbali, G. Riboulet and T. Roncalli [2000], Copulas: an
open field for risk management, Groupe de Recherche Opérationnelle, Crédit Lyonnais,
Working Paper
[5] Breeden, D. and R. Litzenberger [1978], State contingent prices implicit in option prices,
Journal of Business, 51, 621-651
[6] Brown, J.R. [1966], Approximation theorems for markov operators, Pacific Journal of
Mathematics, 16, 13-23
[7] Cherubini, U. and E. Luciano [2000], Multivariate option pricing with copulas, University of
Turin, Working Paper
Copulas in Finance
7-1
References
[8] Coles, S., J. Currie and J. Tawn [1999], Dependence measures for extreme value analyses,
Department of Mathematics and Statistics, Lancaster University, Working Paper
[9] Costinot, A., T. Roncalli and J. Teiletche [2000], Revisiting the dependence between financial
markets with copulas, Groupe de Recherche Opérationnelle, Crédit Lyonnais, Working Paper
[CMMRR] Coutant, S., P. Martineu, J. Messines, G. Riboulet and T. Roncalli [2001], Revisiting the
dependence in credit risk models, Groupe de Recherche Opérationnelle, Crédit Lyonnais,
Working Paper, available soon
[10] Dall’Aglio, G., S. Kotz and G. Salinetti [1991], Advances in Probability Distributions with
Given Marginals (Beyond the Copulas), Kluwer Academic Publishers, Dordrecht
[11] Darsow, W.F., B. Nguyen and E.T. Olsen [1992], Copulas and markov processes, Illinois
Journal of Mathematics, 36-4, 600-642
[12] Deheuvels, P. [1978], Caractérisation complète des lois extrêmes multivariées et de la
convergence des types extrêmes, Publications de l’Institut de Statistique de l’Université de
Paris, 23, 1-36
[13] Deheuvels, P. [1979a], Propriétes d’existence et propriétes topologiques des fonctions de
dependance avec applications à la convergence des types pour des lois multivariées,
Comptes-Rendus de l’Académie des Sciences de Paris, Série 1, 288, 145-148
[14] Deheuvels, P. [1979b], La fonction de dépendance empirique et ses propriétés — Un test non
paramétrique d’indépendance, Académie Royale de Belgique – Bulletin de la Classe des
Sciences – 5e Série, 65, 274-292
[15] Durrleman, V. [2001], Implied correlation, Princeton University, report
[16] Durrleman, V., A. Nikeghbali and T. Roncalli [2000], A simple transformation of copulas,
Groupe de Recherche Opérationnelle, Crédit Lyonnais, Working Paper
[17] Durrleman, V., A. Nikeghbali and T. Roncalli [2000], Which copula is the right one?, Groupe
de Recherche Opérationnelle, Crédit Lyonnais, Working Paper
[18] Durrleman, V., A. Nikeghbali and T. Roncalli [2000], Copulas approximation and new families,
Groupe de Recherche Opérationnelle, Crédit Lyonnais, Working Paper
[DNRa] Durrleman, V., A. Nikeghbali, and T. Roncalli [2000], How to get bounds for distribution
convolutions? A simulation study and an application to risk management, Groupe de
Recherche Opérationnelle, Crédit Lyonnais, Working Paper
[DNRb] Durrleman, V., A. Nikeghbali and T. Roncalli [2000], A note about the conjecture on
Spearman’s rho and Kendall’s tau, Groupe de Recherche Opérationnelle, Crédit Lyonnais,
Working Paper
[19] Embrechts, P., A.J. McNeil and D. Straumann [1999], Correlation and dependency in risk
management : properties and pitfalls, Departement of Mathematik, ETHZ, Zürich, Working
Paper
[20] Genest, C. and J. MacKay [1986], The joy of copulas: Bivariate distributions with uniform
marginals, American Statistician, 40, 280-283
[21] Gupton, G.M., C.C. Finger and M. Bhatia [1997], CreditMetrics — Technical Document,
Morgan Guaranty Trust Co.
[22] Joe, H. [1997], Multivariate Models and Dependence Concepts, Monographs on Statistics and
Applied Probability, 73, Chapmann & Hall, London
[23] Li, D.X. [2000], On default correlation: a copula function approach, Journal of Fixed Income,
March, 43-54
[24] Lindskog, F. [2000], Modelling dependence with copulas and applications to risk management,
RiskLab Research Paper
[25] Lindskog, F. and A.J. McNeil [2000], Common poisson shock models, RiskLab Research Paper
[26] Maccarinelli, M. and V. Maggiolini [2000], The envolving practice of credit risk management in
global financial institutions, Risk Conference, 26/27 September, Paris
[27] Mikusiński, P., H. Sherwood and M.D. Taylor [1991], Probabilistic interpretations of copulas, in
G. Dall’Aglio, S. Kotz and G. Salinetti (Eds.), Advances in Probability Distributions with Given
Marginals (Beyond the Copulas), Kluwer Academic Publishers, Dordrecht
[28] Nelsen, R.B. [1999], An Introduction to Copulas, Lectures Notes in Statistics, 139, Springer
Verlag, New York
[29] Nyfeler, F. [2000], Modelling dependencies in credit risk management, RiskLab Research Paper
[30] Olsen, E.T., W.F. Darsow and B. Nguyen [1996], Copulas and Markov operators, in L.
Rüschendorf, B. Schweizer and M.D. Taylor (Eds.), Distributions with Fixed Marginals and
Related Topics, Institute of Mathematical Statistics, Hayward, CA
[31] Rosenberg, J.V. [2000], Nonparametric pricing of multivariate contingent claims, Stern School
of Business, Working Paper
[32] Rüschendorf, L., B. Schweizer and M.D. Taylor [1993], Distributions with Fixed Marginals and
Related Topics, Institute of Mathematical Statistics, Hayward, CA
[33] Schweizer, B. [1991], Thirty years of copulas, in G. Dall’Aglio, S. Kotz and G. Salinetti (Eds.),
Advances in Probability Distributions with Given Marginals (Beyond the Copulas), Kluwer
Academic Publishers, Dordrecht
[34] Schweizer, B. and E. Wolff [1976], Sur une mesure de dépendance pour les variables
aléatoires, Comptes Rendus de l’Académie des Sciences de Paris, 283, 659-661
[35] Schweizer, B. and E. Wolff [1981], On nonparametric measures of dependence for random
variables, Annals of Statistics, 9, 879-885
[36] Sempi, C. [2000], Conditional expectations and idempotent copulæ, to appear
[37] Sklar, A. [1959], Fonctions de repartition à n dimensions et leurs marges, Publications de
l’Institut de Statistique de l’Université de Paris, 8, 229-231
[38] Song, P. [2000], Multivariate dispersion models generated from Gaussian copula, forthcoming
in Scandinavian Journal of Statistics
[39] Wang, S.S. [1997], Aggregation of correlated risk portfolios: models & algorithms, CAS
Committee on Theory of Risk, preprint

Copulas: A Tool For Modelling Dependence In Finance

Transcription

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