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0 x 2ˇx 2ˇ fI G .xI ˛; ˇ/ D ˆ ˆ : 0 ; if x 0: The corresponding distribution function is: 8 Z x .˛ ˇz/2 ˛ ˆ 3=2 ˆ
0 2ˇz 2ˇ 0 FI G .xI ˛; ˇ/ D ˆ ˆ : 0 ; if x 0: We write then X I G .˛; ˇ/. Lemma 5.1. The standardized central moments of an Inverse Gaussian distributed random variable X I G .˛; ˇ/ are: E.X / D
˛ ˇ
˛ V .X / D E .X E.X //2 D 2 ˇ 0 !3 1 3 X E.X / AD p S.X / D E @ p ˛ V .X /
5.1 The Main Properties of the Normal Inverse Gaussian Distribution
0 X E.X / p V .X /
K.X / D E @
!4 1 AD3C
131
15 : ˛
Proof. See [54, p. 262f] with D ˛=ˇ, and D ˛ 2 =ˇ.
Definition 5.2 (Normal Inverse Gaussian distribution). A random variable X follows a Normal Inverse Gaussian (NIG) distribution with parameters ˛, ˇ, and ı if: X j Y D y N . C ˇy; y/ Y I G .ı; 2 / with WD
p ˛2 ˇ2 ;
with parameters satisfying the following conditions: 0 jˇj < ˛ and ı > 0. We then write X N I G .˛; ˇ; ; ı/ and denote the density and probability functions by fN I G .xI ˛; ˇ; ; ı/ and FN I G .xI ˛; ˇ; ; ı/ correspondingly. The density of the NIG distribution is then: Z fN I G .xI ˛; ˇ; ; ı/ D
1 0
and the distribution function: Z FN I G .xI ˛; ˇ; ; ı/ D
x 1
fN .xI C ˇy; y/ fI G .yI ı; 2 /dy
Z
1
0
(5.1)
fN .tI C ˇy; y/ fI G .yI ı; 2 /dydt (5.2)
with fN .xI ; 2 / the density function of the Gaussian distribution: fN .xI ; 2 / D p
.x /2 exp : 2 2 2 2 1
Lemma 5.2. The density of a random variable X N I G .˛; ˇ; ; ı/ can be also written in the form: fN I G .xI ˛; ˇ; ; ı/ D
p ı˛ exp .ı C ˇ.x // K1 ˛ ı 2 C .x /2 : p ı 2 C .x /2
R1 Where K1 .w/ WD 12 0 exp 12 w.t C t 1 / dt is the modified Bessel function of p the third kind and WD ˛ 2 ˇ 2 .
132
5 Normal Inverse Gaussian Factor Copula Model
Proof. We start with plugging the expressions for the Gaussian and Inverse Gaussian densities into (5.1): fN I G .xI ˛; ˇ; ; ı/ Z 1 D fN .xI C ˇy; y/ fI G yI ı; 2 dy 0 Z 1 .x ˇy/2 1 ı p y 3=2 p exp .ı / D exp 2y 2y 2 0 1 ı2 dy exp C 2y 2 y Z 1 ı 1 .x /2 C ı 2 D y 2 exp exp .ı C ˇ.x // C ˛2 y dy: 2 2 y 0 With substitution y D
p
ı 2 C.x/2 ˛w
we continue:
fN I G .xI ˛; ˇ; ; ı/ Z 1 ˛ 2 w2 ı D exp .ı C ˇ.x // 2 ı 2 C .x /2 0 !! p p ı 2 C .x /2 ı 2 C .x /2 1 ˛w..x /2 C ı 2 / 2 exp p C˛ dw 2 ˛w ˛w2 ı 2 C .x /2 Z 1 ˛ ı p D exp .ı C ˇ.x // 2 2 ı C .x /2 0 1 p 2 1 2 exp ˛ ı C .x / .w C w / d w 2 p ı˛ exp .ı C ˇ.x // D K1 ˛ ı 2 C .x /2 : p ı 2 C .x /2
While the density function of the NIG distribution is quite complicated, its moment generating function has a simple form. Lemma 5.3. The moment generating function of a random variable X N I G .˛; ˇ; ; ı/ is p exp ı ˛ 2 ˇ2 p : MX .t/ D exp .t/ exp ı ˛2 .ˇ C t/2 Proof. Since for any density function f holds: Z
1
1
f .x/dx D 1;
5.1 The Main Properties of the Normal Inverse Gaussian Distribution
133
we have for the NIG density function: Z
p ı˛ exp .ı C ˇ.x // K1 ˛ ı 2 C .x /2 dx p ı 2 C .x /2 1 p Z 1 K1 ˛ ı 2 C .x /2 e ˇx ı˛ D p dx; exp .ı ˇ/ ı 2 C .x /2 1
1D
1
and thus we notice that, 0 @
Z
1
1
p 11 p K1 ˛ ı 2 C .x /2 e ˇx ı˛ p dx A D exp ı ˛ 2 ˇ 2 ˇ : ı 2 C .x /2
Now we have for the moment generating function h i MX .t/ D E e tX p Z 1 K1 ˛ ı 2 C .x /2 e .ˇ Ct /x p ı˛ D p dx exp ı ˛ 2 ˇ 2 ˇ ı 2 C .x /2 1 p p ı˛ exp ı ˛ 2 ˇ 2 ˇ exp ı ˛ 2 .ˇ C t/2 C .ˇ C t/ ı˛ p exp ı ˛2 ˇ 2 p : D exp .t/ exp ı ˛2 .ˇ C t/2
D
Lemma 5.4. The central moments (mean, variance, skewness and kurtosis) of a random variable X N I G .˛; ˇ; ; ı/ are: E.X / D C ı
ˇ
ˇ S.X / D 3 p ˛ ı Proof. Follows from Lemma 5.3.
V .X / D ı
˛2 3
2 ! 1 ˇ . K.X / D 3 C 3 1 C 4 ˛ ı
Figure 5.2 shows the NIG densities for different parameter sets. We change in the first three plots the value of only one of the parameters. In the last plot we show some more densities with more than one parameter changed.
134
5 Normal Inverse Gaussian Factor Copula Model 0.7
0.8 α=3, β=0,μ=0,δ=1 α=1, β=0,μ=0,δ=1 α=0.1,β=0,μ=0,δ=1
0.7 0.6
α=2,β= 0, μ=0,δ=1 α=2,β=−1.5,μ=0,δ=1 α=2,β= 1, μ=0,δ=1 α=2,β= 1.9,μ=0,δ=1
0.6 0.5
0.5 0.4
0.4 0.3
0.3 0.2
0.2
0.1
0.1 0 −4
−2
0
2
4
6
0 −4
−2
(a) ˛ variable
0
2
1 α=1,β=0,μ=0,δ=0.5 α=1,β=0,μ=0,δ=1 α=1,β=0,μ=0,δ=3
0.8
4
6
(b) ˇ variable α=1,β=0, μ=−3,δ=1 α=3,β=0, μ=−3,δ=3 α=1,β=0.9,μ=0,δ=1 α=2,β=1.9,μ=0, δ=1 α=3,β=2.9,μ=0, δ=1
0.5
0.4 0.6 0.3 0.4
0.2
0.2
0 −4
0.1
−2
0
2
4
6
0 −6
(c) ı variable
−4
−2
0
2
4
6
(d) Parameter combinations
Fig. 5.2 Densities of NIG distribution
Table 5.1 Changes in the central moments in dependence on the parameters ˛ ˇ % &
V & V %
K& K%
E% E&
V % V %
S% S&
K% K%
E% E&
ı V % V &
K& K%
We can see the influence of the parameter changes on the distribution density from Fig. 5.2. We can also derive the changes in the central moments of the distribution from Lemma 5.4. We summarize these dependencies in the following table. We change one of the parameters and give the corresponding change in the central moments (Table 5.1). Additionally, the distribution is symmetric only when ˇ D 0. So we have seen so far that the four parameter NIG family contains distributions with heavy tails and non-symmetric skewed distributions. Further very important properties of the NIG distribution are the scaling property and closure under convolution.
5.1 The Main Properties of the Normal Inverse Gaussian Distribution
135
Lemma 5.5 (Scaling and convolution properties of NIG). (i) For NIG distributed random variable X N I G .˛; ˇ; ; ı/ and a scalar c, cX is NIG distributed as well with parameters cX N I G
˛ ˇ ; ; c; cı : c c
(5.3)
(ii) For independent random variables X N I G .˛; ˇ; 1 ; ı1 / and Y N I G .˛; ˇ; 2 ; ı2 /, their sum is NIG distributed as well with parameters X C Y N I G .˛; ˇ; 1 C 2 ; ı1 C ı2 / :
(5.4)
Proof. (i) We consider the moment generating function for cX . From the properties of moment generating functions we have: McX .t/ D MX .ct/: So we get:
p exp ı ˛ 2 ˇ 2 p McX .t/ D exp .ct/ exp ı ˛ 2 .ˇ C ct/2 ! r ˛ 2 ˇ 2 exp cı c c D exp .ct/
q ˛ 2 ˇ 2 exp cı . c C t/ c
which is the moment generating function of N I G ˛c ; ˇc ; c; cı . (ii) From the properties of moment generating functions we have for independent X and Y : MXCY .t/ D MX .t/MY .t/: So for X N I G .˛; ˇ; 1 ; ı1 / and Y N I G .˛; ˇ; 2 ; ı2 / we get MXCY .t/
p p exp ı1 ˛ 2 ˇ 2 exp ı2 ˛ 2 ˇ 2 exp .2 t/ p p D exp .1 t/ exp ı1 ˛ 2 .ˇ C t/2 exp ı2 ˛ 2 .ˇ C t/2 p exp .ı1 C ı2 / ˛ 2 ˇ 2 D exp ..1 C 2 /t/ p exp .ı1 C ı2 / ˛ 2 .ˇ C t/2
which is the moment generating function of N I G .˛; ˇ; 1 C 2 ; ı1 C ı2 /.
136
5 Normal Inverse Gaussian Factor Copula Model
5.2 Efficient Implementation of the NIG Distribution The Normal Inverse Gaussian distribution usually does not belong to the package of standard distributions that are already implemented in programs like Matlab, S-Plus, R and Mathematica. Since the NIG distribution functions are quite complicated we would expect them to be computationally intensive if using the straight forward implementation. The work of Tempes [87] showed that this is indeed the case. Some alternative implementations in Matlab were developed and compared in [87] and in [55]. We use this toolbox in our empirical work on the NIG copula model for CDO pricing as well. Since the computational speed will become especially important to us in the later work on the asset allocation with CDOs, we spend some additional efforts to make this toolbox1 even more efficient. This section describes and compares the alternative implementation algorithms of the NIG distribution. We denote different implementations of the density, distribution and inverse distribution functions of NIG with Funktion NIG. The computation times of various algorithms will be compared by evaluating the functions of the NIG distribution with the standard parameter set .˛; ˇ; ; ı/ D .1; 0; 0; 1/ at m points between 5 and 5. We start with the implementation of the probability density function pdf. The naive approach would be to follow the definition of the NIG distribution and to use the form of the density function in (5.1): Z fN I G .xI ˛; ˇ; ; ı/ D
1 0
fN .xI C ˇy; y/ fI G .yI ı; 2 /dy:
We denote this implementation with f NIG and compute the integral with the Matlab function quad.2 We will use this function further as well to compute the integrals. If the function f NIG should be evaluated at many points together, i.e. at a vector x 2 Rm , a for-loop will be used to compute the integral m times. Evaluating f NIG at m D 10:000 points takes 23.64 s.3 This is more than 1.000 times slower than the Gaussian density function and 100 times slower than Studentt density function. The reason for this extremely long computation time is that the integral must be computed at each of the m points without using any of the already computed values. However one call of the Matlab build-in and highly optimized function quad takes only 0.0002 s. So we would not be able to save any time by using any self written integration routine. Instead, it is more efficient to use a different expression for the density function containing terms with functions that
1
The NIG toolbox can be downloaded from Matlab central file exchange. The function quad approximates the integral of a function on the interval [a, b] with the recursive adaptive Simpson-quadrature. 3 The computation times in this section are taken from [87] that contains no information on the used processor. However, not the absolute numbers but the comparison of different implementation methods are important for us. 2
5.2 Efficient Implementation of the NIG Distribution Table 5.2 CPU times of the NIG probability density functions in seconds
m 1 10 100 1.000 10.000 100.000 1.000.000
137 f NIG 0 0:02 0:21 1:73 17:34 311:47
g NIG 0 0 0 0.01 0.05 0.62 6.64
are already implemented in Matlab and avoiding the integration. The alternative expression for the NIG density function in (5.2) p ı˛ exp .ı C ˇ.x // p K1 ˛ ı 2 C .x /2 ı 2 C .x /2
gN I G .xI ˛; ˇ; ; ı/ D
contains the Bessel function instead of the integral. Although the Bessel function includes an integration itself, this is a build-in Matlab function4 and thus is significantly faster than performing an integration with quad. We denote the implementation of this expression for the NIG density function with g NIG. The computational times for both density functions are compared in Table 5.2. This comparison shows that the implementation g NIG is more than 500 times faster than f NIG. So we choose to use g NIG further on. Next we continue with the implementation of the probability function. There are two expressions for the probability function following from the two expressions for the density function. The first expression arises from the definition in (5.1): Z FN I G .x/ D
x
Z
1
1
0
fN .tI C ˇy; y/ fI G yI ı; 2 dydt:
As we have already learned from the implementation of the density function that explicit integration is very time consuming, so we can eliminate one integration by using the Gaussian probability function instead of the density: Z FN I G .x/ D
1 0
FN .xI C ˇy; y/ fI G yI ı; 2 dy:
The second representation of the NIG probability function follows from the alternative expression for the density function with the Bessel function in (5.2): Z GN I G .x/ D
4
x 1
p ı˛ exp .ı C ˇ.t // K1 ˛ ı 2 C .t /2 dt: (5.5) p ı 2 C .t /2
The Matlab Bessel function is besselk.
138
5 Normal Inverse Gaussian Factor Copula Model
Both expressions for the probability function FN I G and GN I G contain only one integration now. In both implementations, F NIG and G NIG, we compute the integrals with quad and use a for-loop to evaluate the functions at several points. Evaluating the functions at 10.000 points needs 29.41 s for the F NIG and 59.45 s for G NIG that is both very slow. The reason for this is that the integration is performed at every point new. While it is unavoidable for the function F NIG, it is possible to use the already computed values in G NIG. Therefore, we sort the input vector x 2 Rm and get the vector x 2 Rm such that: x1 x2 xm : So the new implementation H NIG of the probability function GN I G still uses a for-loop to evaluate the function at all input points but utilizes the previously computed values: H
NIG
.xi / D H
NIG
.xi1 / C
Z
xi
xi1
g
NIG
.t/dt
(5.6)
with x0 D 1 and H NIG .x0 / D 0. Evaluation of the H NIG at 10.000 points needs only 6.97 s (incl. the sorting of the points) that is an improvement of 70% in comparison to the first implementation (see Table 5.3). Now we work on the implementation of the NIG inverse distribution function. First, we need to show that the NIG distribution function is continuous and strictly monotone for all x 2 R and all feasible parameter ˛ > jˇj; ı > 0; 2 R. Note that x enters only the Gaussian distribution function which is continuous. Since multiplying a continuous function with a constant and integrating the expression gives a continuous result, the NIG distribution function is continuous as well. Further, FN is strictly monotone increasing function and fI G is independent from x. Thus the NIG distribution function is strictly monotone increasing as well. So there exists a unique inverse function: 1 FN I G .u/ D x ” FN I G .x/ D u
” FN I G .x/ u D 0: Table 5.3 CPU times of the NIG distribution function in seconds
m 1 10 100 1.000 10.000 100.000 1.000.000
F NIG 0 0:03 0:32 2:91 29:41 402:71
(5.7)
G NIG 0:01 0:07 0:63 6:01 59:45 599:89
H NIG 0 0:02 0:07 0:75 6:97 70:32 739:01
5.2 Efficient Implementation of the NIG Distribution
139
Equation (5.7) can be solved with the Matlab function fzero. Since we want to evaluate the inverse NIG function in a vector u 2 Œ0; 1m , we need to solve this equation in a for-loop for each uj to get the corresponding xj . So fzero must be called m times with a scalar. fzero evaluates the NIG distribution function FN I G in each iteration of its search. Thus, such an implementation of the NIG inverse function would be very time consuming because of the quite long computation times of the NIG distribution function. The following algorithm of evaluation of the NIG inverse function on a vector turns out to be much more efficient. Given a vector u 2 Œ0; 1m where the function must be evaluated, we first find an interval Œxmi n ; xmax such that all the points of the input vector are: FN I G .xmi n / uj FN I G .xmax /; for all j . Since the interval can be arbitrarily large and its bounds must not be very close to the minimal and the maximal uj , it is very fast to find an appropriate interval. Further, we choose N D 2s with s, e.g., 10–14, equidistant points ti , i D 1; : : : ; N on the interval Œxmi n ; xmax and evaluate the NIG density function on this vector: y D fN I G .t/ D .fN I G .ti //i D1;:::;N : We can compute the NIG distribution function in vector t very fast using the approximation: FN I G .ti / zi D FN I G .xmi n / C
i X kD1
yi
xmax xmi n : 2s
(5.8)
So we have created a table of the NIG distribution function using a very efficient implementation approximating FN I G . This table is then used to find an x with FN I G .x/ D u by using linear interpolation inside the grid for FN I G .t/, i.e. for u D zi C .1 /zi C1 , 1 i N 1, we find x D ti C .1 /ti C1 as the interpolated inverse of the NIG-distribution at point x. The approximation method of the computation of the NIG distribution function described in (5.8) can although be used for the evaluation of FN I G at a very large number of points. Given a sufficiently large vector x, this approximation is much more efficient than the implementation H NIG, and still very exact. We denote this new implementation with H Large NIG. Now only the routines for the generation of NIG distributed random numbers are missing. Clearly, the usual way of generating uniformly distributed random numbers and transforming them to NIG distributed random numbers via applying the inverse NIG distribution function would be very inefficient. Generating NIG distributed random numbers according to the Definition 5.2 via the Gaussian and Inverse Gaussian random numbers proved to be the most efficient way.
140
5 Normal Inverse Gaussian Factor Copula Model
To generate the Inverse Gaussian distributed random numbers, we use the following algorithm developed by [70]. Algorithm 5.2.1 (Simulation of IG random numbers) 1. Generate a realization of Y with Y 21 . 2. Calculate 1 0 s 1 ˛3 A ˇ2 @ Y1 D C 4 Y C Y C 4 2 Y ; ˛ 2˛ ˇ 0 Y2 D
2
1 ˇ C 4 @Y ˛ 2˛
1
s Y C4
˛3 ˇ2
Y A:
3. Generate a realization of U with U U .0; 1/. 4. Set 8 ˛ < Y1 if U ˛CˇY 1 XD : Y2 if U > ˛ : ˛CˇY1 The algorithm for generating NIG random numbers is defined in the following way: Algorithm 5.2.2 (Simulation of NIG random numbers) p 1. Generate random numbers Y1 IG.ı ˛ 2 ˇ 2 ; ˛2 ˇ 2 / with Algorithm 5.2.1. 2. Generate random number Y2p N .0; 1/. 3. Compute X D C ˇY1 C Y1 Y2 . Table 5.4 gives the CPU times of the chosen implementations of the NIG density function g NIG, two implementations of the distribution function H NIG and H Large NIG, inverse distribution function Finv NIG and the function generating NIG distributed random numbers. As before, the standard parameter set .˛; ˇ; ; ı/ D .1; 0; 0; 1/ is used for this comparison. The input vectors have values between 5 and 5 for the density and distribution functions and between 0 and 1 for the inverse distribution function. We compare the exact implementation of the distribution function H NIG with the alternative implementation H Large NIG that evaluates only the first integral exactly and approximates the next parts with the sum. For a sufficiently large vector, m > 100, the second approximative implementation is much more efficient. We will use both implementations of the distribution function depending on the size of the input vector.
5.3 One Factor NIG Copula Model
141
Table 5.4 CPU times of the NIG functions in seconds m g NIG H NIG H Large NIG 1 0 0 0.42 10 0 0.02 0.35 100 0 0.07 0.35 1.000 0.01 0.75 0.35 10.000 0.05 6.97 0.38 100.000 0.62 70.32 0.53 1.000.000 6.64 739.01 2.42
Finv NIG 0.7 1.28 1.86 2.04 2.17 2.55 4.83
Random numbers 0 0 0 0 0.02 0.17 1.25
5.3 One Factor NIG Copula Model Now we want to apply the NIG distribution to the one factor copula model of correlated defaults and to derive the semi-analytic pricing formulas for the large homogeneous portfolio under the NIG copula model. Since the convolution property of the NIG distribution in (5.4) does not hold for two arbitrary NIG random variables, we need to find the right parametrization of the factors M and Xi in the copula model so that Ai follows a NIG distribution as well. We start with M N I G .˛1 ; ˇ1 ; 1 ; ı1 / and Xi N I G .˛2 ; ˇ2 ; 2 ; ı2 /. Then applying the scaling property in (5.3) we get:
˛1 ˇ1 aM N I G ; ; a1 ; aı1 ; (5.9) a a p p p ˇ2 ˛2 1 a 2 Xi N I G p ; p ; 1 a2 2 ; 1 a2 ı2 : (5.10) 1 a2 1 a2 p Further, to be able to apply the convolution property to the expression aM C 1 a2 Xi , the two first parameters in (5.9) and (5.10) must be equal, i.e. ˛1 ˇ1 ˛2 ˇ2 ; : Dp Dp a 1 a2 a 1 a2 Since M is the common market factor, it should not depend on the portfolio correlation parameter a. So we set p
p 1 a2 1 a2 ˛1 D ˛; ˇ1 D ˇ; ˛2 D ˛; ˇ2 D ˇ: a a p Now the random variable Ai D aM C 1 a2 Xi is NIG distributed for any 1 , 2 , ı1 and ı2 . Its parameters are: Ai N I G
p p ˛ ˇ ; ; a1 C 1 a2 2 ; aı1 C 1 a2 ı2 : a a
142
5 Normal Inverse Gaussian Factor Copula Model
Next, we restrict the parameters further in order to standardize the distributions of the both factors, i.e. the third and the fourth parameters are chosen so that the distributions have zero mean and unit variance. Using Lemma 5.4, we thus get: p ˇ ˛2 D 0 and ı1 3 D 1 for M with D ˛ 2 ˇ 2 ; q ˛2 ˇ2 2 C ı 2 D 0 and ı2 23 D 1 for Xi with 2 D ˛22 ˇ22 : 2 2 1 C ı1
The solution of this system of equations is: ˇ 2 3 1 D 2 ; ı1 D 2 ; 2 D ˛ ˛ p with D ˛ 2 ˇ2 . Then, the distribution of Ai is: Ai N I G
p
1 a2 ˇ 2 ; ı2 D a ˛2
˛ ˇ 1 ˇ 2 1 3 ; ; ; a a a ˛2 a ˛2
p 1 a2 3 ; a ˛2
:
Note that it has zero mean and unit variance as well: ˇ
E.Ai / D
1 ˇ 2 1 3 a D 0; C a ˛2 a ˛ 2 a1
1 3 V .Ai / D a ˛2
˛2 a2 1 3 a3
D 1:
Now we can summarize the obtained results and define the one factor NIG copula model. Definition 5.3 (One factor NIG copula model). Consider a homogeneous portfolio of m credit instruments. The standardized asset return up to time t of the i th issuer in the portfolio, Ai .t/, is assumed to be of the form: Ai .t/ D aM.t/ C
p 1 a2 Xi .t/;
with independent random variables5
ˇ 2 3 To simplify notations we denote the distribution function FN I G xI s˛; sˇ; s ˛2 ; s ˛2 with FN I G .s/ .x/. So, for example, the distribution function of the factor M is FN I G .1/ .x/, of the p factor Xi it is F 1a 2 .x/, and of Ai it is FN I G . 1 / .x/.
5
N IG.
a
/
a
5.3 One Factor NIG Copula Model
M.t/ N I G Xi .t/ N I G where D
p
143
ˇ 2 3 (5.11) ; ˛2 ˛2 ! p p p p 1 a2 1 a2 1 a2 ˇ 2 1 a2 3 ; ˛; ˇ; ; a a a ˛2 a ˛2
˛; ˇ;
˛2 ˇ2 .
Under this copula model the variable Ai .t/ is mapped to default time ti of the i th issuer using a percentile-to-percentile transformation, as in Chap. 4. The issuer i defaults before time t if FN I G . 1 / .Ai .t// Q .t/ ;
(5.12)
1 .Q .t// DW C.t/; Ai .t/ FN IG.1/
(5.13)
a
or equivalently
a
where Q.t/ denotes the risk-neutral probability of the instruments to default before time t. Now we can formulate the main result for the semi-analytic CDO pricing under the one factor NIG copula model. Proposition 5.1. Consider an infinitely large homogeneous portfolio with the asset returns following the one factor NIG copula model in Definition 5.3. Then the distribution of the portfolio loss before recovery is given by 0 B F1 .t; x/ D 1 FN I G .1/ @
1 .Q .t// FN IG.1/ a
p
1 a2 F 1
N IG.
a
p
1a 2 a
/
.x/
1 C A;
(5.14) with x 2 Œ0; 1 the percentage portfolio loss and Q.t/ the risk-neutral default probability of each issuer in the portfolio. Proof. Due to the scaling and convolution properties of NIG distribution, the distribution of Ai is: Ai N I G
˛ ˇ 1 ˇ 2 1 3 ; ; ; a a a ˛2 a ˛2
:
The portfolio loss distribution in (5.14) follows immediately from the Theorem 4.1.
144
5 Normal Inverse Gaussian Factor Copula Model
5.4 CDO Valuation Using the One Factor NIG Model We use the expression in (4.8) together with F1 for the tranche expected loss,6 and rewrite it as 1 EL.K1 ;K2 / .t/ D K 2 K1
K Z2
.x K1 /dF1 .t; x/ C .1 F1 .t; K2 // :
(5.15)
K1
To compute the integral we need the density function of the portfolio loss: dF1 .t; x/ dx
f1 .t; x/ D
0
B D fN I G .1/ @ p
p D
1 a2 a
1 .Q .t// FN IG.1/
p
a
1 a2 F 1
p
N IG.
1a 2 a
/
.x/
a ! dF 1
p
1a 2 a
N IG.
dx 0 F 1
B fN I G .1/ @
1/ N IG.a
1 a2 a
/
N IG
p .Q.t // 1a2 F 1 N IG
p
1a 2 a
a
p
1a 2 a
C A
.x/
1
0 f
1
@F 1
N IG
p
1a 2 a
! .x/
1 C A :
.x/A
The integral K Z2
.x K1 /f1 .t; x/dx K1
has no analytical solution and has to be computed numerically. As we have already seen, the inverse distribution function of the NIG distribution is quite computationally intensive. Computing this integral numerically involves the evaluation of the inverse distribution function numerous times. However, it is very easy to avoid this by means of a variable change:
6
A non-zero recovery can be easily taken into account by the corresponding transformation of the attachment points as this was shown in Sect. 4.2.5.
5.4 CDO Valuation Using the One Factor NIG Model
y D F 1
p
1a 2 a
N IG
Then 0
dy D f
p
N IG
1a 2 a
145 .x/:
dx
@F 1
1: p
1a 2 a
N IG
.x/A
So we get F 1 N IG
K Z2
p
1a 2 a
! .K2 /
!
Z
.x K1 /f1 .t; x/dx D
F
N IG
F 1
K1
N IG
p
1a 2 a
fN I G .1/
! .K1 /
C.t/
p
1a 2 a
.y/
K1
!p p 1 a2 y 1 a2 dy: a a
This expression contains the inverse NIG distribution function only in the integration limits. The function under the integral contains only a NIG distribution function and a NIG density that are much faster to compute than the inverse distribution function. The important advantage of the NIG copula model is that the default thresholds are easy and fast to compute due to the convolution property of the NIG distribution. Before we consider the fitting ability of the NIG model in the next section, we want to investigate some of its properties and point out the main difference between the one factor Gaussian and NIG copula models. The Gaussian distribution is fully described by only two moments since the skewness of the Gaussian distribution is always zero and its kurtosis equals three. The NIG distribution is characterized by four moments. In the next proposition we calculate the moments of the NIG parametrization used in the NIG copula model. 2 3 ; s Proposition 5.2. A random variable X N I G s˛; sˇ; s ˇ has the 2 2 ˛ ˛ following central moments: E.X / D 0 ˇ S.X / D 3 2 s with D
p
V .X / D 1
2 ! 2 ˛ ˇ K.X / D 3 C 3 1 C 4 , ˛ s2 4
˛2 ˇ2 .
Proof. Straightforward from Lemma 5.4.
146
5 Normal Inverse Gaussian Factor Copula Model
10 α=0.5 α=1 α=2
8 6
15 β =−0.5 β =−1 β =0.5 β =1
10
4 2
5
0 0
−2 −4
−5
−6 −10
−8 −10 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−15 0
β
2
4
(a) Skewness
α
6
8
10
(b) Skewness
30
30
α=0.5 α=1 α=2
25
β=0 β=0.5 β=1
25
20
20
15
15
10
10
5
5
0 −2
−1.5
−1
−0.5
0 β
0.5
1
1.5
2
0
0
2
4
(c) Kurtosis
α
6
8
10
(d) Kurtosis
ˇ 2 Fig. 5.3 Skewness and kurtosis of X N I G ˛; ˇ; ˛2 ;
3 ˛2
Corollary 5.1. For the one parameter NIG model, i.e. with ˇ D 0, we get the following central moments of the corresponding NIG distribution X N I G .s˛; 0; 0; s˛/: E.X / D 0 S.X / D 0
V .X / D 1
K.X / D 3 1 C
1 . s 2 ˛2
Figure 5.3 shows the skewness and kurtosis of the NIG distribution with s D 1 dependent on the parameters ˛ and ˇ. In particular, we are interested in kurtosis values implied by different values of ˛: kurtosis is higher for smaller values of ˛.
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model
147
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model At this point we want to investigate the fitting ability and other properties of the one factor NIG copula model. We compare them with those of the one factor Gaussian and double-t models. To do so, we use the market quotes from the 12th of April 2006, of the 5th series of the tranched iTraxx Europe with 5 years maturity. The settlement of this series is the 20th of March 2006, and maturity the 20th of June 2011. Recall that the reference portfolio of this index consists of equally weighted credit default swaps of 125 European firms. However, all models under comparison employ the LHP assumption, i.e. that the reference portfolio contains infinitely many firms having the same characteristics. So, since the corresponding 5 year iTraxx index is trading at 32 bp at this day, we assume that all reference firms in the portfolio have a CDS spread of 32 bp. Further, the constant default intensity model (see Sect. 3.2.1) is employed for the marginal default distributions. The default intensity of the large homogeneous portfolio is estimated from the CDS spread (see Sect. 3.2.2). The constant recovery rate is assumed to be 40%. The standard tranches have attachment/detachment points at 3%, 6%, 9%, 12% and 22%. The investors of the tranches receive quarterly spread payments on the outstanding notional and compensate for losses when these hit the tranche they are invested in. The investor of the equity tranche receives an up-front fee that is quoted in the market and an annual spread of 500 bp quarterly. Gaussian and double-t factor copulas have only one parameter, the correlation. We estimate this parameter so that the price of the equity tranche fits the market quote, i.e. we calculate the implied equity correlation. The same correlation is used to price the other tranches. The versions of the NIG factor copula we consider have one parameter ˛ (ˇ D 0) or two parameters ˛ and ˇ besides the correlation parameter. We minimize the sum of the absolute errors over all tranches to estimate these parameters. Table 5.5 presents the market quotes of the iTraxx tranches as well as the prices of the LHP model with the Gaussian one factor copula, the double-t distribution with 3 and 4 degrees of freedom and the NIG factor copula with one and two parameters. In the one parameter NIG copula, the parameter ˇ is set to zero which makes the distribution symmetric. The double-t factor copula models fit only the equity tranche exactly since it has only one continuous valued parameter (correlation). The double-t model with 3 degrees of freedom underprices the second tranche while the double-t model with 4 degrees of freedom overprices it. Since the second model parameter (degrees of freedom) is only integer valued, it is in general impossible to fit the second tranche exactly. The results of the NIG copulas are similar to the results of double-t copulas. The additional free parameter in the NIG copula makes it more flexible: the second tranche can be fitted exactly as well. Surprising is that one more free parameter ˇ doesn’t improve the fitting results in this example. The NIG models overprice the three most senior tranches similar to the double-t model. The overall results of the NIG models are slightly better then those of the double-
148
5 Normal Inverse Gaussian Factor Copula Model
Table 5.5 Pricing iTraxx tranches with the LHP model based on different distributions Market Gaussian t(4)–t(4) t(3)–t(3) NIG(1) 0–3% 23.53% 23.53% 23.53% 23.53% 23.53% 3–6% 62.75 bp 140.46 bp 73.3 bp 53.88 bp 62.75 bp 6–9% 18 bp 29.91 bp 28.01 bp 23.94 bp 27.9 bp 9–12% 9.25 bp 7.41 bp 16.53 bp 15.96 bp 17.64 bp 12–22% 3.75 bp 0.8 bp 8.68 bp 9.94 bp 9.79 bp Absolute error 94.41 bp 32.82 bp 27.82 bp 24.34 bp Correlation 15.72% 19.83% 18.81% 16.21% ˛ 0.4794 ˇ 0 Comp. time 0.5 s 12.6 s 11 s 1.5 s
NIG(2) 23.53% 62.75 bp 27.76 bp 17.42 bp 9.6 bp 23.77 bp 15.94% 0.6020 0.1605 1.6 s
Table 5.6 Pricing iTraxx tranches with the LHP model based on different distributions Market Gaussian t(5)–t(5) t(4)–t(4) NIG(1) NIG(2) 0–3% 23.53% 19.24% 26.85% 27.84% 26.94% 26.93% 3–6% 62.75 bp 175.65 bp 65.07 bp 47.53 bp 62.75 bp 62.75 bp 6–9% 18 bp 55.14 bp 18.33 bp 14.37 bp 19.75 bp 19.80 bp 9–12% 9.25 bp 20.10 bp 8.97 bp 7.75 bp 9.73 bp 9.73 bp 12–22% 3.75 bp 3.75 bp 3.75 bp 3.75 bp 3.75 bp 3.75 bp Absolute error 1st tranche 428.86 bp 332.24 bp 431.36 bp 340.62 bp 339.63 bp 2nd–5th tranches 160.89 bp 2.94 bp 20.35 bp 2.24 bp 2.29 bp Aggregate 589.75 bp 335.18 bp 451.71 bp 342.86 bp 341.92 bp Correlation 22.47% 13.97% 12% 9.85% 9.64% ˛ 0.6678 0.7502 ˇ 0 0.1103 Comp. time 0.5 s 12.4 s 12.6 s 1.5 s 1.6 s
t models. However, the important advantage of the NIG model is the much lower computation time. As was already mentioned in Sect. 4.2.6, the prices of the equity tranche are mainly made by a special kind of the market participants, like hedge funds, and thus can be influenced by some exogenous factors. In contrast, the most senior tranches are traded by different kinds of market participants, and therefore are supposed to be priced fairly. For this reason, we try now to calibrate the models on the most senior tranche. The correlation parameter for the Gaussian and double-t models is found to fit the 12–22% tranche exactly. The NIG models are fitted to minimize the sum of the absolute errors over the four tranches: 3–6%, 6–9%, 9–12% and 12–22%. The results are given in the Table 5.6. In this case, the double-t model with 5 degrees of freedom fits the data better than the double-t models with 3 and 4 degrees of freedom. The fit of the both models, double-t and NIG, to the four upper tranches is very good. The absolute error over the four tranches is only 2.94 bp for the double-t model and 2.24 bp for the NIG(1) model. However, both models show a large deviation from the market quote of the
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model
149
equity tranche. Again, the NIG(2) model does not bring any improvement comparing to the NIG(1) model. The optimization routine even seems to fail finding an optimal solution for the NIG(2) model. We will consider this problem later in this section. Since fitting the models to the senior tranche or to the four upper tranches appeared to yield much better fitting results for all tranches besides the equity tranche, we will choose this fitting method for the rest of this chapter. Besides, we are particularly interested in alternative investment possibilities for a classical investor for whom the equity tranche is typically to risky. Thus, we are basically interested in mezzanine to senior tranches and not in the equity tranche, so an appropriate modeling of the upper tranches is much more important to us than modeling of the equity tranche. Now we want to compare the descriptive properties of the models using the second calibration. We plot density and cumulative distribution functions of portfolio losses from the Gaussian model, double-t model with 5 degrees of freedom and NIG model in Fig. 5.4. The loss distributions of the double-t and NIG models are very similar, they heavily differ from that of the Gaussian model. Figure 5.5 shows the differences between t or NIG and Gaussian densities. It is especially easy to see that both t and NIG models redistribute risk out of the lower end of the equity tranche to its higher end. The Gaussian model allocates more risk to mezzanine tranches than the other models. Figure 5.6 presents the differences between the densities of the NIG and double-t models. NIG models allocate slightly less risk at the 3–6% tranche than the double-t model does. We have plotted the density functions of the asset returns in Fig. 5.7. Please note, that while the asset distributions in the Gaussian and NIG models are standardized, i.e. have zero mean and unit variance, the variance of the asset return in the double-t model is not one. The density of the NIG(2) model is slightly skewed to the right. Further, we investigate the tail dependence of the one factor copulas under study. In particular, the amount of dependence in the lower-left-quadrant is relevant for modeling credit portfolios. Let X1 and X2 be continuous random variables. We consider the coefficient of lower tail dependence: L .x/ D P fX2 xjX1 xg: Random variables X1 and X2 are said to be asymptotically dependent in the lower tail if L > 0 and asymptotically independent in the lower tail if L D 0, where L D lim L .x/: x!1
The coefficient of upper tail dependence is defined as U .x/ D P fX2 > xjX1 > xg;
150
5 Normal Inverse Gaussian Factor Copula Model 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Gaussian copula t(5)−t(5) NIG(1) copula NIG(2) copula
0.2 0.1 0
0
2
4
6
8
10
Loss (%)
(a) Cumulative distribution function 60 Gaussian copula t(5)−t(5) NIG(1) copula NIG(2) copula
50
40
30
20
10
0
0
2
4
6 Loss (%)
(b) Density function Fig. 5.4 Portfolio loss distribution from LHP model
8
10
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model
151
60 t(5)−t(5) − Gaussian NIG(1) − Gaussian NIG(2) − Gaussian
40
20
0
−20
−40
−60
0
1
2
3
4
5
6
7
8
9
Loss (%)
Fig. 5.5 Difference between t or NIG and Gaussian loss density
NIG(1) − t(5)−t(5) NIG(2) − t(5)−t(5)
10
5
0
−5
−10 0
1
2
3
4 5 Loss (%)
Fig. 5.6 Difference between double-t and NIG loss density
6
7
8
9
152
5 Normal Inverse Gaussian Factor Copula Model 0.45 Gaussian copula t(5)−t(5) NIG(1) copula NIG(2) copula
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 −5
0
5
Fig. 5.7 Distribution of asset returns
and X1 and X2 are asymptotically dependent in the upper tail if U > 0, where U D lim U .x/: x!1
We compute the tail dependence coefficients numerically and plot them in Fig. 5.8. The Gaussian copula shows no tail dependence. The tail dependence coefficient of double t(5) copula is significantly larger than that of the Gaussian copula. The tail dependence of the NIG copula lie between those of the Gaussian and double-t copulas. However, none of the models has an asymptotical dependence since the coefficients tend to zero. Note, that the upper and the lower tail dependence structures of Gaussian, double-t(5) and NIG(1) copulas are symmetric. This is not the case for the two parameter NIG copula. This copula has a higher lower tail dependence coefficient similar to that of the NIG(1) copula, and a very small upper tail dependence coefficient. These properties of the copulas under study can be observed in Fig. 5.9 as well. We have simulated pairs of correlated asset returns with Gaussian, double-t with 5 degrees of freedom, NIG with one and two parameter factor copulas and plotted the contours of their joint densities. The double-t factor copula produces more extreme values than the other copulas. The NIG copula with two parameters has more extreme values in the lower left tail than in the upper right tail. Now we return to the calibration of the NIG model. As already mentioned, we minimize the sum of the absolute deviations to fit the model to the market quotes. NIG .a; ˛; ˇ/ the fair spread, calculated with the NIG(2) model Denote with S.K 1 ;K2 /
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model
153
0.25 Gaussian copula t(5)−t(5) NIG(1) copula NIG(2) copula
lambda(x)
0.2
0.15
0.1
0.05
0 −6
−5
−4
−3
−2
−1
x
(a) Lower tail dependence 0.25 Gaussian copula t(5)−t(5) NIG(1) copula NIG(2) copula
lambda(x)
0.2
0.15
0.1
0.05
0
1
2
3
4 x
(b) Upper tail dependence Fig. 5.8 Tail dependence coefficients
5
6
154
5 Normal Inverse Gaussian Factor Copula Model Gaussian
t(5)−t(5)
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3 −2
0
2
−2
NIG(1)
0
2
NIG(2)
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3 −2
0
2
−2
0
2
Fig. 5.9 Density contours of one factor copula
according to (4.4):7 n P R;NIG .a; ˛; ˇ/ D S.K 1 ;K2 /
iD1
R;NIG ELR;NIG .K1 ;K2 / .ti ; a; ˛; ˇ/ EL.K1 ;K2 / .ti1 ; a; ˛; ˇ/ B.t0 ; ti / n P
ti 1 ELR;NIG .K1 ;K2 / .ti ; a; ˛; ˇ/ B.t0 ; ti /
;
iD1
where ELR;NIG .K1 ;K2 / .ti ; a; ˛; ˇ/ is the expected loss of the tranche .K1 ; K2 / computed according to the NIG model with parameters a, ˛ and ˇ and recovery rate R. The up-front fee is calculated as: R;NIG U.K .a; ˛; ˇ/ D 1 ;K2 /
R;NIG ELR;NIG .K1 ;K2 / .ti ; a; ˛; ˇ/ EL.K1 ;K2 / .ti 1 ; a; ˛; ˇ/ B.t0 ; ti /
n X i D1
0:05
n X i D1
7
ti 1 ELR;NIG .K1 ;K2 / .ti ; a; ˛; ˇ/ B.t0 ; ti /:
The expected losses are computed with the assumption of 40% recovery rate.
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model
155
We have plotted the spread and up-front fee surfaces for some values of correlation parameter a in Fig. 5.10. M M We denote the market quotes with S.K for spreads and U.K for the 1 ;K2 / 1 ;K2 / up-front fee. Then the objective function for the first optimization problem, i.e. calibration in all tranches, is R;NIG M .a; ˛; ˇ/ U.0;3/ j Dev1 .a; ˛; ˇ/ D jU.0;3/ R;NIG R;NIG M M .a; ˛; ˇ/ S.3;6/ j C jS.6;9/ .a; ˛; ˇ/ S.6;9/ j CjS.3;6/ R;NIG R;NIG M M .a; ˛; ˇ/ S.9;12/ j C jS.12;22/ .a; ˛; ˇ/ S.12;22/ j: CjS.9;12/
The objective function for the second optimization problem that calibrates the model into the four upper tranches is: R;NIG R;NIG M M Dev2 .a; ˛; ˇ/ D jS.3;6/ .a; ˛; ˇ/ S.3;6/ j C jS.6;9/ .a; ˛; ˇ/ S.6;9/ j R;NIG R;NIG M M .a; ˛; ˇ/ S.9;12/ j C jS.12;22/ .a; ˛; ˇ/ S.12;22/ j: CjS.9;12/
Then the optimization problems are formulated as: min Devi .a; ˛; ˇ/ ;
a;˛;ˇ
(5.16)
with i D 1; 2 for the first or the second calibration. In Fig. 5.11 the surfaces of the first objective function for different values of ˛ and ˇ and for some values of correlation are presented. We have chosen the correlation of 16% that approximately corresponds to the optimal value, and two correlation values around the optimum. For the correlation values smaller than 16% the surface is convex around its minimum, and for correlation values larger than 16% the surface is concave with minimal values on the bounds of its feasible region. The 16% correlation surface has its minimum indeed around a zero value for ˇ. However, it has a complex form being very flat with values similar to the optimum on a complete line close to the left bound ˇ D ˛ (see contours of the objective function in Fig. 5.11d). For this reason the optimizer fails to find the global minimum, but converges in some of the points on this line. The second objective function has a similar behavior around the optimal correlation value of 10% (Fig. 5.12). The objective function for the NIG model with one additional parameter ˛ and ˇ D 0, Devi .a; ˛; 0/, has a more simple convex form (Fig. 5.13). Both optimization problems are solved very fast and there are no convergence problems. For the reason of bad convergence of the optimization for NIG(2) model, and also because the second parameter ˇ seems not to result in any important improvement of the fitting ability of the model, we choose to work with the NIG(1) model in the further study. We have also performed more calibration trials for different dates in the history,
156
5 Normal Inverse Gaussian Factor Copula Model
(a) Fair up-front fee of the 0–3% tranche
(b) Fair spread of the 3–6% tranche
(c) Fair spread of the 6–9% tranche
(d) Fair spread of the 9–12% tranche
(e) Fair spread of the 12–22% tranche Fig. 5.10 Fair prices of the iTraxx tranches from the NIG(2) model in dependence of its parameters
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model
157
1.5
350
300
1 α
250
200
0.5 150 −1
−0.5
0 β
0.5
1
(a) 13% correlation – absolute error surface (b) 13% correlation – absolute error contours 1.5 160 140 120 α
1 100 80 60 0.5
40 −1
(c) 16% correlation – absolute error surface
−0.5
0 β
0.5
1
(d) 16% correlation – absolute error contours
1.5 300 280 260 240 1 α
220 200 180 160
0.5
140 −1
(e) 19% correlation – absolute error surface
−0.5
0 β
0.5
1
(f) 19% correlation – absolute error contours
Fig. 5.11 Absolute pricing error of all tranches from the NIG(2) model in dependence of its parameters
158
5 Normal Inverse Gaussian Factor Copula Model
1.5
70 65 60 55 50
1 α
45 40 35 30 25
0.5
20 −1
(a) 7% correlation – absolute error surface
−0.5
0 β
0.5
1
(b) 7% correlation – absolute error contours 1.5 60 50 40
α
1
30 20 10
0.5 −1
(c) 10% correlation – absolute error surface
−0.5
0 β
0.5
1
(d) 10% correlation – absolute error contours 1.5 60 55 50 45 1 α
40 35 30 25 20
0.5
15 −1
(e) 13% correlation – absolute error surface
−0.5
0 β
0.5
1
(f) 13% correlation – absolute error contours
Fig. 5.12 Absolute pricing error of the 2nd–5th tranches from the NIG(2) model in dependence of its parameters
and always faced the same convergence problem in the optimization and a similar objective function. Next we want to investigate the influence of the recovery assumption on the pricing and fitting results. It is a common praxis in CDS valuation to assume a recovery
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model
159
1.5
900 800 700 600 1 α
500 400 300 200
0.5
100 5
(a) All tranches: absolute error surface
10 15 correlation (%)
20
25
(b) All tranches: absolute error contours 1.5
140 120 100 80
α
1
60 40 20
0.5 5
(c) 2nd–5th tranches: absolute error surface
10 15 correlation (%)
20
25
(d) 2nd–5th tranches: absolute error contours
Fig. 5.13 Absolute pricing error of the NIG(1) model in dependence of its parameters
of 40%. So far we have also done this to calibrate the models into the data. However, the following questions arise: what deviation in price will be caused by a different recovery value? How will the calibrated model parameters change with a different recovery assumption? First, we have calculated the fair tranche spreads using the same model parameters but different recovery values. Figure 5.14 shows that changes in the results are very small. The reason for this is that given a different recovery value we get a different value for the default intensity such that the fair spread of the reference single name CDS is still the same, i.e. 32 bp. Figure 5.15a presents the values of the default intensity corresponding to different recovery values. The default intensity grows with the recovery, i.e. given a larger recovery value, the default probability has to be larger to get the same fair CDS spread. To answer the second question, we have performed a new parameter calibration of the NIG(1) model for different recovery assumptions. We have done this for both calibration methods: in all tranches and in the four upper tranches. The correlation grows slightly with growing recovery, while the value of the ˛ parameter
160
5 Normal Inverse Gaussian Factor Copula Model 0
10
20
30
40
50
70
70
25.5
0−3% (right axis)
3−6% 6−9% 9−12% 12−22% (left axis)
80
60
25 24.5
60 50
23.5
40
23
30
22.5
20
22
10
21.5
0
0
10
20
30 40 recovery (%)
50
60
70
up−front fee (%)
spread (bp)
24
21
(a) Parameter set from the calibration in all tranches 70
0
10
20
30
40
50
60
70
29
0−3% (right axis) 28.5 3−6% 6−9% 9−12% 12−22% (left axis)
50
28
spread (bp)
27.5 40 27 30 26.5 20
26
10
0
up−front fee (%)
60
25.5
0
10
20
30 40 recovery (%)
50
60
70
25
(b) Parameter set from the calibration in 2nd–5th tranches Fig. 5.14 Fair spreads/up-front fee of the iTraxx tranches from the NIG(1) model with fixed a and ˛ parameters in dependence from recovery assumption
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model 11
161
x 10−3
10
default intensity
9 8 7 6 5 4 3 0
10
20
30 40 recovery (%)
50
60
70
(a) Default intensity 0.56
18.5
0.54
18
0.5 17
alpha
implied correlation (%)
0.52 17.5
16.5 16
0.46 0.44
15.5
0.42
15
0.4
14.5
0.38 0
10
20
30 40 recovery (%)
50
60
0
70
10
20
30 40 recovery (%)
50
60
70
(c) Optimal ˛ parameter of the NIG(1) model (calibration in all tranches)
(b) Optimal correlation parameter of the NIG(1) model (calibration in all tranches)
0.72
13 12.5
0.7
12
0.68
11.5
0.66
11
alpha
implied correlation (%)
0.48
10.5
0.64 0.62
10 0.6
9.5
0.58
9 8.5 0
10
20
30 40 recovery (%)
50
60
(d) Optimal correlation parameter of the NIG(1) model (calibration in 2nd–5th tranches)
70
0
10
20
30 40 recovery (%)
50
60
70
(e) Optimal ˛ parameter of the NIG(1) model (calibration in 2nd–5th tranches)
Fig. 5.15 NIG(1) calibration under different recovery assumptions
decreases (Fig. 5.15b–e). Table 5.7 and Fig. 5.16 present the fair spreads for the calibrated model parameters with different recovery assumptions. Here, calibration in all tranches was performed. In general, the fitting ability is very similar. The absolute
162
5 Normal Inverse Gaussian Factor Copula Model
Table 5.7 Fair spreads/up-front fee of the iTraxx tranches from the NIG(1) model with optimal a and ˛ parameters in dependence from recovery assumption (calibration in all tranches) Recovery
0%
10%
20%
30%
40%
50%
60%
70%
0–3% 3–6% 6–9% 9–12% 12–22% Abs. error Correlation ˛
23.53% 62.74 bp 26.69 bp 16.44 bp 8.94 bp 21.08 bp 14.86% 0.5200
23.53% 62.75 bp 26.90 bp 16.65 bp 9.10 bp 21.65 bp 15.13% 0.5126
23.53% 62.75 bp 27.16 bp 16.91 bp 9.28 bp 22.35 bp 15.44% 0.5038
23.53% 62.75 bp 27.48 bp 17.23 bp 9.51 bp 23.22 bp 15.80% 0.4930
23.53% 62.75 bp 27.90 bp 17.64 bp 9.79 bp 24.33 bp 16.21% 0.4794
23.53% 62.75 bp 28.48 bp 18.22 bp 10.17 bp 25.86 bp 16.72% 0.4612
23.53% 62.75 bp 29.33 bp 19.05 bp 10.63 bp 28.01 bp 17.32% 0.4357
23.53% 62.75 bp 30.74 bp 20.39 bp 11.06 bp 31.19 bp 18.06% 0.3954
0
10
20
30
40
50
60
90
70
24.5
0−3% (right axis)
3−6% 6−9% 9−12% 12−22% (left axis)
80
70
24
50 23.5 40
up−front fee (%)
spread (bp)
60
30 23
20 10 0
0
10
20
30 40 recovery (%)
50
60
70
22.5
Fig. 5.16 Fair spreads/up-front fee of the iTraxx tranches from the NIG(1) model with optimal a and ˛ parameters in dependence from recovery assumption (calibration in all tranches)
pricing error over all tranches grows from 21.08 bp for zero recovery to 31.19 bp for 70% recovery. In Table 5.8 and Fig. 5.17 the results of calibration in the four upper tranches are summarized. The absolute pricing error over the four tranches is with 1.61 bp the lowest for the zero recovery and increased monotonically with increasing recovery. It is 4.67 bp for 70% recovery. Thus, if we would let recovery be a free parameter in the optimization, the optimal solution would be for zero recovery what is quite unrealistic.
5.5 Calibration and Descriptive Statistics of the One Factor NIG Model
163
Table 5.8 Fair spreads/up-front fee of the iTraxx tranches from the NIG(1) model with optimal a and ˛ parameters in dependence from recovery assumption (calibration in 2nd–5th tranches) Recovery 0–3% 3–6% 6–9% 9–12% 12–22% Abs. error 1st tr. 2nd–5th tr. Correlation ˛
90
0
0%
10%
20%
30%
40%
50%
60%
70%
27.14% 62.75 bp 19.36 bp 9.51 bp 3.75 bp
27.11% 62.75 bp 19.42 bp 9.54 bp 3.75 bp
27.07% 62.75 bp 19.49 bp 9.58 bp 3.75 bp
27.01% 62.75 bp 19.61 bp 9.65 bp 3.75 bp
26.94% 62.75 bp 19.75 bp 9.73 bp 3.75 bp
26.82% 62.75 bp 19.98 bp 9.86 bp 3.75 bp
26.64% 62.75 bp 20.37 bp 10.10 bp 3.75 bp
26.30% 62.75 bp 21.25 bp 10.66 bp 3.75 bp
361 bp 1.61 bp 8.55% 0.7006
358 bp 1.71 bp 8.79% 0.6948
354 bp 1.82 bp 9.07% 0.6880
348 bp 2.00 bp 9.42% 0.6791
341 bp 2.23 bp 9.85% 0.6678
329 bp 2.59 bp 10.42% 0.6521
311 bp 3.22 bp 11.23% 0.6280
277 bp 4.67 bp 12.50% 0.5838
20
30
40
50
60
10
70
29
0−3% (right axis)
3−6% 6−9% 9−12% 12−22% (left axis)
80
70
28.5 28
spread (bp)
27.5 50 27 40 26.5
up−front fee (%)
60
30 26
20
25.5
10 0
0
10
20
30 40 recovery (%)
50
60
70
25
Fig. 5.17 Fair spreads/up-front fee of the iTraxx tranches from the NIG(1) model with optimal a and ˛ parameters in dependence from recovery assumption (calibration in 2nd–5th tranches)
.
Part III
Term-Structure Models
.
Chapter 6
Term Structure Dimension
First, the most quantitative research has been focused on improving the Gaussian copula model to fit different tranches of one CDO simultaneously, so that it would be possible to price off-market tranches as well. We have discussed this modeling dimension in the previous section. Since 2005 CDX and iTraxx tranches started trading more actively also in other maturities besides of 5 years, namely 7 and 10 years. Increase of liquidity in different maturities turned the research interest into the term-structure dimension of the models. The factor copula models, described in the previous section, do not incorporate the term-structure dimension. They just average the correlations and other model parameters over the complete lifetime of the tranche. The distributions of the factors are assumed to be the same over arbitrary time horizons up to maturity of the tranches: recall, that the loss distributions for each payment date (i.e. quarterly for iTraxx tranches) are needed for valuation. Thus, applying the model to the longdated tranches is not consistent with the short-dated ones. In this section we present the extension of the base correlation to the term-structure dimension, that is very popular among the practitioners. We also extend the NIG model to describe different maturities simultaneously.
6.1 Extension of the Base Correlation The base correlation skews in Fig. 6.1 are computed for 5, 7 and 10 years maturity as described in Sect. 4.2.7. As we can see these curves do not coincide in general. So it is not really clear which correlation should be used to valuate a tranche with some non-standard maturity. The problem of the factor copula models is that it does not take into account certain information. For example, to calibrate the 7 years tranches, it only uses the 7 years spreads and ignores the information one could extract from the 5 years spreads. The practitioners solved this problem by extending the base correlation approach into the maturity dimension (see e.g. [76]). This is done in a bootstrap procedure.
A. Schl¨osser, Pricing and Risk Management of Synthetic CDOs, Lecture Notes in Economics and Mathematical Systems 646, DOI 10.1007/978-3-642-15609-0 6, c Springer-Verlag Berlin Heidelberg 2011
167
168
6 Term Structure Dimension 60 5 years 7 years 10 years
55
base correlation (%)
50 45 40 35 30 25 20 15
3
6
9
.
12 tranche (%)
22
Fig. 6.1 Base correlation curves for iTraxx with 5, 7 and 10 years maturity
TS (i) First, the base correlation skew a.0;K .5/, i D 1; : : : ; 5 with K1 D 3%, K2 D i/ 6%, K2 D 9%, K2 D 12% and K2 D 22%, is solved for the tranches with the shortest maturity (e.g. 5 years) as it is described in Sect. 4.2.7. (ii) Further, the base correlation skew is estimated for a longer maturity (7 years) taking into account the information for the first 5 years. In the standard t were computed using the same corapproach the expected losses ELR .0;Ki / j relation a.0;Ki / .7/ for any time tj . In the new approach the expected losses for TS tj 5 are computed using the 5 years correlation a.0;K .5/. The expected i/ TS loss for tj D 7 is computed with the term-structure correlation a.0;K .7/. The i/ expected loss for 5 < tj < 7 is computed using linearly interpolated values. So the correlation parameter is now time dependent:
TS a.0;K .tj / D i/
8 ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ :
TS .5/ a.0;K / i
TS TS .7tj /a.0;K .5/C.tj 5/a.0;K .7/ i/ i/
2
TS .7/ a.0;K i/
if tj 5 if 5 < tj < 7 if tj D 7:
As in the standard base correlation approach, the following equation
(6.1)
6.1 Extension of the Base Correlation
169
Ki TS P V 0; Ki ; S.Ki 1 ;Ki / ; a.0;K .7/ / i Ki Ki 1 Ki 1 TS P V 0; Ki 1 ; S.Ki 1 ;Ki / ; a.0;K .7/ D0 i 1 / Ki Ki 1 TS .7/ recursively for i D 1; : : : ; 5, where present values must be solved for a.0;K i/ are computed as
TS .7/ P V 0; Ki ; S.Ki 1 ;Ki / ; a.0;K i/ D U.Ki 1 ;Ki / C
n X
tj S.Ki 1 ;Ki / 1 ELR B.t0 ; tj / .0;Ki / tj
j D1
n X
j D1
R ELR B.t0 ; tj /; t EL t .0;Ki / j .0;Ki / j 1
and TS .7/ P V 0; Ki 1 ; S.Ki 1 ;Ki / ; a.0;K / i 1 D U.Ki 1 ;Ki / C
n X
B.t0 ; tj / tj S.Ki 1 ;Ki / 1 ELR .0;Ki 1 / tj
j D1
n X R ELR B.t0 ; tj /; .0;Ki 1 / tj EL.0;Ki 1 / tj 1 j D1
where U.Ki 1 ;Ki / are the upfront payments and S.Ki 1 ;Ki / the market spreads of the 7 years tranches. In contrast to the standard base correlation approach, the expected losses ELR .0;Ki / tj in the present values are computed not with the constant correlation parameter a.0;Ki / .7/, but with the time dependent term-structure TS correlation parameter a.0;K .tj / as defined in (6.1). i/ TS .10/ are (iii) Afterwards, the values of the term-structure base correlations a.0;K i/ estimated in the same way to fit the spreads of the 10 year tranches using the TS TS already known values of a.0;K .5/ and a.0;K .7/, and extending (6.1) up to i/ i/ 10 years:
TS a.0;K .tj / i/
D
8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
0 such that the asset return in the second state can be written as: q b ij .t/; db Aij .t/ D aj d c M .t/ C 1 aj2 d X (8.5) with the distributions of the factors given by 3 ˇ 2 ˛; ˇ; k 2 dt; k 2 dt ; ˛ ˛ 0q q 1 aj2 1 aj2 b ij .t/ N I G B dX ˛; ˇ; @ aj aj dc M .t/ N I G
1 kaj2 1 aj2
1 q q 2 2 1 aj2 ˇ 2 1 a 3 1 kaj j C dt; dt A ; 2 2 2 aj ˛ aj ˛ 1 aj
and the distribution of d b Aij .t/ is the same as dAij .t/.
(8.6)
(8.7)
188
8 Regime-Switching Extension of the NIG Factor Copula Model
Proof. We start with general NIG distribution for the factors: dc M .t/ N I G ˛ CM ; ˇ CM ; CM ; ı CM ; d Xij .t/ N I G ˛ CX ; ˇ CX ; CX ; ı CX :
b
(8.8) (8.9)
It follows from the requirement .i / and the convolution property of the NIG distribution, that the first two parameters of the distributions must be equal to those in the first state: q q 1 aj2 1 aj2 ˛CM D ˛; ˇ CM D ˇ; ˛ CX D ˛; ˇ CX D ˇ: (8.10) aj aj Besides, the requirement .i i i / means that: CM D ı CM
ˇ CX ˇ ; D ı CX :
(8.11)
Now we consider the distribution of the asset return in the second state. This is the distribution of the sum of two NIG random variables: ˇ ˛ ˇ b aj d c (8.12) a j ı CM ; M .t/ N I G ; ; b aj ı CM ; b b aj b aj 0 q q q 1 aj2 1 aj2 B 2 1 b aj d Xij .t/ N I G @ q ˛; q ˇ; (8.13) aj 1 b a 2j aj 1 b a2j q q 2 CX ˇ 2 CX aj ı aj ı : ; 1 b 1 b
b
The both distributions are stable under convolution if the first two parameters are equal. This is the case when q
1 aj2 1 D q ; b aj aj 1 b a2j that is equivalent to
q
1 b a2j b aj
q D
1 aj2 aj
:
This is only possible if b aj D aj :
(8.14)
8.2 Crash-NIG Copula Model
189
So the distribution of the asset return in the second state is given by
dAij .t/ N I G aj ı CM
q ˇ ˛ ˇ ; ; aj ı CM C 1 aj2 ı CX ; aj aj q C 1 aj2 ı CX :
(8.15)
According to the requirement .i i /, the distribution must be the same as in the first state, i.e. the third and the fourth parameter must be: q ˇ 1 ˇ 2 dt; D aj ı CM C 1 aj2 ı CX aj ˛ 2 q 1 3 aj ı CM C 1 aj2 ı CX D dt aj ˛ 2
(8.16) (8.17)
The two equations are actually the same. So we have only one equation to solve for two variables ı CM and ı CX . We also have the last requirement .i v/ that still is not satisfied. So we look at the parameter ı CM : q ı CM D
3
1 dt aj2 ˛ 2
1 aj2 aj
ı CX :
ı CM can be independent of aj only if it has the form ı CM D k
3 dt; ˛2
for some constant k > 0. Then corresponding ı CX is ı CX D
1 kaj2 1 aj2
q 1 aj2 3 dt; aj ˛2
and we come up with the distributions in (8.6) that completes the proof.
t u
Remark 8.1. Recall, that in the first correlation regime, the variance of all factor changes is dt. Now, the variance of the factors in the second regime is given by V .d c M / D kdt;
b ij / D V .d X
1 kaj2 1 aj2
dt:
Thus, the correlation of asset returns of an issuer i1 from the rating cell j1 and an issuer i2 from the rating cell j2 is
190
8 Regime-Switching Extension of the NIG Factor Copula Model
C orr dAi1 j1 .t/; dAi2 j2 .t/ D p
M/ aj1 aj2 V .d c V .dAi1 j1 .t//V .dAi2 j2 .t//
D aj1 aj2 k:
The higher correlation in the second regime is implied by the higher variance of the market factor, i.e. by choosing k > 1. The variance of the idiosyncratic factor is then lower than normal. a ˇ
Factor Aij .t/ has zero mean, variance t, skewness 3 2jpt and kurtosis 2 a2 ˛2 j 3 C 3 1 C 4 ˇ˛ . 4t Definition 8.1 (Crash-NIG copula model). The asset return of the i th issuer in cell j for j D 1; : : : ; J , Aij .t/, is assumed to be of the form: dAij .t/ D aj dM.t/ C
q 1 aj2 dXij .t/;
(8.18)
where M.t/; Xij .t/; i D 1; : : : ; m are independent processes with the following distributions: ˇ 2 3 (8.19) dM.t/ N I G ˛; ˇ; 2t 2 dt; 2t 2 dt ; ˛ ˛ 0q q 2 1 a 1 aj2 j B ˛; ˇ; (8.20) dXij .t/ N I G @ aj aj 1 2t aj2 1 aj2
1 q q 2 2 2 1 aj2 ˇ 2 1 a 3 1 t aj j C dt; dt A : 2 2 aj ˛ aj ˛2 1 aj
t is a Markov process with state space f1; g, an initial distribution D f1 ; 2 g and a .2 2/ transition function fP .h/gh0 . The distribution of the increment of the asset return is dAij .t/ D N. 1 / .t/, i.e., aj
dAij .t/ N I G
1 1 ˇ 2 1 3 1 ˛; ˇ; dt; dt : aj aj aj ˛ 2 aj ˛ 2
Remark 8.2. The Crash-NIG copula model can be easily extended to a higher number of regimes. Then, the Markov process t has the state space f1; 1 ; : : : ; n1 g, an initial distribution D f1 ; 2 ; : : : ; n g and a .n n/ transition function fP .h/gh0 . The next Lemma (see e.g. [6] and [60]) defines the intensity matrix O of the Markov process which is sufficient to define the complete transition function fP .h/gh>0 and stays in one to one correspondence with it. The Lemma summarizes the results in Sect. 3.4.
8.2 Crash-NIG Copula Model
191
Lemma 8.1. Let fP .h/gh>0 D f pij .h/i;j D1;:::;n gh>0 be a .n n/ transition function. Then it holds: (i) There exist the limits pij .h/ 1fi Dj g ; h#0 h
oij D li m
for all states i; j D 1; : : : ; n. Denote the limit matrix O D li m h#0
P .h/ I : h
(ii) It holds: oij 0; i ¤ j;
n X
oij D 0:
j D1
(iii) It holds for all h 0 P .h/ D expfhOg D
1 k X h kD0
kŠ
Ok:
(iv) The other way around, given an arbitrary matrix O such that oij 0 for i ¤ j n P and oij D 0. Then the matrix exponential fexpfhOggh0 is a transition j D1
function with intensity O. t u
Proof. See e.g. [6] and [60].
Proposition 8.2. Consider an asset return of the i th issuer in cell j for j D 1; : : : ; J , Aij .t/, as defined in Definition 8.1. Assume, the process t was in state 0 r 2 f1; 2g at the time 0. Let T r .t/ WD T1r .t/; T2r .t/ be a stochastic process giving the duration of the stay in state i starting from the state r at time t D 0: Zt Tir .t/
D
1fstate i at time sg ds:
(8.21)
0
Then the distributions of M.t/ and Xij .t/, the cumulated returns on Œ0; t, conditional on the realization of T r .t/, are NIG with the following parameters: ˇ 2 ˛; ˇ; T1r .t/ C 2 T2r .t/ ; ˛2 r 3 T1 .t/ C 2 T2r .t/ 2 ; ˛
M.t/jT r .t/ N I G
(8.22)
192
8 Regime-Switching Extension of the NIG Factor Copula Model
0q B Xij .t/jT .t/ N I G @ r
1 aj2 aj
q ˛;
1 aj2 aj
ˇ;
(8.23)
q 1 aj2 ˇ 2 ; aj ˛2 1 aj2 1 q r 2 2 2 r 1 a 3 t aj T1 .t/ C T2 .t/ j C A: 2 aj ˛2 1 aj t aj2 T1r .t/ C 2 T2r .t/
The distribution of Aij .t/ is as before Aij .t/ N I G
1 1 ˇ 2 1 3 1 ˛; ˇ; t; t : aj aj aj ˛ 2 aj ˛ 2
Proof. Due to convolution property of the NIG distribution, we just have to integrate the third and fours parameters in (8.19). To do so, the following integral must be calculated: Zt
Zt 2s ds
D
0
1fstate 1 at time sg C 2 1fstate 2 at time sg ds D T1r .t/ C 2 T2r .t/: (8.24)
0
t u
Remark 8.3. All the results are analogue for more than two states changing T1r .t/C 2 T2r .t/ for a suitable expression, e.g. T1r .t/ C 21 T2r .t/ C 22 T3r .t/ for three states. Remark 8.4. Let fT r .t / W ˝t ! R with ˝T D Œ0; t2 be the density function of the duration of the stay in some state starting from state r 2 f1; 2g. Then the unconditional densities of the factors M.t/ and Xij .t/ are Z fM.t / .x/ D
fN I G ˝t
ˇ 2 3 2 2 ; z1 C z2 2 xI ˛; ˇ; z1 C z2 ˛2 ˛
fT r .t / .z1 ; z2 /d.z1 ; z2 / fXij .t / .x/ Z D
0
B fN I G @xI
˝t
t
aj2
q
1 aj2 aj
z1 C 2 z2
1
aj2
q
q ˛;
1 aj
1 aj2 aj
aj2
ˇ ;
t
aj2
z1 C z2 2
1 aj2
1 3 C A fT r .t / .z1 ; z2 /d.z1 ; z2 /: ˛2
q 1 aj2 ˇ 2 ; aj ˛2
8.2 Crash-NIG Copula Model
193
Unfortunately, the distributions of the duration of stay T r .t/ are very complicated even for two states (see [66]). To our knowledge it is impossible to compute the unconditional densities of the factors analytically. A numerical integration would be very time and memory consuming and could only make sense if a very exact pricing on a single day is needed and can be performed on a high-end machine. However, the four moments of the unconditional distributions can be computed quite easily. So, an approximation of the unconditional distributions with a NIG distribution matching the four moments seems to be a good alternative to the exact computation. The next proposition gives the formulas for the four moments. Proposition 8.3. The moments of the unconditional distribution of the factor M.t/ are: E .M.t// D 0 V .M.t// D E T1r .t/ C 2 T2r .t/
! 3ˇ 1 S .M.t// D 2 E p r T1 .t/ C 2 T2r .t/ 2 2 ! ˛ 1 ˇ : E K .M.t// D 3 C 3 1 C 4 r r 2 ˛ T1 .t/ C T2 .t/ 4
The moments of the unconditional distribution of the factor Xij .t/ are: E Xij .t/ D 0
V Xij .t/ D
t aj2 E T1r .t/ C 2 T2r .t/ 0
1 aj2
1
1 3ˇaj B C S Xij .t/ D E @q A 2 2 r r 2 t aj T1 .t/ C T2 .t/ ! 2 2 2 ! aj ˛ ˇ 1 r K Xij .t/ D 3 C 3 1 C 4 : E 2 r 2 ˛ 4 t aj T1 .t/ C T2 .t/ Proof. For the i th central moment of the unconditional distribution of M.t/ we have: Z1 x i fM.t / .x/dx 1
Z1 Z
D
x i fN I G 1˝t
ˇ 2 3 2 xI ˛; ˇ; z1 C 2 z2 ; z C z 1 2 ˛2 ˛2
fT r .t / .z1 ; z2 /d.z1 ; z2 /dx
194
8 Regime-Switching Extension of the NIG Factor Copula Model
3 21 Z Z 2 3 ˇ D 4 x i fN I G xI ˛; ˇ; z1 C 2 z2 ; z1 C 2 z2 2 dx 5 ˛2 ˛ ˝t
1
fT r .t / .z1 ; z2 /d.z1 ; z2 /: So we must just integrate the moments of M.t/jT r .t/, i.e. the moments of a NIG distribution, over the distribution of the duration stays. Since the expectation of the conditional distribution of M.t/jT r .t/ is zero, the unconditional expectation is zero as well, i.e. E .M.t// D 0. Further, the variance of M.t/ is:
3 ˛2 V .M.t// D E T1r .t/ C 2 T2r .t/ 2 3 ˛
D E T1r .t/ C 2 T2r .t/ :
The skewness of M.t/ is: 0
1
3ˇ B S.M.t// D E @ q p ˛ T1r .t/ C 2 T2r .t/
! 1 3ˇ C : A D 2E p r 3 T1 .t/ C 2 T2r .t/ ˛2
And finally, the kurtosis of M.t/ is: ! 2 ! ˇ 1 K .M.t// D 3 C 3 1 C 4 E 3 ˛ T1r .t/ C 2 T2r .t/ ˛2 2 2 ! ˛ ˇ 1 D 3C3 1C4 E : ˛ T1r .t/ C 2 T2r .t/ 4 t u
The derivation of the moments for Xij .t/ is analogue and straightforward.
Remark 8.5. To approximate the unconditional distribution of M.t/ with a NIG distribution: q b b M.t/ ' N I G b ˛ .t/; ˇ.t/; b .t/; ı.t/ ; with b .t/ D b ˛ 2 .t/ b ˇ 2 .t/ (8.25) the following system of four equations has to be solved for b ˛ .t/, b ˇ.t/, b .t/, b ı.t/: b ˇ.t/ D0 b .t/ b ˛ 2 .t/ b D E T1r .t/ C 2 T2r .t/ ı.t/ 3 b .t/
b .t/ C b ı.t/
8.2 Crash-NIG Copula Model
195
!
1 3ˇ 3b ˇ.t/ q D 2E p r T1 .t/ C 2 T2r .t/ b ˛ .t/ b ı.t/b .t/ 0 !2 1 2 ! 2 b ˇ ˇ.t/ ˛ 1 1 @1 C 4 A D 1C4 : E r r 2 b b ˛ .t/ ˛ T1 .t/ C T2 .t/ 4 ı.t/b .t/ Analogue, to approximate the unconditional distribution of Xij .t/ with a NIG distribution: Xij .t/ ' N I G b ˛ .t/; b ˇ.t/; b .t/; b ı.t/ ; (8.26) the following system of four equations has to be solved for b ˛ .t/, b ˇ.t/, b .t/, b ı.t/: b ˇ.t/ D0 b .t/ C b ı.t/ b .t/ 2 t aj2 E T1r .t/ C 2 T2r .t/ b ˛ .t/ b D ı.t/ 3 b .t/ 1 aj2 1 0 b 3ˇaj B 3ˇ.t/ 1 C D E @q q A r 2 2 r 2 t aj T1 .t/ C T2 .t/ ı.t/b .t/ b ˛ .t/ b 0 1 !2 ! 2 ! b ˇ.t/ 1 1 ˇ @1 C 4 A E D 1C4 b b ˛ .t/ ˛ t aj2 T1r .t/ C 2 T2r .t/ ı.t/b .t/
aj2 ˛ 2 4
:
In general, these systems of equations cannot be solved analytically. In the special case of ˇ D 0, however, the systems are easy to solve. The parameter for M.t/ are: q b ˛ 2 .t/ b ˇ 2 .t/ D b ˛ .t/; ˇ.t/ D 0; b .t/ D 0 and, using b .t/ D b ˛ b ˛ .t/ D r ; 1 E T1r .t/ C 2 T2r .t/ E T r .t /C 2 T r .t / 1 2 v u E T r .t/ C 2 T r .t/ u 2 b 1 : ı.t/ D ˛ t 1 E T r .t /C2 T r .t / 1
2
The parameters for Xij .t/ are: b ˇ.t/ D 0; b .t/ D 0 and, using b .t/ D
q b ˛ 2 .t/ b ˇ 2 .t/ D b ˛ .t/;
196
8 Regime-Switching Extension of the NIG Factor Copula Model
q b ˛ .t/ D
aj q
b ı.t/ D
1 aj2
1 aj2 aj
˛ s t aj2 E T1r .t/ C 2 T2r .t/ E t a2 v u aj2 E T1r .t/ C 2 T2r .t/ ˛ u u t : u 1 aj2 t E t a2 T r .t1/C2 T r .t / / 2 j. 1
;
1 r 2 r j .T1 .t /C T2 .t //
Remark 8.6. The expectation E T1r .t/ C 2 T2r .t/ can be easily computed as: E T1r .t/ C 2 T2r .t/ D E T1r .t/ C 2 E T2r .t/ D hr1 .t/ C 2 hr2 .t/; Zt hr1 .t/
D
.exp .sO//r;1 ds
(8.27)
.exp .sO//r;2 ds:
(8.28)
0
Zt hr2 .t/ D Proof.
0
0t 1 Z r E T1 .t/ D E @ 1fstate 1 at time sg ds A 0
Zt D
Q Œstate 1 at time s ds 0
Zt D
.exp .sO//r;1 ds: 0
Besides, we have E T2r .t/ D t E T1r .t/ .
t u
The last remark shows that the variance of the distributions of M.t/ and Xij .t/ is easy to compute. The computation for the skewness and kurtosis is not that straightforward. For applications where the computation speed is a more important issue than the accuracy, the simple fitting of the variance of the distributions may be a better choice. In this case, the parameters of the approximating NIG distribution can be chosen as described in the next remark. Remark 8.7. Approximation of M.t/ and Xij .t/ with r ˇ 2 r 3 2 r 2 r M.t/ ' N I G ˛; ˇ; h1 .t/ C h2 .t/ ; h1 .t/ C h2 .t/ 2 ˛2 ˛
8.2 Crash-NIG Copula Model
0q B Xij .t/ ' N I G @
197
1 aj2 aj
˛;
q 1 aj2 aj
ˇ;
q 1 aj2 ˇ 2 ; aj ˛2 1 aj2 1 q r 2 2 2 r 1 a 3 t aj h1 .t/ C h2 .t/ j C A 2 aj ˛2 1 aj t aj2 hr1 .t/ C 2 hr2 .t/
fits the first two moments of the exact distributions. The third and the fourth moments of the approximate distribution are not higher than those of the exact distribution. In the special case of a non-skewed distributions, i.e. ˇ D 0, the skewness is zero for the approximate and the exact distributions. Proof. The computation of the first two moments of the approximate distributions are straightforward. They are equal to the first two moments in the Proposition 8.3. The skewness of the approximate distribution of M.t/ is 0
1
! 3ˇ B 1 1 C 3ˇ @q A 2 E pT r .t/ C 2 T r .t/ : 2 1 2 E T1r .t/ C 2 T2r .t/ The inequality is given by Jensen’s inequality since f .x/ D tion. The proof for the kurtosis is analogue.
p1 x
is a convex funct u
Remark 8.8. All the results are analogue for more than two states changing hr1 .t/ C 2 hr2 .t/ for a suitable expression, e.g. hr1 .t/ C 21 hr2 .t/ C 22 hr3 .t/ for three states. We have simulated some examples of the model and computed the unconditional distribution functions in order to get a feeling of how good the described approximations are. The first example is a two-state model with following parameter: OD
0:9962 0:0038 0:0038 0:0038 ; P .1/ D 0:0119 0:9881 0:0120 0:0120
D .0; 1/; ˛ D 0:4; D 2; aj D 0:5; t D 3: The states of the Markov process are simulated over a 3-years period. The durations of the stays corresponding to the simulated paths were used to simulate the unconditional distribution of M.t/ and Xij .t/. Figure 8.1 shows the histogram of T1r .t/ C 2 T2r .t/ as well as the unconditional distribution function of the factor
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(b) Distribution function of M.t / Fig. 8.1 Example of a two-state model
M.t/. The distributions of the approximation of the factor M.t/ fitting two and four moments are also plotted to compare the approximation error. The three distribution functions are nearly the same for a model with two states. The results for the factor Xij .t/ are similar. We get even similar results if we increase the transition probability of one state into another.
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(b) Distribution function of M.t / Fig. 8.2 Example of a three-state model with no absorbing states
Next, we consider two examples of a three-state model. The first example has the following parameter: 0
1 0 1 0:0139 0:0140 0:0001 0:9863 0:0137 0 O D @ 0:0142 0:0231 0:0089 A ; P .1/ D @ 0:014 0:9773 0:0087 A 0:0127 0:0001 0:0126 0:0125 0 0:9875 D .0; 1; 0/; ˛ D 0:4; 1 D 0:25; 2 D 1:75; aj D 0:5; t D 3:
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The approximation of the unconditional distribution of the factors with a NIG distribution matching two or four moments is also very exact (Fig. 8.2). Another three-state model under consideration is a model with an absorbing state: once we are in state three, it is not possible to escape from it. The parameter of the model are:
8.3 Valuation of CDO Tranches with the Crash-NIG Copula Model
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0 1 1 0:9953 0:0047 0 0:0048 0:0048 0 O D @ 0:0165 0:0216 0:0051 A ; P .1/ D @ 0:0163 0:9787 0:005 A 0 0 1 0 0 0 0
D .0; 1; 0/; ˛ D 0:4; 1 D 0:25; 2 D 1:75; aj D 0:5; t D 3: For this case, the distribution of T1r .t/ C 21 T2r .t/ C 22 T3r .t/ is quite different than in the previous examples (Fig. 8.3). The approximations of the distribution of factor M.t/ are not that exact anymore. However, they are still very accurate. The difference between the exact unconditional distribution function of Xij .t/ and the approximations is much smaller.
8.3 Valuation of CDO Tranches with the Crash-NIG Copula Model Now all distributions necessary to describe the portfolio loss distribution are available. It depends on the application if the more exact four moment matching approximating distributions for M.t/ and Xij .t/ or the two moment matching approximations that are easier to compute should be used. Since the approximation matching only two moment appeared to be very good, we choose to use it in the future analysis. The next lemma updates the formulas of the distribution of a large homogeneous cell portfolio for the Crash NIG extension. Lemma 8.2. The approximate loss distribution of an infinitely large homogeneous cell portfolio with the asset returns following a Crash-NIG copula model is given by ˇ 2 LHC F1 .t; x/ D 1 FN I G lt1 .x/I ˛; ˇ; hr1 .t/ C 2 hr2 .t/ ; (8.29) ˛2 3 r h1 .t/ C 2 hr2 .t/ 2 ; ˛ with x 2 Œ0; 1 denoting the percentage portfolio loss. The function lt .M.t// is the portfolio loss conditional on the realization of the systematic factor M.t/ and is given by: 0 q q 2 J 1 a 1 aj2 X j B Cj .t/ aj M.t/ lt .M.t// D .1 R/ wj FN I G @ q I ˛; ˇ; aj aj 1 a2 j D1
t aj2 hr1 .t/ C 2 hr2 .t/
q
j
1 aj2 ˇ 2 ; aj ˛2 1 aj2 1 q r 2 2 2 r 1 a 3 t aj h1 .t/ C h2 .t/ j C A: 2 aj ˛2 1 aj
(8.30)
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The default thresholds are computed as Cj .t/ D
1 FN IG
1 1 1 ˇ 2 1 3 ˇ; t; t : Qj .t/I ˛; aj aj aj ˛ 2 aj ˛ 2
Proof. Analogue to Lemma 7.3 with the corresponding distributions for M.t/, Xij .t/ and Aij .t/. t u As it was already the case for the single-regime LHC NIG copula model, there exist no analytical expressions for the expected tranche loss. They have to be computed numerically by approximating the corresponding integrals over the portfolio loss distribution function. We have implemented two versions for CDO valuations using the Crash-NIG copula model. First, the inverse conditional loss function lt1 .x/ was implemented with the help of a look-up table. For all NIG distribution and inverse distribution functions the corresponding routines where called in each evaluation point. Such implementation is quite time consuming taking approximately 25 s1 in Matlab for one trading day. Of course, it is not acceptable for the calibration of a 4 year history and a simulation. So we have implemented a second, vector version with additional look-up tables. Since it is necessary to call the NIG distribution and inverse distribution functions with different parameters and for different time increments for a calibration, this is exactly the part of the pricing function that takes most of the computation time. A possibility to avoid this is the creation of look-up tables for the NIG distribution and the NIG inverse distribution functions. This implementation takes us only 3 s to price the iTraxx tranches on one day, and 25 s for a simultaneous pricing on 200 days on the same computer. Now, the n-dimensional interpolation in the look-up tables is taking most of the computation time. We have not spent any additional effort trying to accelerate this and have simply used the Maltab function “interpn”. However, we believe that this is not the fastest possible implementation and Matlab in general is not the fastest programming environment. Nevertheless, this computation speed is acceptable for the research of this thesis.
8.4 Calibration of the Crash-NIG Copula Model 8.4.1 Data Description Collecting and preparing the relevant data is a very important and sensitive issue in the calibration of the Crash-NIG copula model. The reasons for this are, first, the complexity of the financial instrument we are going to price, and, second, the application of the more detailed information by the large homogeneous cell model 1
On a computer with Intel Core Duo with 2.2 GHz Processor.
8.4 Calibration of the Crash-NIG Copula Model
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in comparison to the large homogeneous portfolio model. Note that the LHC model needs the information on the rating cells of the portfolio while, the average default probability deduced from the iTraxx index spread is enough for the LHP model. We use the complete history of the iTraxx Europe tranched index since its origination on the 21st of June 2004 until the 6th of May 2008. The 7-year maturity became available only since the end of March 2005. The data is presented in Figs. 8.4–8.6. Figure 8.7 presents the data on the index spread for the three maturities. We used the data downloaded from the MorganMarkets,2 the internetbased data source of JP Morgan. These are the proprietary quotes of JP Morgan and not the official market quotes. The official quotes were not available for us. However, we don’t find this a big problem since the bid-ask spreads on iTraxx are very small. The data described so far is the basic iTraxx data necessary for any valuation model. This data would be even enough for the LHP model. For LHC model, a more specific rating-based data is needed. First of all, we need to know the rating composition of the iTraxx portfolio at any point in history. Unfortunately, such data is not available for download. We had to create the rating composition manually. All issuers ever been in the iTraxx Europe portfolio can be found in a convenient Excel format on the website of Markit.3 We have created a rating history for all issuers and finally computed the rating composition for the iTraxx Europe portfolio
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Fig. 8.4 Market quotes of the 5-year iTraxx tranches
2
https://mm.jpmorgan.com/redirect/bankone. Indices matrix available to download as an Excel sheet from http://www.markit.com/information/ products/category/indices/itraxx/asia/matrix.html.
3
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Fig. 8.6 Market quotes of the 10-year iTraxx tranches
on every trading day in the history. This rating history is presented in Fig. 8.8. Most of the time the iTraxx Europe portfolio contained only issuers with ratings AA, A and BBB. The quote of the AA rating in the portfolio varied from 10 to 20%. Rating A was fluctuating around 40% and rating BBB around 45%. The rating AAA and BB were present in the portfolio only for short periods of time with a very small percentage.
8.4 Calibration of the Crash-NIG Copula Model
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Fig. 8.7 Market quotes of the iTraxx index spread 100 AAA AA A BBB BB
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Fig. 8.8 Historical rating composition of the iTraxx portfolio
The last data building block necessary for the LHC model are the average rating spreads. They are required to compute the rating specific default probabilities. As already discussed in the Sect. 7.3, the market average rating spreads are not appropriate for the LHC model since the rating cells of the iTraxx Europe portfolio are not
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8 Regime-Switching Extension of the NIG Factor Copula Model 550 AAA AA A BBB BB
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Fig. 8.9 Average 5-year spreads of the rating cells of the iTraxx portfolio
large enough. Since the most tradable and liquid issuers are selected for the iTraxx Europe portfolio, they tend to be in the upper segment of the particular rating. So the average spreads of the iTraxx rating cells tend to be lower than the average European rating spreads. As there exist no traded iTraxx Europe rating sub-indices, we had to compute them from the issuers’ CDS data. In particular, we downloaded the data for the senior secured CDS with maturities of 5, 7 and 10 years for each issuer and constructed the average spreads for every rating at each point in time. The results are presented in Figs. 8.9–8.11. Note that the average rating spreads are computed based on the issuers with the corresponding rating for the point in time. This is the reason why the AAA and BB spreads are available only partially. This concludes the presentation of the data that is needed to calibrate the large homogeneous cell model. Besides of it, we also used the base correlation data for our analysis. It is also available from MorganMarkets. We do not provide any further explanation and figures of it at this point and will return to it in the next section.
8.4.2 Calibration of the Model with Two States To calibrate the Crash-NIG copula model to the data, we have to estimate the following parameters: ˛; ˇ; aj ; j D 1; : : : ; 5;
8.4 Calibration of the Crash-NIG Copula Model AAA AA A BBB BB
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Fig. 8.10 Average 7-year spreads of the rating cells of the iTraxx portfolio 700 AAA AA A BBB BB
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Fig. 8.11 Average 10-year spreads of the rating cells of the iTraxx portfolio
and
P D
p11 p12 p21 p22
:
We choose to fix the parameter ˇ D 0 since it was shown in Sect. 5.5 to be the best choice. So we can also keep the computation and the computer memory low.
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The estimation of the parameters will be performed in two steps. (i) The Hidden Markov Model (HMM) is estimated separately. Since the correlation values are not observable, we have to use some other relevant observable process to estimate the model. This process follows two probability distributions depending on the state of the Markov chain. It is obvious from Fig. 8.7 of the historical data, that the two states of the market have an impact not only on the correlations, but also on the iTraxx index spread. The spreads were very high during the observed crisis. We will use the 5 and 10 years iTraxx index spreads to derive the parameters of the Hidden Markov Model. We try to estimate the HMM with two distribution assumptions for the spread: normal and log-normal. If the Markov chain is in state one, the distribution of the spread or the log-spread is assumed to be N.1 ; 1 /. In state two, the distribution is N.2 ; 2 /. We also employed another process instead of the iTraxx spread, the base correlation of the equity tranche to calibrate the HMM. However, this data is only available since September 2004. We assume base correlation to be normally distributed. Note, that the base correlation is the implied correlation of the Gaussian copula model. The important difference between the iTraxx spread and the implied correlation of the equity tranche is that the iTraxx spread does not contain the correlation information directly. This information comes into the iTraxx spread indirectly, e.g., the correlation is typically high in the turbulent markets with high credit spreads. Since the implied correlation is a product of another credit portfolio model, we do not actually tend to use it in our model, but we employ this data series in the Hidden Markov Model analysis of this chapter for the comparison and better understanding of the market. The transition matrix P is estimated with the help of the Baum–Welch Algo-
rithm 3.3.1 described in Sect. 3.3.4 The algorithm provides also the initial distribution and the parameters of the distribution of the observable process. Afterwards, we use the function Viterbi of the R package, that computes the most likely sequence of the states of the Markov chain given the estimated transition matrix, the initial distribution and the distribution parameters of the observable process. This function uses the Viterbi Algorithm 3.3.2 described in Sect. 3.3. Using the transition matrix and the sequence of the most likely states, the probabilities hr1 .t/ and hr2 .t/ of the states one and two respectively on the time segments Œ0; t, with r 2 1; 2 denoting the initial state at time 0, are computed according to (8.27). These probabilities are calculated for all t, the time points of the premium payments. (ii) In the second step, the probabilities hr1 .t/ and hr2 .t/ are used for the valuation of the iTraxx tranches and the optimization of the other model parameters. For this we use weekly data of the 5, 7 and 10 years iTraxx Europe tranches and compute the sum of the absolute error between the quoted and the model prices.
4
We use the R package “HiddenMarkov”.
8.4 Calibration of the Crash-NIG Copula Model
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For the tranches 3–6%, 6–9%, 9–12% and 12–22% the spreads are expressed in bp and the errors weighted with the weight 1. For the equity tranche 0–3%, the upfront fee is expressed in % and the error is weighted with the weight 0.1 to avoid its domination over the other tranches. For the optimization we use the Matlab function “fminsearch”. Although, this is a local minimization algorithm, the convergence of the optimization problem is very good. We have tested it with different starting points and found the algorithm always converging to the same values. The results of the estimation of the two-state model with the 5 and 10 year iTraxx spread and equity tranche base correlation data with the Baum–Welch algorithm are presented in the Table 8.1. In all cases, the probabilities of staying in the same state are very high with more than 98%. The probability of changing from the first state to the second state is approximately 0.4% for the iTraxx spread data and even lower with 0.14% for the correlation data. The probability of changing from the second state to the first is with over 1% higher for the spread data. The model estimated with the base correlation data do not return from the second state to the first. For all versions, the first state is associated with the lower values of the spread and the base correlation, and the second state with the higher values.5 This can also be seen in Fig. 8.12 presenting the Viterbi most likely sequence of the states based on the three model estimations with the 5 and 10 year data. Besides of the three state sequences we have also plotted the iTraxx index spread and the base correlation of the equity tranche that were used to calibrate the Hidden Markov Model. Thus, we can better compare the change of the states with the evolution of
Table 8.1 Parameter of the two-state model with 5-year data Data Transition matrix 0:9962 0:0038 5-year spread normal 0:0119 0:9881 0:9959 0:0041 5-year spread lognormal 0:0131 0:9869 0:9986 0:0014 5-year 0–3% correlation normal 0 1 0:9961 0:0039 10-year spread normal 0:0103 0:9897 0:9961 0:0039 10-year spread lognormal 0:0099 0:9901 0:9986 0:0014 10-year 0–3% correlation normal 0 1
Initial distribution 10
01 10 10 10 10
Distribution parameters 32:78I 6:48 74:79I 30:43 3:44I 0:19 4:13I 0:38 0:15I 0:04 0:36I 0:09 52:10I 6:25 86:18I 25:64 3:94I 0:12 4:41I 0:26 0:125I 0:03 0:35I 0:10
5 Distribution parameters in the table are organized as follows: the first column contains and the second of the distribution, the first row represents the first state and the second row the second state.
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8 Regime-Switching Extension of the NIG Factor Copula Model 160
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(b) Calibration with the 10-year data Fig. 8.12 Viterbi states in the two-states model
the observed processes. As already mentioned, we use the base correlation data only for the comparison of the states of the estimated HMM. The state sequences based on the normal 5 and 10 year spreads and the lognormal 10-year spreads are similar. They all detect the second state during the market turbulences after the downgrade of Ford and General Motors in May 2005, and during the sub-prime crisis starting in July 2007, with a short break in
8.4 Calibration of the Crash-NIG Copula Model
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September–October 2007. The major difference of the three sequences are in the length of the turbulence in May 2005. It is longer in the sequence corresponding to the 10-year log-normal spread than to the 10-year normal spread data, and that of the sequence corresponding to the 5-year normal spread data is even shorter. The 5-year log-normal spread yields an additional period of the second state at the very beginning of the history, in June–August 2004. The state sequences of the 5 and 10 year base correlations are identical and quite unspectacular with the only change from the first to the second state in July 2007. This also explains the zero probability of returning to the first state based on this data. Before the calibration of the parameters of the other Crash NIG factor copula model can be performed, the segment state probabilities must be computed. Those are compared in Figs. 8.13 and 8.14. These graphs are to be read in the following manner. The first row of plots assumes the first state at the beginning of the time segment, and the second row the second state at the beginning of the time segment. The three Hidden Markov Models calibrated to the three data series are organized in the three columns: the calibration to the iTraxx spread with normal and log-normal distribution, and to the base correlation with normal distribution assumptions. For each time t on the x-axis of the plots, the black area gives the overall probability of the first state during this time segment, and the white area, showing the probability of the second state, fills it up to one. The formulas for the probabilities are: hri .t/ ; t
(8.31)
with r denoting the initial state and i the state during the time period. Figure 8.13 corresponds to the calibration of the models with the 5-year data and Fig. 8.14 with the 10-year data. The results of the four normal and log-normal spreads models are similar, with the probabilities of the first state a bit lower for the 10-year data. Note that the plots for the different initial states differ significantly only for the time segments up to 5 years. For the initial state one, it is first more probable to be in the state one. For the longer time segments, this probabilities decreases to some asymptotic level. For the initial state two this is the other way around. The asymptotic probability of the first state is a bit higher for the initial state one than for the initial state two. These asymptotic probabilities a slightly higher for the 5-year data than for the 10-year data. For the base correlation data the behavior of these segment probabilities is quite different. Given the initial state one, the probability of being in the state one is decreasing very fast in time. Given the initial state two, the probability of the state one is zero since the probability to return there is zero. Table 8.2 reports the calibration results of the six two-state Crash NIG models. Although the calibration was performed with the weekly data to save the computation time, the absolute pricing errors were computed for all 971 data points. This error is the daily average of absolute deviations of the market quoted spread and the model spread in % over the five tranches and three maturities. An average absolute
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8.4 Calibration of the Crash-NIG Copula Model
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Table 8.2 Parameters of the two-state Crash-NIG copula model Data 5-year 5-year 10-year normal log-normal normal spread spread spread ˛ 0.1717 0.1753 0.2121 a1 0.3869 0.2038 0.4545 0.4493 0.4623 0.5850 a2 0.4494 0.4623 0.5448 a3 0.2820 0.2870 0.2791 a4 a5 0.2541 0.2236 0.1498 2.1141 2.0551 1.6240 Average error (%)
25.76
25.79
25.27
10-year log-normal spread 0.2100 0.3523 0.5137 0.5137 0.2466 0.1648 1.8495 25.08
Table 8.3 Average absolute deviations of tranche model spreads from the market spreads for the time period from 21 March 2005 to 8 May 2008: Crash NIG with 2 states estimated with the 5-year log-normal spread Maturity 0–3% 3–6% 6–9% 9–12% 12–22% 5 years 7.1101% 41.60 bp 15.18 bp 14.30 bp 8.69 bp 7 years 7.8196% 81.42 bp 25.95 bp 16.65 bp 10.23 bp 10 years 6.5018% 128.71 bp 52.35 bp 28.24 bp 12.50 bp
deviation of approximately 25% for 15 quotes seem to be quite high, however the greatest part of it is due to the three equity tranches that are quoted in the term of the up-front fee. We see this on the example of the 5-year log-normal spread model in Table 8.3, that reports the average absolute error for each tranche individually. All models estimated with the HMM calibration on the iTraxx spread produce similar results. First of all, we want to look at the calibration errors and investigate the reason for these. Figure 8.15 shows the errors of the 5-year tranches of the two-state Crash-NIG copula model estimated with the 5-year log-normal spread data. In particular, the attention should be paid to numerous points in time, where the differences between the market and model spreads of all tranches are negative. This means that the model spreads are higher than the market quotes for all tranches. To give an example, we pick up the values for the 20th of August 2007, and present them in Table 8.4. All model spreads are much higher for all tranches with all maturities. Actually, we would not be able to fit the market quotes with the model even with an individual calibration for this day. It is impossible to produce such low prices with any set of parameters. The reason for this is far too high default probabilities implied by the rating spreads we have used. On this point, it is time to note that so far we have not incorporated any liquidity premium into the model. Actually, the risk-neutral default probabilities were calculated from the average rating CDS spreads without any deduction for the liquidity premium. Valuation of the liquidity premium is a very complex subject with no good data sources. Figure 8.16 shows a graph with Libor credit liquidity premium data
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8 Regime-Switching Extension of the NIG Factor Copula Model Sep04
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Fig. 8.15 Deviations of the 5-year tranche model spreads from the market spreads: Crash NIG with 2 states estimated with the 5-year log-normal spread Table 8.4 Market quotes vs. two-state Crash NIG one factor copula model prices for the 20th of August 2007 Maturity 0–3% 3–6% 6–9% 9–12% 12–22% Market quotes 5 years 25.75% 100.00 bp 54.50 bp 34.00 bp 17.50 bp 7 years 36.50% 180.00 bp 93.00 bp 56.00 bp 26.50 bp 10 years 45.75% 462.50 bp 170.00 bp 89.00 bp 49.00 bp Model prices
5 years 7 years 10 years
33.49% 50.59% 58.66%
130.80 bp 299.95 bp 605.88 bp
59.19 bp 102.45 bp 228.36 bp
Fig. 8.16 Libor credit liquidity premium (Source: Lehman Brothers)
40.35 bp 60.25 bp 116.79 bp
26.03 bp 33.47 bp 54.55 bp
8.4 Calibration of the Crash-NIG Copula Model
215
available from Lehman Brothers. Unfortunately, Lehman Brother does not provide the data sheet of it but only the graph. It shows the high liquidity premiums during the sub-prime crisis. However, liquidity premiums are usually quite different for various market segments, single CDS market, iTraxx index and iTraxx tranches. Our findings below reflect that the liquidity premium in the single CDS market is higher than that of the iTraxx tranches. Liquidity premiums can be incorporated into the model in a simple way by deducing a fixed percentage of the credit spread. So we introduce the liquidity indicators lr , with r D 1; 2 the current state, such that the part of the credit spread representing the credit quality is lr times the spread. Then the liquidity premium is .1 lr / times the spread. Now the default probabilities that are used in the CrashNIG copula model are not computed based on the complete spread but only on the part of it cleaned from the liquidity premium. We assume these liquidity indicators to be constant in the same state of the market and estimate them together with the other model parameters. Table 8.5 presents the calibration results of the two-state Crash-NIG copula model with liquidity coefficients. The detailed errors for the case of 5-year normal spreads are given in Table 8.6. The calibration error could be reduced significantly with the help of the liquidity coefficients. Still it is quite high, and Fig. 8.17 shows that the errors are especially high after December 2007. So two states seem to be not enough to describe the history of the last 4 years. This gives us a hint that the two
Table 8.5 Parameters of the two-state Crash-NIG copula model with liquidity coefficients Data 5-year 5-year 10-year 10-year normal log-normal normal log-normal spread spread spread spread ˛ a1 a2 a3 a4 a5 l1 l2
0.3917 0.1680 0.4275 0.4275 0.1767 0.1705 2.2220 0.9439 0.7330
0.3669 0.1624 0.4226 0.4225 0.1781 0.1870 2.2482 0.9523 0.7678
0.3538 0.3798 0.4564 0.4366 0.1823 0.2041 2.0816 0.9541 0.7597
0.3957 0.4151 0.4459 0.4410 0.1630 0.1926 2.1305 0.9547 0.7691
Average error (%)
18.13
19.01
18.99
19.20
Table 8.6 Average absolute deviations of tranche model spreads from the market spreads for the time period from 21 March 2005 to 08 May 2008: Crash NIG with 2 states and liquidity estimated with the 5-year normal spread Maturity 0–3% 3–6% 6–9% 9–12% 12–22% 5 years 4.6751% 23.67 bp 16.35 bp 14.46 bp 9.22 bp 7 years 5.5094% 44.13 bp 24.09 bp 17.22 bp 11.64 bp 10 years 4.7840 % 79.25 bp 39.89 bp 24.21 bp 12.06 bp
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Fig. 8.17 Absolute calibration errors of the two-states Crash-NIG copula model with liquidity coefficients
observed crises are actually of a different nature and have different characteristics. We can find another confirmation of this in Fig. 8.12 if we compare the evolutions of the iTraxx spread and the base correlation. The correlations were moving in the opposite direction than the spreads until July 2007. Also, during the small crisis in May 2005, the correlation was falling as the spread was growing. Since the beginning of the sub-prime crisis in July 2007, the correlation changed its behavior. Since then it was growing or falling together with the spread. On this point we can make a conclusion of the calibration results of the two-state Crash-NIG copula model. All two-state Hidden Markov Models estimated with the iTraxx spread data determined the second state, corresponding to the crisis in May 2005, and from July 2007. This turned to be not a good choice since the two events have different characteristics. So, two states are actually not enough to describe the 4 year iTraxx history from June 2004 to May 2008. The two crisis that took place during this time were of different nature, the first one was a smaller branch-specific crisis while the second one has become a huge global market crisis. In the next section we calibrate the Crash-NIG copula model with three possible states.
8.4.3 Calibration of the Model with Three States Analogue to the calibration procedure of the two-state Crash-NIG copula model, the first step is the estimation of the Hidden Markov Model on the 5 and 10 year iTraxx spreads with the assumptions of normal and log-normal distributions.
8.4 Calibration of the Crash-NIG Copula Model Table 8.7 Parameters of the three-state model with 5-year data Data Transition Initial matrix distribution 0 1 0:9953 0:0047 0 @ 0:0163 0:9787 0:005 A 5-year spread normal 010 0 0 1 0 1 0:9863 0:0137 0 @ 0:014 0:9773 0:0087 A 5-year spread lognormal 001 0:0125 0 0:9875 0 1 0:9925 0:005 0:0025 @ 0:0069 0:9931 0 A 5-year 0–3% correlation 100 normal 0 0 1 0 1 0:9843 0:0157 0 @ 10-year spread normal 010 0:0110 0:9823 0:0067 A 0 0:0101 0:9899 1 0 0:9843 0:0157 0 @ 0:0112 0:9820 0:0068 A 010 10-year spread lognormal 0 0:0101 0:9899 0 1 0:9986 0:0014 0 @0 10-year 0–3% correlation 100 0:9908 0:0092 A normal 0 0 1
217
Distribution parameters 0 1 31:25I 5:54 @ 46:50I 5:92 A 99:34I 26:69 0 1 3:28I 0:14 @ 3:60I 0:05 A 4:10I 0:37 0 1 0:17I 0:01 @ 0:11I 0:02 A 0:36I 0:09 0 1 45:61I 3:22 @ 56:54I 2:99 A 85:72I 25:56 0 1 3:82I 0:07 @ 4:03I 0:05 A 4:41I 0:12I @ 0:28I 0:45I 0
0:26 1 0:03 0:04 A 0:06
Again, we estimate the model with the base correlation time series of the 0–3% equity tranche as well. The transition probabilities, initial distribution and the distribution parameters estimated with the Baum–Welch algorithm are reported in the Table 8.7. In all cases the probabilities of staying in the same state are very high with more than 97%. The probability to stay in state three is equal to one in the case of the 5-year normal spread as well as 5 and 10 year correlation. All models except of the 5-year log-normal spread can change from state one only to state two directly. All models estimated with the spread data have the first state when the expected value of the spread is the lowest. The expected value of the spread is the highest in the third state. So the first state is the quiet state of the market, the second state is a bit turbulent and the third is the crisis state. This can also be seen in Fig. 8.18 where the most likely states estimated with the Viterbi algorithm are plotted. However, only the states of the 5 years normal spread are exactly as discussed in the previous section: May 2005, is recognized as state two, together with June–August 2004 and July and September 2007. The rest of the history since December 2007 is estimated as the third state. The 5 and 10 year log-normal spreads and the 10 year normal spread put May 2005 to the same, third, state with the crisis after July 2007. The states of the 5-year correlation results have exactly the characteristics we have discussed in the previous section: the mean of the correlation is the lowest in the second state and the highest in the third state. Indeed, the Viterbi state in May 2005 of the 5 year correlation model is the second. However, different to the 5 year normal spread, this model has the second state also over a long time segment from
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8 Regime-Switching Extension of the NIG Factor Copula Model 160
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(b) Calibration with the 10-year data Fig. 8.18 Viterbi states in the three-states model
September 2005 until October 2006. The third state starts already in July 2007. The 10-year correlation model states are different. Here, May 2005 is considered as a first, normal, state together with the complete time segment from August 2004 until July 2007. July to December 2007 is classified as the second state. And the spread explosion after December 2007 is recognized as the third state.
8.4 Calibration of the Crash-NIG Copula Model Lognormal spread 1
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Fig. 8.19 Probabilities of the states on increasing time segments: three-state models on the 5-year data
The effects of the transition matrices of the estimated Hidden Markov Models are clearer when looking at the segment state probabilities in the Figs. 8.19 and 8.20. These plots are organized in the following way. The first row gives the probabilities of the three states over the time segment Œ0; t with increasing t starting from state one at time zero. The second row assumes that the initial state at time zero is two, and the third row state three. First, we examine the probabilities of the 5 and 10 year log-normal spreads and the 10 year normal spread. Here, the segment probabilities of the three states, given different initial states, differ from one another only during the first 2 years. Afterwards, they converge very fast to asymptotic levels that are very similar for the three initial states. This means that the three states probably cannot have a big impact on the CDO pricing. In contrast, the segment probabilities of the 5-year normal spread as well as of the both correlations are absolutely different, given different initial states. Starting in the first state, the probability of being in the first state during the time segment Œ0; t is decreasing continuously. The probability of the second state is almost the same,
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8 Regime-Switching Extension of the NIG Factor Copula Model Lognormal spread
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Fig. 8.20 Probabilities of the states on increasing time segments: the three-state models on the 10-year data
quite low, for any time segment, while the probability of the third state continuously increases. Given the initial state is the second, the picture is quite similar with a much higher probability of the second state at the beginning and the overall level of the first (third) state being lower (higher). For the 10-year correlation, the probability of the first state is even zero while the probability of the third state increases very fast up to almost one. Conditional on the third state as the initial one, the three models stay in this third state with probability one. Since the conditional state probabilities are so different for these models, we would expect the prices of the CDO tranches to be quite different from those of the model with only one state. The estimated values of the parameters of the three state models are presented in Table 8.8. The parameter ˛ of the NIG distribution is similar for all versions of the model. The five correlation parameters a1 ; : : : ; a5 do not differ much as well, having the highest value for a2 , the correlation parameter in the rating cell AA, and the lowest for a5 , in the rating cell BB. The two factors 1 and 2 giving the reduction
8.4 Calibration of the Crash-NIG Copula Model
221
Table 8.8 Parameters of the three-state Crash-NIG copula model Data 5-year 5-year 10-year normal log-normal normal spread spread spread ˛ 0.3274 0.3287 0.3476 a1 0.2562 0.3926 0.3167 0.5437 0.4247 0.4534 a2 0.3429 0.4247 0.4461 a3 0.2130 0.2150 0.2006 a4 a5 0.0828 0.1795 0.1739 0.2353 0.6773 0.7375 1 1.7443 2.2369 2.0920 2 0.9679 0.9122 0.9971 l1 l2 0.8827 0.9867 0.9394 0.7361 0.7679 0.7717 l3 Average error (%)
14.80
18.99
19.08
10-year log-normal spread 0.3728 0.4001 0.4168 0.4168 0.1899 0.1917 0.6350 2.2790 0.9792 0.9198 0.7589
One State model 0.3615 0.2476 0.9607 0.4975 0.3256 0.1161
19.05
23.98
0.9562
or increase in correlation in the second and the third states vary across the models. The value of 1 is below one for all models meaning the reduction in correlation in the second state. The lowest value of 0.23 has the model with the 5-year normally distributed spread. The other three versions have much higher values of 0.63–0.73. The liquidity coefficients l1 ; l2 ; l3 are similar for all the models. All the models, except of the 5 year log-normal spread, have the highest liquidity of 0.95–0.99 in the first state. The liquidity is a bit lower with 0.88–0.92 in the second state. The third state representing the global crash has the lowest liquidity of 0.73–0.77. In particular, this value means that around 23–27% of the credit spread is the liquidity premium and only 73–77% represent the price for the default protection. The model estimated with the 5-year normally distributed spreads performs much better than the others. This is actually exactly what we expected, since the probabilities of being in different states, reported in Figs. 8.19 and 8.20, for 5-year normal spread are very different for different initial states, while these probabilities for the other three models, estimated with the spread data, are only different for the first 2 years converging afterwards to the asymptotical values, that are very similar for all initial states. We have also calibrated the LHC NIG model with only one state to be able to evaluate the added value of the Crash-NIG copula model. The absolute error of 23.98% is much higher for this model than the error of all three-state models. The problem of the one-state model to deal with a crash scenario can be also seen on the values of the estimated correlation parameters. The correlation parameter a2 of the AA rating cell is estimated to be 0.96, which is rather typical for the crash state and not for the normal correlation state. The average absolute deviations for each tranche and maturity are reported in Table 8.9 for the best performing model estimated with 5-year normal spreads. This is actually the breakdown of the overall error of 14.80% over the 15 tranches with different maturities. For comparison, the breakdown of the error over the 15 tranches with different maturities is also presented for the one-state model in Table 8.10. As
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Table 8.9 Average absolute deviations of tranche model spreads from the market spreads for the time period from 21 March 2005 to 8 May 2008: Crash-NIG copula model with 3 states and liquidity estimated with the 5-year normal spread Maturity 0–3% 3–6% 6–9% 9–12% 12–22% 5 years 3.36% 19.62 bp 13.93 bp 10.52 bp 4.64 bp 7 years 4.43% 42.33 bp 19.34 bp 13.73 bp 7.62 bp 10 years 4.35% 61.36 bp 39.87 bp 23.21 bp 8.33 bp Table 8.10 Average absolute deviations of tranche model spreads from the market spreads for the time period from 21 March 2005 to 8 May 2008: NIG copula model with one state Maturity 0–3% 3–6% 6–9% 9–12% 12–22% 5 years 6.45% 36.09 bp 12.46 bp 12.11 bp 7.44 bp 7 years 7.48% 74.62 bp 20.80 bp 12.29 bp 8.50 bp 10 years 6.36% 116.41 bp 42.30 bp 17.02 bp 9.79 bp 150 normal spread 5y one state model
error (%)
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Fig. 8.21 Absolute calibration errors of the three-states Crash NIG model with liquidity coefficients: comparison of the best three-state model with one-state model
we can see, the average absolute deviation is higher for the one-state model almost for all tranches. Besides, we also present the comparison of the two models for the 12th of April of 2006 (Table 8.11), for which we compared the Gaussian and LHC NIG model in Table 7.1. The overall absolute error is twice higher for the one state model. Figure 8.21 shows the absolute calibration error of the models during the history. The plot compares the error of the Crash-NIG copula model estimated with the 5-year normal spread data with the error of the plain NIG model without crash.
8.4 Calibration of the Crash-NIG Copula Model
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Table 8.11 Pricing iTraxx tranches with different maturities with the Crash-NIG model with 3 states and the NIG model with one state on 12th of April 2006 Maturity (years) 5 7 10 iTraxx spread 32 bp 41 bp 52 bp AAA spread 10.19 bp 13.75 bp 17.00 bp AA spread 14.51 bp 19.40 bp 24.74 bp A spread 24.68 bp 32.95 bp 41.41 bp BBB spread 44.68 bp 62.09 bp 68.51 bp Market
0–3% 3–6% 6–9% 9–12% 12–22%
23.53% 62.75 bp 18 bp 9.25 bp 3.75 bp
36.875% 189 bp 57 bp 26.25 bp 7.88 bp
48.75% 475 bp 124 bp 56.5 bp 19.5 bp
Crash NIG
0–3% 3–6% 6–9% 9–12% 12–22%
20.60% 79.70 bp 27.79 bp 14.85 bp 6.41 bp
37.39% 225.25 bp 74.04 bp 36.22 bp 14.08 bp
48.01% 434.67 bp 171.97 bp 83.06 bp 30.16 bp
Absolute error All tranches 2nd–5th tranches
328.03 bp 35.01 bp
120.89 bp 69.47 bp
199.94 bp 125.62 bp
0–3% 3–6% 6–9% 9–12% 12–22%
17.58% 75.97 bp 31.39 bp 22.515 bp 13.96 bp
35.06% 195.08 bp 66.41 bp 39.28 bp 23.61 bp
47.28% 414.40 bp 148.74 bp 70.54 bp 35.20 bp
Absolute error All tranches 2nd–5th tranches
644.82 bp 50.09 bp
225.83 bp 44.24 bp
262.11 bp 115.07 bp
One state NIG
648.86 bp 230.10 bp
1132.76 bp 209.40 bp
The Crash model performs better during almost all the history. In particular, the advantage of the model is especially high during the crash period. Figure 8.22 splits the aggregate error up for the individual tranches with 5-year maturity. Comparing the figure to the case of the two-state model without liquidity coefficient in Fig. 8.15, we can see that there are no days anymore when all tranches were overpriced simultaneously. The errors of the higher tranches are very small with only some basis points during almost all the time. During the normal correlation state, the error of the equity tranche is also not very high with up to 5%. The errors are higher during the crash period for all tranches. Of course, the quoted spreads are also much higher during the crash. For this reason we take a look on the relative errors for all tranches and maturities in Fig. 8.23. Relative errors are calculated as a deviation of the model tranche fair spread from the quoted one divided with the quoted spread. So, for example, the relative error for a model spread of 11 bp of a tranche quoted with 10 bp would be 10%. The relative errors are in
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8 Regime-Switching Extension of the NIG Factor Copula Model Sep04
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Fig. 8.22 5-year tranche errors of the Crash-NIG copula model with three states estimated with the 5-year normal spread 5 year tranches relative error (%)
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Fig. 8.23 Relative tranche errors of the Crash-NIG copula model with three states estimated with the 5-year normal spread
8.4 Calibration of the Crash-NIG Copula Model Sep04
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Fig. 8.24 5-year iTraxx quotes vs. three-state Crash-NIG copula model prices
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Fig. 8.25 7-year iTraxx quotes vs. three-state Crash-NIG copula model prices
absolute value not higher for the crash period in comparison to the normal correlation period. The relative errors of the two most junior tranches are even very often lower then those of the more senior tranches. Of course, the errors do not look like white noise. However, there are no errors that are systematically to high or too low. Figures 8.24–8.26 show the quoted versus the model spreads. Here we also can see
8 Regime-Switching Extension of the NIG Factor Copula Model
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Fig. 8.26 10-year iTraxx quotes vs. three-state Crash-NIG copula model prices
that the fitting ability of the Crash-NIG copula model is very good. A model with more states could possibly fit the data even better, but one should also be aware of potential overfitting. We will employ the three-state Crash-NIG copula model with the parameters estimated with the 5 year normally distributed spread data for our further asset allocation analysis in the next chapter.
Chapter 9
Simulation Framework
In this section we develop a simulation framework that includes all factors necessary to model a portfolio of traditional credit instruments and synthetic CDOs: Rating migration and default model Credit spread model Interest rate model.
Such a simulation framework is useful in the application fields like portfolio optimization and asset-liability management. First, a risk-factor scenarios are simulated. Afterwards, the present values of various instruments can be computed along the simulated paths. Finally, taking into account the development of present values and cash flows of the instruments along the simulated paths, a portfolio optimization can be performed and an optimal asset allocation can be determined. As we have already discussed, the possibility of different correlation regimes in the Crash-NIG copula model is not that important for a pure pricing application. For pricing of CDO tranches, the correlation coefficients can be simply adjusted on every pricing day by a new calibration. In contrast, different correlation regimes are very important for a scenario simulation. Such scenarios take into account the possibility of market turbulence with correlation shifts like we could observe currently. The big advantage of our simulation framework is that the same model can be used for both simulation and pricing: the Crash-NIG copula model is powerful enough to be able to generate realistic simulation paths and at the same time admits a semi-analytical solution for the fair value computation. The simulation framework allows to use both a one-period setting in the case when only the distributions at some final horizon are of interest, and a multi-periodsetting when the complete simulation path is used, e.g. for an investment strategy. We consider time discretisation 0 t0 < t1 < < tk1 < tk < < tn and denote the time increments by tk D tk tk1 :
A. Schl¨osser, Pricing and Risk Management of Synthetic CDOs, Lecture Notes in Economics and Mathematical Systems 646, DOI 10.1007/978-3-642-15609-0 9, c Springer-Verlag Berlin Heidelberg 2011
227
228
9 Simulation Framework
9.1 Rating Migration and Default Model The migration of credit instrument i with a rating j within the time increment dt is driven by the realization of the asset return that follows the one-factor Crash-NIG copula model q dAij .t/ D aj dM.t/ C
1 aj2 dXij .t/;
(9.1)
where dM.t/ is the return of the market factor and dXij .t/ the return of the idiosyncratic factor. aj is the correlation coefficient of the asset return to the market factor for the rating cell j . All factors are assumed to be independent and NIG distributed with following parameters ˇ 2 3 ˛; ˇ; 2t 2 dt; 2t 2 dt ; ˛ ˛ 0q q 2 1 aj2 B 1 aj ˛; ˇ; dXij .t/ N I G @ aj aj dM.t/ N I G
1 2t aj2 1 aj2
(9.2)
1 q q 2 2 2 1 aj2 ˇ 2 1 a 3 1 t aj j C dt; dt A : (9.3) 2 2 2 aj ˛ aj ˛ 1 aj
t is a Markov process with the state space f1; 1 ; 2 g, an initial distribution D f1 ; 2 ; 3 g and a transition function P .u/. So the NIG model without regimeswitching extension is a special case of this model, if only one regime with t D 1 is considered. The distribution of the increment of the asset return is 1 1 ˇ 2 1 3 1 ˛; ˇ; dt; dt : dAij .t/ N I G aj aj aj ˛ 2 aj ˛ 2 Recall, that the Crash-NIG copula model was defined in the Sect. 8.2 in such way that it can be discretized in an arbitrary way and the distributions of the increments of all factors are stable under convolution. The parameters of the model are the same for any time horizon since the time component is taken into account by dt. To simplifynotations we denote the distribution function FN I G .xI s˛; sˇ; 2 3 s ˇ t; s ˛2 t with FN.s/ .t / .x/. So the distribution function of the asset return ˛2 Ai .tkC1 / on the time increment tkC1 is FN 1 .tkC1 / .x/. .a
j
/
Now we consider the rating migration of the credit instrument i within the time increment tk . Given its rating Rj at time tk , the rating migration of the credit instrument depends on the realization of the variable Aij .tkC1 /. The rating at time tkC1 is Rh if the variable Aij .tkC1 / lies in the bucket .bjh ; bjh1 . The migration thresholds bjh , h D 1; : : : ; J 1 are calibrated to the migration probabilities:
9.1 Rating Migration and Default Model
229
Fig. 9.1 Determination of the rating migration buckets
P ŒAij .tkC1 / > bj1 D 1 FN
P Œbjh1
::: Aij .tkC1 / > bjh D FN
.tkC1 / . a1 / j
.tkC1 / . a1 / j
.bj1 / D P .Rj ;R1 / .tkC1 /
.bjh1 / FN
.tk / . a1 / j
.bjh / (9.4)
D P .Rj ;Rh/ .tkC1 / ::: P ŒbjK1 Aij .tkC1 / D FN
.tkC1 / . a1 / j
.bjK1 / D P .Rj ;RK / .tkC1 /
The real-world migration probabilities P .Rj ;Rh / .tkC1 / can be computed from the migration matrices as described in Sect. 3.4. The idea of the determination of the rating transition bounds is visualized in Fig. 9.1. The buckets are defined so that the corresponding areas under the distribution function fN 1 .tkC1 / .x/ equal the .a
j
/
migration probabilities. Simulation of the rating migrations is performed in the following way. First, the states of the Markov process t are simulated on a daily grid. Then the duratkC1 R tion stays for the time segments Tir .tkC1 / D 1fstate i at time sg ds are computed tk
for each path. Afterwards, the increments of the market factor M.tkC1/ and the idiosyncratic factor Xij .tkC1 / are simulated from the NIG distributions given by Proposition 8.2 and Remark 8.3:
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9 Simulation Framework
ˇ 2 ˛; ˇ; T1r .tkC1 / C 21 T2r .tkC1 / C 22 T3r .tkC1 / ; ˛2 ! 3 r 2 r 2 r T1 .tkC1 / C 1 T2 .tkC1 / C 2 T3 .tkC1 / 2 ; ˛ 0q q 1 aj2 1 aj2 B Xij .tkC1 / N I G @ ˛; ˇ; aj aj M.tkC1 / N I G
q 1 aj2 ˇ 2 t aj2 T1r .tkC1 / C 21 T2r .tkC1 / C 22 T3r .tkC1 / ; aj ˛2 1 aj2 1 q 1 aj2 3 t aj2 T1r .tkC1 / C 21 T2r .tkC1 / C 22 T3r .tkC1 / C A: aj ˛2 1 aj2
Now, the asset return can be computed according to Aij .tkC1 / D aj M.tkC1 / C
q
1 aj2 Xi .tkC1 /;
(9.5)
and a new rating can be now assigned, depending on the bucket where the return lies. For the correct understanding of the factor copula models, it is important to note the important difference to a structural models like Merton’s. In the structural model, the asset process is considered. A default occurs if the asset process goes below the default threshold. In the factor copula model, the market and individual factors as well as the asset return are only fictive variables that are used only for defining a correlation structure for a portfolio of issuers. Here, not the asset process is considered but the asset return, and the default threshold is implied by the marginal default probability. On each time increment of the simulated path, we consider the incremental asset return and decide on the rating transition or default according to the probability buckets given by the transition matrix. If the asset return of a certain issuer indicated default, the asset is removed from the portfolio. On the next time increment, the same procedure is applied to the survived portfolio assets. So fraction of defaulted assets up to each time step is exactly as given by the default term structure of the transition matrix. We choose to simulate the states on a fine grid and compute the duration stays from them since the distributions of the duration stays are too complex to be able to simulate them directly on the discretisation ftk gkD1;:::;n . Besides, we use the paths of states for the credit spread simulation described below as well. Before we continue with presenting the models for the other risk factors and the case study, we want to investigate the difference between the Crash-NIG and the one-state NIG models. We assume ˛ D 0:35 and the correlation parameters for all ratings in thefirst regime to be 0.2. We consider a two-state model with 0:98 0:02 and D 1; 2; 3; 4; 5. Please note, that with D 1 the model P .1/ D 0:02 0:98
9.1 Rating Migration and Default Model
231
1 λ=1 λ=2 λ=3 λ=4 λ=5
0.9 0.8
default probability
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4 6 tranche attachment point (%)
8
10
Fig. 9.2 Tranche default probability in a simulation with the Crash-NIG and one-state models
has actually only one state, and with D 5 the portfolio is perfectly correlated in the second regime. We simulate the rating transition of a large portfolio (600 issuers) with five equally weighted rating cells from AAA to BB over a time horizon of 5 years starting in the first correlation regime. To compare the five simulated portfolios, we calculate the probability to default for senior tranches with different attachment points (from zero to 10%) (see Fig. 9.2). This means that for each attachment point on the x-axe, we compute a quote of paths where the tranche suffered defaults. Since the overall average default probability in the portfolio is the same for all models and is implied by the rating migration matrix, the influence of different correlation values can be seen on the probability of simultaneous defaults. As we would expect, there are more simultaneous defaults that hit upper tranches (with attachment point over 3%) for the model with the higher . For the lower attachment points, this probability is lower for higher since the probability of no defaults is higher. We have also plotted the histograms of the portfolio default rate (Fig. 9.3) for the two extreme cases: for D 1, which is the one-state model, and for D 5, which leads to the perfect correlation in the second regime. Here we also can see, that the probability of no defaults is higher for the D 5 case as well as the probability of a high number of simultaneous defaults.
232
9 Simulation Framework 4000 λ=1 λ=5
3500 3000 2500 2000 1500 1000 500 0
0
10
20
30 40 50 Default rate of the portfolio(%)
60
70
80
Fig. 9.3 Portfolio default rate distribution after 5 years
9.2 Interest Rate Model The dynamics of the risk free nominal short rate r is described by the Vasicek model: dr.t/ D r . r r.t// dt C r d Wr .t/;
(9.6)
with kr and r positive constants, r a non-negative constant and Wr .t/ a Brownian motion. The Brownian motion can be correlated to the market factor M.t/: r D C orr.d Wr .t/; dM.t//:
(9.7)
Since the distribution of the short rate in Vasicek’s model is known for any t, the simulation can be performed for an arbitrary time discretisation, in our case ftk gkD1;:::;n , and the grid should not necessarily be fine for exact results. The short rate r.tkC1 / conditional on Ftk is normally distributed with mean and variance given by E r.tkC1 /jFtk D r.tk /e r tkC1 C r 1 e r tkC1
2 V r.tkC1 /jFtk D r 1 e 2r tkC1 : 2r
(9.8)
9.3 Credit Spread Model
233
The short rate is simulated according to the formula: C r 1 e r tkC1 C
s
r2 1 e 2r tkC1 kC1 ; 2r (9.9) where kC1 are independent standard normal random variables. r.tkC1 / D r.tk /e
r tkC1
9.3 Credit Spread Model As we have seen already from the spread historical data, spreads have very different distributions during different market regimes. During a crises, spreads rise very fast to a much higher level and are very volatile. The affine models cannot incorporate these properties. We extend the Vasicek model to a regime-switching model by introducing an individual parameter set for each regime: r r r d Wj;i .t/; j;i sj;i .t/ dt C j;i dsj;i .t/ D kj;i
(9.10)
r r r with kj;i and j;i positive constants and j;i a non-negative constant, r D 1; : : : ; 3 the states of the economy and Wj;i .t/ Brownian motions. j D 1; : : : ; J is the index corresponding to the rating and i D 1; : : : ; L enumerate the different maturities. Spreads for different maturities Ti ; i D 1; : : : ; L must be modeled with an own process. The Brownian motions for different ratings and maturities are correlated according to
s D fC orr.d Wj1 ;i1 .t/; d Wj2 ;i2 .t//g1j1 ;j2 K;1i1 ;i2 L :
(9.11)
It is natural to assume that the Brownian motion of the spread Wj;i .t/ is negatively correlated to the market factor M.t/. In the periods of economic growth when then asset returns are high, the credit spreads are typically low since the credit quality of the firms is good. On the other hand, the asset returns are negative and the spreads high during the economic recession. The corresponding correlation is denoted by: s;M D fC orr.d Wj;i .t/; dM.t//g1j K;1i L :
(9.12)
Of course, the formulas for the mean and variance in (9.8) are not valid for the regime switching version of spreads. However, simulation on the finer grid can be avoided by approximating the exact mean and variance by the values corresponding to simulated paths for the states of the Markov process. For example, consider the finer discretisation tj1 ; tj2 ; : : : ; tjm of the time period Œtk ; tkC1 such that tj1 D tk and tjm D tkC1 . We assume that the economy is in state ri during the time period Œtji ; tji C1 , i D 1; : : : ; m 1. We omit the indices corresponding to the maturity and rating of the spread for simplicity of notations and consider a spread process s.t/. Starting with the value s.tj1 / D s.tk / in i D 1, we would compute s.tj2 / according
234
9 Simulation Framework
to (9.9): r1 C r1 1 e tj2 C
s
r1 2 2 r1 tj2 1 e 2 ; 2 r1 (9.13) where 2 is standard normal random variable. Repeating this recursion for i D 2, we get: s.tj2 / D s.tj1 /e
r1 tj2
0 s.tj3 / D @s.tj1 /e
e
r1 t
r2 t j3
C
j2
C
r2
r1
1e
1 e
r1 tj2
r2 t j3
s C
s C
1
r1 2 r 1 1 e 2 tj2 2 A 2 r1
r2 2 r 1 e 2 2 tj3 3 ; r 2 2
(9.14)
with 3 standard normal. So, the mean of s.tj3 / conditional on s.tj1 / and r1 and r2 can be computed recursively: E s.tj3 /js.tj1 /; r1 ; r2 (9.15) r r r r 1 1 2 2 D s.tj1 /e tj2 C r1 1 e tj2 e tj3 C r2 1 e tj3 r2 r2 D E s.tj2 /js.tj1 /; r1 e tj3 C r2 1 e tj3 : The variance of s.tj3 / conditional on s.tj1 / and r1 and r2 is: V s.tj3 /js.tj1 /; r1 ; r2 (9.16) 2 2
r2
r1 r1 r2 r2 D r 1 e 2 tj2 e 2 tj3 C r 1 e 2 tj3 2 1 2 2 r2 2 2r2 t
r2 j3 C r 1 e 2 tj3 : D V s.tj2 /js.tj1 /; r1 e 2 2 We can continue this recursion until tkC1 is reached and compute E.s.tkC1 /js.tk /; r1 ; : : : ; rm1 / and V .s.tkC1 /js.tk /; r1 ; : : : ; rm1 /. Then the value of the spread at tkC1 is simulated according to s.tkC1 / D E .s.tkC1 /js.tk /; r1 ; : : : ; rm1 / C
p
V .s.tkC1 /js.tk /; r1 ; : : : ; rm1 / kC1 ; (9.17)
where kC1 is standard normally distributed and correlated to the return of the market factor according to (9.12). In this way, the realisations of the states of the simulation on the daily grid can be taken into account without simulating daily paths of the spread processes. This trick makes the simulation procedure faster.
9.4 Case Study
235
9.4 Case Study 9.4.1 Model Calibration 9.4.1.1 Rating Migration and Default Model The simulation will be performed with the three state Crash NIG model with parameters estimated in the previous chapter. The best performing model with the Hidden Markov Model calibrated with the 5 year iTraxx spread under the assumption of a normal distribution is chosen for the simulation. Recall, that the calibrated transition matrix has a probability zero of getting out the third state which is the crash state. Since the data history available to us ends in the state three, the simulation results would not be very exciting if we start from this date. The simulated paths would just stay in the third state over the complete simulation. For this reason we choose to start our example simulation on the 20th of September 2007, when the model was in the second state. The Crash-NIG copula model parameters are given in the Table 9.1. The transition matrix P given in the table is for a one day period. We assume the real-world parameters are the same as the risk-neutral. The real-world rating transition and default probabilities are given by the S&P global transition matrix in Table 9.2. The threshold boundaries for the asset returns, computed from the rating transition matrix as described in the previous section, are presented in Table 9.3. The table has to be read in the following way. Given the current rating AAA, the firm stays AAA in the next time step (after a 3 month period) if its asset return is higher than 1:0899. The firm will be downgraded to AA if its asset return is between 3:1243 and 1:0899. The firm defaults if its asset return is below 10:3611.
9.4.1.2 Interest Rate Model In the Vasicek model, the link between the real-world and risk-neutral processes is given via the market price of risk r , such that r D r r
r : r
(9.18)
The parameters r and r are the same for the both measures. The parameters of the Vasicek short-rate model are given in Table 9.4. First we estimated the real-world parameters from the overnight historical rates from July 2004 to July 2008. Then the market price of risk was calibrated to the zero curve on 20th of September 2007.
236 Table 9.1 Parameter of the three-state Crash-NIG copula model
9 Simulation Framework 0 P
1 0:9953 0:0047 0 @ 0:0163 0:9787 0:005 A 0 0 1
˛
0.3274
a1
0.2562
a2
0.5437
a3
0.3429
a4
0.2130
a5
0.0828
1
0.2353
2
1.7443
l1
0.9679
l2
0.8827
l3
0.7361
Table 9.2 Global average 1-year transition rates, 1981–2007 AAA AA A BBB BB AAA 91.39% 7.95% 0.48% 0.09% 0.09% AA 0.62% 90.99% 7.62% 0.56% 0.06% A 0.04% 2.17% 91.49% 5.62% 0.41% BBB 0.01% 0.18% 4.24% 90.07% 4.31% BB 0.02% 0.06% 0.23% 5.90% 83.88% B 0.00% 0.06% 0.18% 0.32% 6.73% CCC/C 0.00% 0.00% 0.28% 0.42% 1.18% D 0.00% 0.00% 0.00% 0.00% 0.00%
B 0.00% 0.10% 0.17% 0.77% 7.93% 83.01% 13.60% 0.00%
CCC/C 0.00% 0.02% 0.03% 0.17% 0.87% 4.50% 54.89% 0.00%
D 0.00% 0.01% 0.06% 0.25% 1.11% 5.20% 29.64% 100.00%
9.4.1.3 Credit Spread Model To calibrate the parameters of the regime-switching credit spread model, the average credit spreads of the constituents of the iTraxx portfolio with ratings AAA, AA, A, BBB and BB are used. The data was described in Sect. 8.4. The parameter calibration is performed for the 5 and 10 year data. Since the calibration of a regimeswitching spread model is not straightforward, we set the mean-reversion level to the mean of the historical data in this state. The mean-reversion speed is set to 3
9.4 Case Study
237
Table 9.3 Rating migration boundaries of the Crash-NIG copula model AAA AA A BBB BB AAA 1.0899 3.1243 3.8377 4.3112 7.6247 AA 3.1094 1.0618 3.2093 4.4675 4.7996 A 4.8290 1.8477 1.2609 2.7696 3.4087 BBB 4.1760 2.7775 1.3468 1.2648 1.9484 BB 3.7857 3.1930 2.7024 1.2266 1.0480 B 1 3.3268 2.6596 2.4203 1.1685 CCC/C 1 1 2.5361 2.1334 1.7775 Table 9.4 Short rate Vasicek model parameters
r r
r r r.0/
B 9.2366 6.6708 4.2074 2.3933 1.7541 1.0615 0.8524
Real world 1.3331 0.0557 0.0136
CCC/C 10.3611 8.9032 4.5556 2.6743 2.0500 1.3616 0.6425 Risk neutral 0.0476 0.7940
0.0403
Table 9.5 Five-year credit spread parameter State AAA AA 1 5.8776 13.2783 2 9 21.3611 3 25 64.5190
A 21.9964 29.8321 63.7510
BBB 45.5698 54.4007 97.4304
BB 181.7816 186.5635 210
Table 9.6 Ten-year credit spread parameter State AAA AA 1 9.5511 24.2406 2 15 34.7418 3 26 67.0590
A 38.9602 48.9437 76.8028
BBB 70.1553 76.7142 111.1784
BB 213.0992 224.2761 260
in each state, a typical value for the mean-reversion speed for spreads. The volatility parameter is computed as the annualized standard deviation of the absolute spread returns in each state. The parameter values are reported in Tables 9.5–9.8. The cursive entries in the tables correspond to extrapolated values. These parameters are impossible to estimate due to lack of data. Further, we need a correlation matrix of spreads with different maturities and ratings, and the correlations between the spreads and the market factor M . The correlation matrix is assumed to be the same for all states of the market. We use the absolute spread returns to estimate the correlation. The results are given in Table 9.9. For the correlation between the spreads and the systematic migration factor M , we use the values estimated by [30]. These value were estimated with the spreads of Lehman Euro Corporate Bond Indices and DJ Euro Stoxx 50. The starting values of the credit spreads for different ratings and maturities are given in Table 9.10. These values are also used to price the corresponding single
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9 Simulation Framework
Table 9.7 Five-year credit spread parameter
State AAA AA 1 12.9622 19.3861 2 25 34.2298 3 50 73.1025
A 18.5510 33.1457 63.6117
BBB 21.4499 46.3581 88.3875
BB 50 100 150
Table 9.8 Ten-year credit spread parameter
State AAA AA 1 20.8544 23.8436 2 30 53.3234 3 55 79.1013
A 25.9443 50.8085 71.3535
BBB 63.3332 76.2667 89.8982
BB 120 140 160
Table 9.9 Correlation matrix for spreads and market factor 5y AAA 5y 5y 5y 5y 5y 10y 10y 10y 10y 10y M
AAA AA A BBB BB AAA AA A BBB BB
5y AA
5y A
1 0:2669 0:2779 0:2669 1 0:9441 0:2779 0:9441 1 0:2396 0:8961 0:8779 0:0496 0:4848 0:5051 0:1120 0:0227 0:0342 0:2147 0:6657 0:6482 0:0926 0:5079 0:6411 0:1427 0:4672 0:4483 0:0027 0:1500 0:1195 0:02 0:06 0:23
5y BBB
5y BB
10y AAA
10y AA
10y A
0:2396 0:0496 0:1120 0:2147 0:0926 0:8961 0:4848 0:0227 0:6657 0:5079 0:8779 0:5051 0:0342 0:6482 0:6411 1 0:6139 0:0646 0:6747 0:3664 0:6139 1 0:0100 0:4516 0:2223 0:0646 0:0100 1 0:0127 0:0064 0:6747 0:4516 0:0127 1 0:3741 0:3664 0:2223 0:0064 0:3741 1 0:5334 0:3595 0:0161 0:4746 0:2556 0:1662 0:4653 0:0034 0:0581 0:0193 0:32 0:36 0:02 0:06 0:23
Table 9.10 Credit spread starting values on the 20th of September 2007 Maturity AAA AA A BBB 5 years 9 25.9136 28.0531 44.0323 7 years 11 30.3917 35.4494 56.0619 10 years 15 39.5771 46.7494 69.7117
10y BBB
10y BB
0:1427 0:0027 0:4672 0:1500 0:4483 0:1195 0:5334 0:1662 0:3595 0:4653 0:0161 0:0034 0:4746 0:0581 0:2556 0:0193 1 0:0849 0:0849 1 0:32 0:36
BB 186 198 224
name credit default swaps. The quotes of the iTraxx tranches for different maturities on the 20th of September 2007 are presented in Table 9.11.
9.4.2 Simulation of the Economic Factors and Pricing of the Credit Instruments As already described in Sect. 9.1, in a first step the states of the Markov process are simulated on the daily grid for a time horizon of 5 years. Then the duration times are computed for each time increment of 3 month on each simulation path. Afterwards, only 3 month time increments are considered. Using these values the
9.4 Case Study
239
Table 9.11 iTraxx spreads on the 20th of September 2007 Maturity 0–3% 3–6% 6–9% 5 years 18.1880% 83.50 bp 38.50 bp 7 years 28.2500% 150.50 bp 72.00 bp 10 years 38.5000% 376.00 bp 146.75 bp
9–12% 23.00 bp 43.00 bp 79.50 bp
12–22% 14.00 bp 26.50 bp 45.50 bp
3-month increments of the market factor M.t/ are simulated. The 3-month increments for the idiosyncratic processes Xij .t/ are simulated for the 125 issuers of the iTraxx portfolio and five additional single name CDS issues with different initial ratings. At each time step the corresponding asset returns are computed. The new ratings of all 128 issuers are verified according to the rating migration boundaries in Table 9.3. Besides the rating migrations of the credit portfolio, the interest rate and the credit spreads are also simulated on the 3 month grid. We simulate the interest rate according to the Vasicek model independent from the other factors ( r D 0). The 5 and 10 year credit spreads with ratings AAA, AA, A, BBB and BB are simulated as Vasicek processes correlated to the market factor M.t/ as described above. We do not consider the ratings B, CCC/C since there is no historical credit spread data for these ratings. We assume that the BB spread represents the spread for all non-investment grade ratings. However, we keep all non-investment rating classes separate in the simulation of the rating migrations to prevent the Markov property of the rating migrations. Now we have all factors and rating migrations necessary to price the credit instrument on the 3-month grid of the simulation. We compute the prices of the following instruments: credit default swaps with initial ratings AAA, AA, A, BBB, BB and maturities on the 20th of September 2012, 2014 and 2017, i.e. 5, 7 and 10 years; the 5, 7 and 10 year iTraxx index and five tranches with maturities on the 20th of December 2012, 2014 and 2017. In particular, the following return ingredients have to be computed: Present values:
– The present values of the CDS with different maturities Ti , i D 1; 2; 3 and ratings Rj , j D 1; : : : ; 5 at time steps tk , k D 1; : : : ; 20 P VCDS.Ti ;Rj / .tk /I – The present values of the iTraxx index with different maturities Ti , i D 1; 2; 3 at time steps tk , k D 1; : : : ; 20 P VCDOindex.Ti / .tk /I – The present values of the iTraxx tranches with different maturities Ti , i D 1; 2; 3 and tranches t rj , j D 1; : : : ; 5 at time steps tk , k D 1; : : : ; 20 P VCDO.Ti ;t rj / .tk /:
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9 Simulation Framework
Outstanding notionals:
– The outstanding notionals of the CDS with different ratings Rj , j D 1; : : : ; 5 at time steps tk , k D 1; : : : ; 20 (the outstanding notionals are the same for CDS with different maturities) NCDS.Rj / .tk / D
1 not defaulted 0 defaulted
– The outstanding notionals of the iTraxx index at time steps tk , k D 1; : : : ; 20 NCDOindex .tk / D 1
nD I 125
with nD number of defaulted instruments in the iTraxx portfolio – The outstanding notionals of the iTraxx tranches t rj , j D 1; : : : ; 5 at time steps tk , k D 1; : : : ; 20 0 NCDO.t rj / .tk / D 1 mi n @
max
nD 125
.1 R/ K1j .t0 /; 0
K2j .t0 / K1j .t0 /
1 ; 1A :
Spreads payments:
– The spread payments of the CDS with different maturities Ti , i D 1; 2; 3 and ratings Rj , j D 1; : : : ; 5 at time steps tk , k D 1; : : : ; 20 SpreadCDS.Ti ;Rj / .tk / D NCDS.Ti ;Rj / .tk / sCDS.Ti ;Rj / .t0 / tk I – The spread payments of the iTraxx index with different maturities Ti , i D 1; 2; 3 at time steps tk , k D 1; : : : ; 20 SpreadCDOindex.Ti / .tk / D NCDOindex.Ti / .tk / sCDOindex.Ti / .t0 / tk I – The spread payments of the iTraxx tranches with different maturities Ti , i D 1; 2; 3 and tranches t rj , j D 1; : : : ; 5 at time steps tk , k D 0; : : : ; 20 SpreadCDO.Ti ;t rj / .tk / D NCDO.t rj / .tk / sCDO.Ti ;t rj / .t0 / tk : To compute the present values of the instruments, the new default intensities are bootstrapped from the simulated 5 and 10 year spreads for the particular rating as described in Sect. 3.2.3. For the computation of the present values of the iTraxx tranches, the new rating composition is used besides of the new default intensities. j j Note, that the lower and upper attachment points K1 .tk / and K2 .tk / of the tranche j , that are used in the pricing formulas, are expressed in percent of the outstanding notional, and so must be updated at every time step. For the computation of
9.4 Case Study
241 j
j
outstanding notional, the initial lower and upper bounds K1 .t0 / and K2 .t0 / are used. The new notionals are also taken into account for the computation of the present values. The spread payments represent the amounts paid at the particular time step according to the accrual time and the outstanding notional. The up-front fee of the equity tranches is paid at time t0 and is taken into account by the entry SpreadCDO.Ti ;1/ .t0 / D NCDO.1/ .t0 / upf rontCDO.Ti ; 1/.t0 /. The entries of this variable for the time t0 are zero for mezzanine and senior tranches. Now we compute cumulated payments for all instruments at each time point of the simulation. We compound them to the next time step with the corresponding interest rate R.tk1 ; Rk /, that is calculated according to the Vasicek model and the simulated short rate: The cumulated payments of the CDS with different maturities Ti , i D 1; 2; 3 and
ratings Rj , j D 1; : : : ; 5 at time steps tk , k D 1; : : : ; 20
PaymentCDS.Ti ;Rj / .tk / D NCDS.Ti ;Rj / .tk / NCDS.Ti ;Rj / .tk1 / .1 R/ C SpreadCDS.Ti ;Rj / .tk / CPaymentCDS.Ti ;Rj / .tk1 /e R.tk1 ;tk /tk ; where R .tk1 ; tk / is the zero rate observed at time tk1 with maturity tk .
The cumulated payments of the iTraxx index with different maturities Ti , i D
1; 2; 3 at time steps tk , k D 1; : : : ; 20
PaymentCDOindex.Ti / .tk / D NCDOindex.Ti / .tk / NCDOindex.Ti / .tk1 / .1 R/ CSpreadCDOindex.Ti / .tk / CPaymentCDOindex.Ti / .tk1 /e R.tk1 ;tk /tk : The cumulated payments of the iTraxx tranches with different maturities Ti , i D
1; 2; 3 and tranches t rj , j D 1; : : : ; 5 at time steps tk , k D 0; : : : ; 20 PaymentCDO.Ti ;t rj / .tk / D NCDO.t rj / .tk / NCDO.t rj / .tk1 / CSpreadCDO.Ti ;t rj / .tk /
CPaymentCDO.Ti ;t rj / .tk1 /e R.tk1 ;tk /tk : Finally, the profit and loss (P&L) of the credit instruments is computed as the sum of the cumulated payment and the present value of the instrument at time tk (note that the value of a CDS at time zero, that should be substracted, is zero): The profit and loss of the CDS with different maturities Ti , i D 1; 2; 3 and ratings
Rj , j D 1; : : : ; 5 at time steps tk , k D 1; : : : ; 20
PLCDS.Ti ;Rj / .tk / D PaymentCDS.Ti ;Rj / .tk / C P VCDS.Ti ;Rj / .tk /:
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The profit and loss of the iTraxx index with different maturities Ti , i D 1; 2; 3 at
time steps tk , k D 1; : : : ; 20
PLCDOindex.Ti / .tk / D PaymentCDOindex.Ti / .tk / C P VCDOindex.Ti / .tk /: The profit and loss of the iTraxx tranches with different maturities Ti , i D 1; 2; 3
and tranches t rj , j D 1; : : : ; 5 at time steps tk , k D 0; : : : ; 20
PLCDO.Ti ;t rj / .tk / D PaymentCDO.Ti ;t rj / .tk / C P VCDO.Ti ;t rj / .tk /:
9.4.3 Asset Allocation Results Recall that all credit instruments we consider are unfunded. The investors that are not allowed to act as a protection seller in unfunded portfolio or single name credit swaps or tranches, may choose the funded version of these credit instruments that are also available in the market. The funded instruments can also be seen as a combination of the unfunded product with a risk-free floating rate note. So the P&Ls of the unfunded instruments can be interpreted as the excess returns over the risk-free rate. Using the P&L distributions of the considered credit instruments, the optimal portfolios can be determined using mean-variance or CVaR optimization approaches described in Sect. 3.5. However, before we start with the optimization, we take a look at the histograms and statistics of the P&L distributions. Figure 9.4 contains some examples of histograms of the P&L distributions of the CDS, the iTraxx tranche and the iTraxx index for the 5 years investment horizon. The most of the probability weight is distributed around the mean. For the P&L distribution of the CDS, some little weight on very extreme left points is very typical. While the variance of the main part of the P&L distribution is explained with the variation in the interest rates and the present value of the CDS, the extreme points correspond to the default event which happens with a low probability. The points at the most left tail correspond to very early defaults: the defaulted amount that is 0.6 is compounded up to the considered investment horizon. In case of the later defaults, the loss is a bit lower since the investor receives the spread payments before the firm defaults. The P&L distribution of the iTraxx tranches is quite different from those of the single name CDS. The left tail spreads over the complete interval between 1 to 0. The upper part of the negative returns is generated by a relative small number of defaults and the deviation in the present values. The lower part is due to more defaults. Besides, the distribution function is slowly increasing on the left part since the present value of the tranche decreases as defaults in subordinated tranches occur. The distributions of the equity tranches have a different form. In particular, the right tail of the equity tranche P&L distribution is not capped in contrast to that of the mezzanine and senior tranches. Thanks to the high spread of the equity tranche, much higher returns are possible in the scenarios with no defaults in the portfolio.
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243
a 1 year horizon
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(a) 10 year CDS with initial rating A
b 1 year horizon
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c 1 year horizon
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d
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(d) 10 year iTraxx index Fig. 9.4 P&L distributions of credit instruments for the time horizons of 1, 3 and 5 years
0
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9 Simulation Framework
Table 9.12 P&L statistics of the 5-year CDS for the 5 year investment horizon Statistic AAA AA A BBB Mean (%) 0:5232 1:3335 1:3456 1:2348 Median (%) 0:5228 1:5051 1:6293 2:5556 Std. deviation (%) 0:0165 3:3996 4:4138 9:6178 VaR (%) 0:4881 1:4028 1:5165 66:2866 CVaR (%) 0:4846 15:8956 26:4219 73:2672 Min (%) 0:4757 72:0215 78:5192 84:5722 Max (%) 0:6061 1:7451 1:8892 2:9653 Skewness 0:3164 19:6594 15:5349 7:1922 Kurtosis 3:1993 389:2875 244:5369 53:2748
BB 7:5212 10:7863 14:9387 65:7747 71:6827 85:3481 12:5258 4:3809 20:5658
Table 9.13 P&L statistics of the 7-year CDS for the 5 year investment horizon Statistic AAA AA A BBB Mean (%) 0:0423 0:9253 1:0639 0:9786 Median (%) 0:2467 1:4216 1:7233 2:7838 Std. deviation (%) 0:9599 3:5887 4:5906 9:7187 VaR (%) 3:3723 3:2639 4:6458 65:8944 CVaR (%) 4:2188 20:3367 31:2966 73:0291 Min (%) 7:5413 71:9942 78:5192 84:4404 Max (%) 0:9223 2:5483 2:9724 4:6487 Skewness 1:8052 16:3497 13:5316 6:7896 Kurtosis 7:8800 303:4154 202:9187 49:4482
BB 8:3517 11:7509 15:2270 65:4902 71:4683 85:3481 16:3800 4:1981 19:5410
Table 9.14 P&L statistics of the 10-year CDS for the 5 year investment horizon Statistic AAA AA A BBB Mean (%) 0:4974 1:0334 1:3416 1:2285 Median (%) 0:1367 1:8971 2:4201 3:5883 Std. deviation (%) 2:1476 4:3338 5:3198 10:3630 VaR (%) 7:7900 8:1905 11:0650 65:4495 CVaR (%) 9:5542 24:9459 35:5858 72:7589 Min (%) 16:2675 71:9383 78:5192 84:2909 Max (%) 1:6549 4:3665 5:1578 7:6266 Skewness 1:6801 9:5071 8:9109 5:5923 Kurtosis 6:9959 141:9653 112:7417 38:0828
BB 11:1967 14:6115 16:3707 65:0412 70:9063 85:3481 24:3487 3:6357 16:4002
The main difference of the P&L distributions of the iTraxx index to those of the single name CDS and the iTraxx tranches is the very light left tail. This feature of the iTraxx index is due to the high diversification of the default risk in the portfolio. One default in the portfolio makes only approximately 0.5% loss of the notional.
9.4 Case Study
245
Table 9.15 P&L statistics of the 5-year iTraxx index and tranches for the 5 year investment horizon Statistic Index 0–3% 3–6% 6–9% 9–12% 12–22% Mean (%) Median (%) Std. deviation (%) VaR (%) CVaR (%) Min (%) Max (%) Skewness Kurtosis
1:2197 1:4144 25:3200 1:0899 6:2785 49:2589 2:3178 22:5404 765:5153
25:6104 3:4863 31:0747 4:9013 25:3200 10:2092 57:7998 66:1395 71:7561 85:1482 107:7067 106:7024 60:9567 5:5737 1:0910 8:0195 4:6798 69:3853
1:6942 2:2246 5:0664 18:4211 43:5611 91:0295 2:5699 12:1482 164:7355
0:5466 1:3058 6:8785 30:2798 63:8573 93:4781 1:5353 10:4520 116:4823
0:4159 0:7889 9:2213 60:6619 84:3547 96:8785 0:9345 8:5784 77:8712
Table 9.16 P&L statistics of the 7-year iTraxx index and tranches for the 5 year investment horizon Statistic Index 0–3% 3–6% 6–9% 9–12% 12–22% Mean (%) 0:9483 19:7398 4:5635 2:7008 1:1109 0:1034 Median (%) 1:2715 21:5442 9:2644 4:4672 2:4891 1:5124 Std. deviation (%) 31:6918 31:6918 14:2168 8:2085 7:5991 9:3131 VaR (%) 3:6007 52:7275 71:1998 38:6875 40:3172 62:1790 CVaR (%) 8:6810 61:4328 85:5544 67:9394 68:2475 83:3786 Min (%) 49:0511 93:2323 103:8978 95:9329 96:4581 97:5039 Max (%) 3:5670 86:1033 12:9465 6:1937 3:6990 2:2796 Skewness 9:1704 0:1882 4:1369 8:3661 9:2280 8:2327 Kurtosis 212:7107 2:5291 23:3526 83:9359 95:9282 73:2876
Table 9.17 P&L statistics of the 10-year iTraxx index and tranches for the 5 year investment horizon Statistic Index 0–3% 3–6% 6–9% 9–12% 12–22% Mean (%) 1:0875 21:1478 15:1625 5:8179 2:4818 0:8444 Median (%) 1:7063 16:0346 22:2583 10:0288 5:0608 2:8661 Std. deviation (%) 33:2251 33:2251 23:6050 12:7470 8:9761 8:1476 VaR (%) 7:5679 37:7244 58:5571 49:4597 36:7589 49:5259 CVaR (%) 12:1298 47:5123 69:5417 69:5103 63:3696 66:9396 Min (%) 48:6122 78:4875 94:4585 91:7277 94:3406 92:8224 Max (%) 5:8833 114:8403 42:0709 16:4200 8:8215 5:0488 Skewness 2:9789 0:5728 1:2674 3:2175 4:9770 6:8328 Kurtosis 35:7005 2:8436 4:3828 17:8058 37:9990 59:4782
The corresponding statistics of the P&L distribution of CDS are summarized in Tables 9.12–9.14. The expected values of all CDS are positive after 5 years. The expected means of the BBB CDS returns are lower than those of the higher rated A CDS. This means that the difference between the BBB and A credit spreads is not high enough to compensate for a higher default probability of a BBB firm. The same
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9 Simulation Framework 12 5 year 7 year 10 year
10
return (%)
8 6 4 2 0 −2
0
2
4
6 8 10 12 standard deviation (%)
14
16
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(a) Mean-variance optimization 12 5 year 7 year 10 year
10
return (%)
8 6 4 2 0 −10
0
10
20
30 40 CVaR (%)
50
60
70
80
(b) CVaR optimization Fig. 9.5 Efficient frontiers of the portfolios of CDS with different maturities for the 5 year investment horizon
observation can be made for BBB CDS with 7 and 10 years maturity. The expected returns of AA and A CDS are very similar for the 5 years maturity. The difference between them is higher for longer maturities. The expected returns as well as the standard deviations of BB CDS are much higher than those of the investment grade CDS. For the iTraxx tranches, returns and variances decrease with the increasing tranche seniority (Tables 9.15–9.17). We compute the 99% VaR and CVaR for the P&L distributions. The negative values of those indicate that the highest of the worst
9.4 Case Study
247 Maturity 5y, Mean−Var
Maturity 5y, CVaR 100 portfolio position (%)
portfolio position (%)
100 80 60 40 20 0
4 6 return (%) Maturity 7y, Mean−Var portfolio position (%)
portfolio position (%)
60 40 20 0
2
4 6 return (%) Maturity 10y, Mean−Var
20 2
4 6 return (%) Maturity 7y, CVaR
portfolio position (%)
80 60 40 20
80 60 40 20 0
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8
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4 6 return (%) Maturity 10y, CVaR
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0 0
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10
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Fig. 9.6 Composition of the efficient portfolios of CDS with different maturities for the 5 year investment horizon
1% of the CDS returns is positive. In general, for more risky instruments VaR and CVaR are higher. Now we start with the optimization analysis of the credit portfolio. First, we allow only the traditional CDS, and perform an optimization for CDS with different ratings. We consider three maturities, 5, 7 and 10 years, separately. For each set of credit instruments we consider the 5 years investment horizon. Figure 9.5 shows the efficient frontiers of the mean-variance and the CVaR optimization. Longer maturities allow to get more return by accepting more risk. Five year maturity CDS portfolios dominate portfolios of longer maturities in the low risk part. Afterwards,
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9 Simulation Framework 30 5 year 7 year 10 year
25
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20 15 10 5 0
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(a) Mean-variance optimization 30 5 year, long 5 year, long and short 7 year, long 7 year, long and short 10 year, long 10 year, long and short
25
return (%)
20 15 10 5 0
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30
40 50 CVaR (%)
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(b) CVaR optimization Fig. 9.7 Efficient frontiers of the portfolios of the iTraxx index and tranches with different maturities for the 5 year investment horizon
the efficient frontiers of different maturities cross and portfolios with 10 years maturity dominate the others for higher risk budgets. The portfolio composition for both optimization approaches is presented in Fig. 9.6 and is very similar for the two approaches. It starts with a high quote of AAA CDS, exchanging it for lower rated CDS in portfolios with higher risk. BBB CDS are almost not present in portfolios. Now we consider portfolios with only alternative credit investments: the iTraxx index and its tranches. Besides the mean-variance optimization, we perform two
9.4 Case Study
249 Maturity 5y, Mean−Var
Maturity 5y, CVaR long
60 40 20 10 15 20 25 return (%) Maturity 7y, Mean−Var
40 20 5
10 15 20 25 return (%) Maturity 7y, CVaR long
50 0 −50 −100 −150
5 10 15 20 25 return (%) Maturity 7y, CVaR long and short 100
60 40 20 10 15 return (%) Maturity 10y, Mean−Var
40 20 10 15 return (%) Maturity 10y, CVaR long
40 20 5
10 15 return (%)
20
0−3% tranche
80 60 40 20 0
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5 10 15 return (%) Maturity 10y, CVaR long and short 100
portfolio position (%)
portfolio position (%)
60
iTraxx index
60
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100 portfolio position (%)
60
100 portfolio position (%)
portfolio position (%)
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80
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Maturity 5y, CVaR long and short 100
100 portfolio position (%)
portfolio position (%)
100
50
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−50 5
3−6% tranche
10 15 return (%)
20
6−9% tranche
5
10 15 return (%)
9−12% tranche
20
12−22% tranche
Fig. 9.8 Composition of the efficient portfolios of the iTraxx index and tranches with different maturities for the 5 year investment horizon
versions of the CVaR optimization for the alternative credit portfolios. In the first version, only long positions are allowed. This means that it is only allowed to buy protection. In the second version, both long and short positions are permitted, i.e. the investor can also act as protection seller. The dashed lines in Fig. 9.7 represent the second approach with long and short positions. As we would expect, the additional possibility of short positions creates portfolios with higher expected return for the same risk. The mean-variance approach finds portfolios with instruments of 5 year maturity to be dominating. In general, the alternative credit instruments can produce portfolios with much higher returns than the traditional single-name CDS and, correspondingly, higher standard deviations. Figure 9.8 gives the compositions of the alternative credit portfolios. The iTraxx index and the equity tranche are the most dominating in these portfolios. The index represents the investment with the lowest risk. The equity tranche has the highest risk compared to the other tranches, and thus contributes 100% to the most risky portfolios. Other tranches are included in the mean-variance portfolios with risk-return profiles in between the two extremes. The portfolios with the minimal risk start with 100% of the iTraxx index, that is first replaced with the less risky mezzanine tranches. Further, with higher possible risk, these tranches are replaced with more risky tranches. This explains the curves in the plots. The compositions of
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9 Simulation Framework 30 5 year 7 year 10 year
25
return (%)
20 15 10 5 0 −5
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10 15 20 25 standard deviation (%)
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(a) Mean-variance optimization 30 5 year, long 5 year, long and short 7 year, long 7 year, long and short 10 year, long 10 year, long and short
25
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20 15 10 5 0 −10
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30 40 CVaR (%)
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(b) CVaR optimization Fig. 9.9 Efficient frontiers of the portfolios of CDS and iTraxx index and tranches with different maturities for the 5 year investment horizon
both CVaR versions are quite unspectacular including almost only the most senior tranche 12–22% besides of the two main positions: the iTraxx index and the equity tranche. For the portfolios with both long and short positions, the long, protection seller, positions are summed up in the positive part of the scale, while the short, protection buyer, positions are summed up in the negative part of the scale. As in the version with only long positions, the less risky portfolios include at most long iTraxx index position. For more risky portfolios, the share of the long iTraxx index is
9.4 Case Study
251
Fig. 9.10 Composition of the efficient portfolios of CDS and iTraxx index and tranches with different maturities for the 5 year investment horizon
reduced and the tranches, especially the equity tranche, are added. Buying protection on some mezzanine or senior tranches allows to limit loss for a small premium. Last, we mix the traditional and alternative credit investments. The efficient frontiers of the mean-variance and CVaR optimization are presented in Fig. 9.9. Again, we consider two cases: the solid lines represent portfolios with only long instruments, the dashed lines correspond to portfolios with both, long and short instruments. We allow to sell protection on iTraxx instruments. Again, the meanvariance optimization prefers the 5 year instruments while CVaR optimization invests in the 10 year instruments.
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9 Simulation Framework Maturity 5y, Mean−Var
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0
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30
iTraxx, long and short
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CDS & iTraxx, long
60
80
CDS & iTraxx, long and short
Fig. 9.11 Efficient frontiers of the portfolios of CDS and iTraxx index and tranches for the 5 year investment horizon
Figure 9.10 shows the portfolio compositions for CDS and iTraxx instruments. The dominating portfolios with lower risk are represented by traditional CDS, with the highest quote of AAA. Further, with increasing risk, iTraxx index, AA, A and BB CDS are added. The highest risk segment is represented with the iTraxx equity tranche. The middle risk segment additionally includes the senior and mezzanine iTraxx tranches for the mean-variance optimization, and mainly only the senior iTraxx tranche for the CVaR optimization. The long positions for the long-short portfolios of the CVaR approach are similar to the long only portfolios. As we see, additional return can be achieved by buying protection on senior iTraxx tranches and so limiting the loss. Figure 9.11 presents all the efficient frontiers, considered above, once more. Now, we have ordered them in different ways. Traditional CDS investments and alternative iTraxx investments as well as their mixture are compared in one plot, for different maturities. The alternative credit instruments generate more return than the traditional ones. Combining both types of credit instruments allows to construct investment portfolios dominating all portfolios with only one of the instrument types.
Chapter 10
Conclusion
In the first part of the dissertation, all necessary and important background information is presented. It gives the introduction to credit derivatives and markets and mathematical preliminaries for the further results. The second part is devoted to the static types of copula models where the distributions of the factors are given by the same, (0,1)-distributions. The third chapter gives details on the basic one factor model for credit portfolios, namely the one using the Gaussian distribution for all factors. The section contains the complete definitions of the model and derivation of all related formulas. The result for the portfolio loss distribution is generalized also for arbitrary distributions of the factors. After the theoretical results are presented, we also analyzed fitting abilities of the model empirically. We discussed the problem of the correlation smile and the effort of fixing it with the help of base correlation. We showed that base correlation is not arbitrage-free. This chapter is concluded with an overview of existing extensions of the Vasicek model. We did not perform the comparative analysis of these extensions since it was already done by several authors. Extending the model by using a heavy-tailed distribution for factors proved to bring the most improvement into the model for the fixing of the correlation smile. In particular, the double-t copula achieved the best empirical results being, however, not very convenient and fast to handle because of the instability under convolution of Student-t distribution. Motivated by the findings presented in the third chapter, we decided to concentrate our research into the direction of extending the Vasicek model with the help of another heavy-tailed distribution having more convenient properties than the Student-t distribution. So we have chosen the Normal Inverse Gaussian distribution and presented the straightforward extension in the second chapter of the second part. We performed an empirical analysis of the NIG factor copula model in this section: it showed that the second free parameter of the NIG distribution, ˇ, did not bring any improvement into the fitting ability of the model. The optimal fitting results could be achieved with a zero value of the ˇ, that means that the market quotes of the iTraxx tranches do not assume any skew for the distribution of the asset returns. The calibration ability of the NIG factor copula model proved to be
A. Schl¨osser, Pricing and Risk Management of Synthetic CDOs, Lecture Notes in Economics and Mathematical Systems 646, DOI 10.1007/978-3-642-15609-0 10, c Springer-Verlag Berlin Heidelberg 2011
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10 Conclusion
especially good when using only the data of the four upper tranches. This can be explained by the fact that the market for the equity tranche differs from the market for the upper tranches. Since recovery is assumed to be an input parameter in the model, we performed sensitivity analysis. We found that: (1) the fair tranche spreads are not very sensitive to change in recovery, (2) recovery cannot be used as a free parameter for calibration since the optimal value would be zero. The third part of the thesis presents a term-structure model, that takes the time component into account for modeling the factors for different time horizons, as well as some further model features. In the third part we show, that, in contrast to the Gaussian model, it is possible to include a time dependence into the NIG factor copula model and to model the asset returns with a NIG process with a zero mean and variance t. However, calibration of the term-structure NIG copula model proved to be not so successful. The major reason for this is the fact that it is difficult to describe 15 or more market quotes with only two free parameters. Nevertheless, we decided to further track this model due to its good qualities as a simulation model: model factors described by such NIG processes can be discretized in an arbitrary way. Besides, the term-structure NIG copula model combined with the next extension, the large homogeneous cell, considered in the third section, achieved good calibration results. The second chapter of the third part presents a further extension of the NIG factor copula model. As we have seen in the previous chapter, it is difficult to fit all quotes with only two parameters. So a model extension relaxing some assumption and introducing some further parameters could help. We applied the Large Homogeneous Cell assumption instead of the Large Homogeneous Portfolio. In our case, we introduced four rating cells representing four parts of the underlying portfolio with the same rating. Using the default probabilities, that are averaged over each rating cell, and four correlation parameters instead of only one, we could achieve really good calibration results, comparing with both the LHC Gaussian model and the LHC term-structure NIG copula model. This model extension is even more suitable for a simulation framework, which is the goal of this thesis, because so we can naturally model the changes in the portfolio credit spread that are not only due to the usual market fluctuations but also due to the rating migration in the portfolio. Still the model is very convenient for a simulation and can be arbitrarily discretized. The third and the last extension of the NIG factor copula model is introduced in the third chapter of the third part. So far, we still had constant over time correlations in our model. Especially after the market development of 2008, it became obvious that this assumption is very unrealistic. We incorporated a regime switching component into the model. However, we did not do it simply by allowing two (or more) states for the correlation parameter, but by deriving the distributions of the factors in such a way that their increments are still stable under convolution for different states. This property allows not only an arbitrary discretisation of the model for a simulation, but also semi-analytical pricing. However, the pricing cannot be performed exactly since the distributions of the durations of a stay in a particular regime are not known. It turned out that a good moment matching approximation for the distributions of the factors is possible. The approximating distributions are NIG distributions again and the pricing can be performed in a similar way as in the
10 Conclusion
255
previous version of the model. Now we believe to have a model that must be able to fit not only the spreads of all tranches with all maturities on a single day, but also to describe the complete history of the quotes. The third part of the dissertation is concluded with the calibration of the CrashNIG copula model. During the calibration we detected also the existence of different liquidity regimes, since the implied default probabilities from the market single CDS spreads appeared to be too high to fit the tranche prices. We also found that the history of 2004–2008 of the iTraxx quotes can be explained with three correlation regimes: a normal regime, a regime with lower correlation and a bit lower liquidity during the crisis of May 2005, and a regime with higher correlation and much lower liquidity during 2008. The calibration of the Crash-NIG copula model turned out to be very unproblematic giving reasonable parameter values. The data preparation is of course quite work intensive since the rating distribution of the iTraxx portfolio is not directly available over time and the average rating spreads should also be computed from the individual spreads of the portfolio constituents. The reason for this is that the number of issuers in each rating cell of the iTraxx portfolio is rather small. So the average EUR rating spreads are not representative for them. Finally, we developed a simulation framework, which is consistent to the CDO pricing model. The rating migration and default model is the same LHC CrashNIG copula model, that was also used for pricing. We used a Vasicek process, that can be correlated to the market factor, for the short interest rate. Credit spreads for different ratings and maturities are modeled with correlated Vasicek processes, that are also correlated to the market factor, with regime switching coefficients. In the case study, we showed how to perform the simulation and compute the total returns of single-name CDS and CDS index and tranches. Using the distributions of the total returns, an asset allocation optimization can be performed, e.g., with mean-variance or CVaR approach. To summarize, we want to point out once again the new developments of this thesis that contribute to the literature of factor copula models for CDO pricing: The NIG factor copula model was introduced in the standard setting for large
homogeneous portfolio (LHP) assumption analogue to the Gaussian copula model (Chap. 5). The term-structure NIG copula model introduced in Sect. 6.2 allows to model CDO tranches with different maturities in a consistent way. The properties of NIG distribution make this extension possible while the term-structure dimension cannot be modeled in such way with the Gaussian copula model. In Chap. 7 we showed how the large homogeneous cell (LHC) setting, introduced for the Gaussian model by [30], can be applied to the NIG model. The calibration results demonstrated a much better ability of the term-structure LHC NIG copula model to fit the market quotes than of the Gaussian LHC model. We find the contribution of the Chap. 8 even more important. The Crash NIG copula model introduced here allows for different correlation regimes. Besides, it has a number of properties that make it possible to discretize the model for a Monte Carlo simulation and admit a semi-analytic pricing of CDO tranches.
256
10 Conclusion
To our knowledge, literature on factor copula models did not propose a regime switching model so far. Chapter 9 demonstrated the application of the Crash NIG copula model in a Monte Carlo simulation together with a number of other risk factors that are necessary to model a portfolio of traditional and structured credit instruments. The case study, closing the thesis, gave an example of a portfolio optimization with mean-variance and CVaR approaches based on the simulation.
Appendix A
Some Results in Chapter 4
A.1 Proof of Proposition 4.1 Proposition 4.1. The expected tranche loss can be written as 0 EL.K1 ;K2 / .t/ D
1 B @ K2 K 1
Z1
1
Z1
C .x K2 /dF .t; x/A :
.x K1 /dF .t; x/ K1
K2
Proof. 1 EL.K1 ;K2 / .t/ D K2 K 1
Z1 .min.x; K2 / K1 /dF .t; x/ K1
0
D
1 B @ K2 K 1 0
1 B D @ K2 K 1 Z1 C K2
D
K Z2
1
Z1
C .K2 K1 /dF .t; x/A
.x K1 /dF .t; x/ C K1
K2
Z1
Z1 .x K1 /dF .t; x/
K1
1
.x K1 /dF .t; x/ K2
C .K2 K1 /dF .t; x/A 0
1 B @ K2 K 1
Z1
Z1 .x K1 /dF .t; x/
K1
1 C .x K2 /dF .t; x/A
K2
A. Schl¨osser, Pricing and Risk Management of Synthetic CDOs, Lecture Notes in Economics and Mathematical Systems 646, DOI 10.1007/978-3-642-15609-0, c Springer-Verlag Berlin Heidelberg 2011
257
258
A Some Results in Chapter 4
A.2 Proof of Proposition 4.2 Proposition 4.2. For any p and x in .0; 1 it holds: lim
Œmx X
m!1
kD0
0; if x < p m k mk D p .1 p/ 1; if x > p k
Proof. Let us consider Sm number of ones in m independent Bernoulli trials in which 1 comes with probability of p. Then, according to the law of large numbers, Sm ! p stochastically as m ! 1. m Consider the distribution function of Smm Sm x : Fm .x/ WD P m
Since the stochastic convergence implied the convergence in distribution, Fm .x/ ! F .x/ in distribution as m ! 1, where F .x/ is the distribution function of the random variable xp p: F .x/ WD P xp x D
0; if x < p 1; if x > p
Note that Fm .x/ D
m X m p k .1 p/mk k
kD0
k m x
bmxc
D
X
kD0
m p k .1 p/mk ; k
where bac denotes the integer part of a.
A.3 Lemma on Change of Limit and Integration Order Lemma A.1. For fn a convergent sequence of measurable functions and given the following conditions: Rb Rb (i) g is a function such that g C .x/dx < 1 and g .x/dx < 1. a
a
(ii) 9C a constant such that 8n 1 and 8x 2 Œa; b: jfn .x/j C , the lim fn g is integrable and n!1
A.4 Proof of Lemma on Expected Tranche Loss
Zb
259
Zb fn .x/g.x/dx D
lim
n!1 a
. lim fn .x//g.x/dx:
(A.1)
n!1
a
Proof. Using, e.g., Theorem 19.6 in [48, p. 119], we consider a sequence of measurable integrable functions fn g, for which the following condition is satisfied:
jfn gj C jgj D C g C .x/ C g .x/ ; with C jgj an integrable function. Besides, fn g is a sequence convergent to lim fn g. n!1
So the lim fn g is integrable and (A.1) holds. n!1
A.4 Proof of Lemma on Expected Tranche Loss Lemma 4.2. In the Vasicek model, the expected loss at time t of the mezzanine tranche taking losses from K1 to K2 percent of the overall portfolio assuming zero recovery is given by:
˚2 ˚ 1 .K1 / ; C.t/; ˚2 ˚ 1 .K2 / ; C.t/; EL.K1 ;K2 / .t/ D ; K2 K 1 where ˚2 is the bivariate normal distribution function and the covariance matrix D
! p 2 1 a 1 p 1 a2 1
Proof. Using integration by parts we get Z1
Z1 .x K/dF .t; x/ D F .t; x/.x K/ j F .t; x/dx 1
K
K
K
Z1 D 1K
F .t; x/dx
(A.2)
K
and thus for the function F .t; x/ D ˚ Z1
p
K
p
Z1 .x K/dF .t; x/ D .1 K/
˚ K
1a2 ˚ 1 .x/C.t / a
! 1 a2 ˚ 1 .x/ C.t/ dx: a
(A.3)
260
A Some Results in Chapter 4
We can rewrite (A.3) as Z1
p
Z1 .x K/dF .t; x/ D
1˚
K
K
1 a2 ˚ 1 .x/ C.t/ a
!! dx:
(A.4)
Note that p 1˚
1
a2 ˚ 1
.x/ C.t/
a
p
1a 2 ˚ 1 .x/C.t / a
!
Z
D 1 Z1
1
D p
1a 2 ˚ 1 .x/C.t / a
y2 1 p e 2 dy 2
y2 1 p e 2 dy: 2
Returning to (A.4) we get for the right-hand expression Z1 K
Z1 p
1a 2 ˚ 1 .x/C.t / a
y2 1 p e 2 dydx: 2
(A.5)
With the variable change ˚ 1 .x/ D x 0 , which is equivalent to x D ˚.x 0 /, we get: 1 x 02 dx D ˚ 0 .x 0 /dx 0 D p e 2 dx 0 : 2 The integration limits are from ˚ 1 .K/ to
˚ 1 .1/ D 1: So we get for (A.5) 1 Z
Z1
˚ 1 .K/
p
1a 2 x 0 CC.t / a
1 Z
2 1 0 0 y2 p ˚ .x /e dydx 0 2 1 x02 Cy 2 2 e dydx 0 2
Z1
D ˚ 1 .K/
p
1a 2 x 0 CC.t / a
˚ 1 Z .K/
Z1
D 1
p
1a 2 x 0 CC.t / a
1 x02 Cy 2 2 dydx 0 : e 2
(A.6)
A.4 Proof of Lemma on Expected Tranche Loss
261
p p 0 2 0 Now, changing the variable y 0 D ay C 1 a2 x 0 , i.e. y D 1aax Cy , we get dy 0 dy D a and the integration limits from
p a
1 a2 x 0 C C.t/ a
!
p
1 a2 x 0 D C.t/
p a1 C 1 a2 x 0 D 1:
to
Then (A.6) can be written as ˚ 1 Z .K/ 1 Z
1
C.t /
0 B 1 B @ 2a e
˚ 1 .K/C.t /
Z
Z
D 1
1
x 02 C
2 p 1a 2 x 0 Cy 0 2
1 x02 C2 e 2a
a2
1 C 0 0 C dy dx A
p
1a 2 x 0 y 0 Cy 02 2a 2
dy 0 dx 0 :
(A.7)
Recall that a bivariate normal distributed vector .X Y / with covariance matrix 1 : ˙D 1 has a distribution function written as follows: '2 .x; y; ˙/ D
p
1
2 1
2
e
x
2 2xyCy 2 2 1 2
.
/ dydx:
(A.8)
Then the function under the double integral in (A.7) is thepdensity function of a bivariate normal distributed vector .X 0 Y 0 / with D 1 a2 , and so the expression in (A.7) can be written as ˚ 1 Z .K/C.t Z /
'2 .x 0 ; y 0 ; /dy 0 dx 0 D ˚2 ˚ 1 .K/ ; C.t/;
1
(A.9)
1
with covariance matrix D
! p 2 1 1 a p : 1 a2 1
.
Appendix B
Normal Inverse Gaussian Process
Let X D X.t/; t 0 be a stochastic process defined on a probability space .˝; F ; P /, X W ˝ ! R. The process has: Independent increments if for each n2 N and each 0 t1 < t2 < ::: < tnC1 < 1
the random variables X.ti C1/ X.ti /; 1 i n are independent.
Stationary increments if each X.ti C1 /X.ti / is distributed as X.ti C1 ti /X.0/.
Definition B.1. X is called a L`evy process if: (i) X.0/ D 0 (a.s.) (ii) X has independent and stationary increments (iii) X is stochastically continuous, i.e. for all a > 0 and for all s 0 lim P .jX.t/ X.s/j > a/ D 0:
t !s
Here we are going to denote a standard Brownian motion with B D B.t/; t 0 and a Brownian motion with drift with D D D.t/; t 0. For the standard Brownian motion we have B.t/ N.0; t/. The Brownian motion with drift can be written as D.t/ D bt C B.t/. Then each D.t/ N.tb; 2 t/. Definition B.2. A subordinator T D T .t/; t 0 is a one-dimensional L`evy process that is non-decreasing (a.s), i.e. T .t/ 0 a.s. for each t 0 T .t1 / T .t2 / a.s. whenever t1 t2 :
(B.1) (B.2)
A subordinator can be thought of as a random model of time evolution. Definition B.3. The Inverse Gaussian subordinator is defined as T .ı; / .t/ D inffs > 0jD . / .s/ D ıtg;
(B.3)
with ı > 0, D . / .t/ D t C B.t/ and 2 R.
263
264
B Normal Inverse Gaussian Process
Definition B.4. The Normal Inverse Gaussian process can be defined as Z.t/ D t C D .ˇ / .T .ı; / .t//; for each t 0;
(B.4)
D .ˇ / .t/ D ˇt C B.t/;
(B.5)
where and T
.ı; /
.t/ is an inverse Gaussian subordinator such that T .ı; / .t/ D inffs > 0jD . / .s/ D ıtg; b D . / .t/ D t C B.t/;
(B.6) (B.7)
p and ˇ 2 R, D ˛2 ˇ 2 , ˛ 2 R with ˛ 2 ˇ2 and the standard Brownian b are independent. motions B.t/ and B.t/ Each Z.t/ has a density given by fZ.t / .xI ˛; ˇ; ; ı; t / D
p ıt ˛ exp .ıt C ˇ.x t // K1 ˛ ı 2 t 2 C .x t /2 ; p ı 2 t 2 C .x t /2 (B.8)
R1 where K1 .w/ WD 12 0 exp 12 w.t C t 1 / dt is the modified Bessel function of p the third kind and WD ˛ 2 ˇ 2 , which is exactly the density of a N I G .˛; ˇ; t; ıt/. Z.t/ has following moments: E.X / D t C ıt S.X / D 3
˛
ˇ p
ˇ
ı t
V .X / D ıt
˛2 3
2 ! ˇ 1 K.X / D 3 C 3 1 C 4 . ˛ ı t
The NIG process is by construction a time changed Brownian motion and is a L`evy process. The NIG process N.s/ .t/ defined in Sect. 6.2 is a special case of the general NIG process with parameters chosen in the way to have zero mean and variance t.
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