The Normal inverse Gaussian distribution for synthetic CDO

pricing

Anna Kalemanova∗ Bernd Schmid† Ralf Werner‡

1st August 2005

Abstract

This paper presents an extension of the popular Large Homogeneous Portfolio(LHP) approach to the pricing of CDOs. LHP (which has already become a standardmodel in practice) assumes a flat default correlation structure over the reference creditportfolio and models default using a one factor Gaussian copula. However, this modelfails to fit the prices of different CDO tranches simultaneously which leads to the wellknown implied correlation smile. Many researchers explain this phenomenon with thelack of tail dependence and propose to use a Student t copula. Incorporating the effectof tail dependence into the one factor portfolio credit model yields significant pricingimprovement. However, the computation time increases dramatically as the Student tdistribution is not stable under convolution. This makes it impossible to use the modelfor computationally intensive applications such as the determination of the optimalasset allocation in an investor’s portfolio over different asset classes including CDOs.We present a modification of the LHP model replacing the Student t distribution withthe Normal inverse Gaussian (NIG) distribution. We compare the properties of ournew model with those of the Gaussian and the double t copulas. The employment ofthe NIG distribution does not only speed up the computation time significantly butalso brings more flexibility into the dependence structure.

∗risklab germany, Nymphenburger Str. 112-116, 80636 Munich, Germany, anna.kalemanova@risklab.de†Senior Vice President, Algorithmics Incorporated, 101 Finsbury Pavement, London EC2A 1RS, UK,

bernd.schmid@fitchrisk.com‡Allianz, GRC Risk Methodology, Koniginstr. 28, 80802 Munchen, Germany, ralf.werner@allianz.de

1

Introduction

The calculation of loss distributions of the portfolio of reference instruments over differ-ent time horizons is the central problem of pricing synthetic CDOs. The factor copulaapproach for modelling correlated defaults has become very popular. Unfortunately, com-putationally intensive Monte Carlo simulation techniques have to be used if the correlationstructure is assumed to be completely general. However, the concept of conditional in-dependence yields substantial simplification: If it is assumed that defaults of differenttitles in the credit portfolio are independent conditional on a common market factor itis much simpler to compute the aggregate portfolio loss distributions for different timehorizons. The factor (or conditional independence) approach allows to use semi-analyticcomputation techniques avoiding time consuming Monte Carlo simulations. Examples arethe approaches described by Gregory and Laurent [8] who use fast Fourier transformationtechniques, as well as Hull and White [7] and Andersen, Sidenius and Basu [2] who applyan iterative numerical procedure to build up the loss distribution for the pool of referenceinstruments.

Going one step further and making the additional simplifying assumption of a large ho-mogeneous portfolio (LHP), i.e. we assume that it is possible to approximate the realreference credit portfolio with a portfolio consisting of a large number of equally weightedidentical instruments (having the same term structure of default probabilities, recoveryrates, and correlations to the common factor), we get a closed-form analytic syntheticCDO pricing formula. This LHP limit approximation employing the Law of Large Num-bers was first proposed by Vasicek [14], [15].

In the case of the one factor Gaussian copula all integrals in the pricing formulas canbe computed analytically (see [9]). Due to its simplicity, this model has become marketstandard. However, it appears to have a fundamental problem quite similar to the prob-lem in option pricing with the Black-Scholes model. Implied correlations that have beencomputed with the LHP model to fit observable market prices of different tranches of thesame CDO are not the same over the whole structure but exhibit a correlation smile. Themain explanation of this phenomenon is the lack of tail dependence of the Gaussian cop-ula. Various authors have proposed different ways to bring more tail dependence into themodel. One approach is the introduction of additional stochastic factors into the model.Andersen and Sidenius [1] extended the Gaussian factor copula model to random recov-ery and random factor loadings. Trinh et al [13] allowed the possibility of idiosyncraticand systematic jumps to default. Many authors proposed a different approach - to usea copula that exhibits more tail dependence: Examples are the Marshall-Olkin copula inAndersen and Sidenius [1], the Student t copula in O’Kane and Schloegl [10], and thedouble t distribution in Hull and White [7].

Burtschell, Gregory and Laurent [5] performed a comparative analysis of a Gaussian cop-ula model, stochastic correlation extension to Gaussian copula, Student t copula model,double t factor model, Clayton copula and Marshall-Olkin copula. By pricing the tranchesof DJ iTraxx they showed that Student t and Clayton copula models provided results verysimilar to the Gaussian copula model. The Marshall-Olkin copula lead to dramatic fatten-

2

ing of the tail. The results of the double t factor model and stochastic correlation modelwere closer to the market quotes, and the factor loading model of Andersen [1] performedsimilar to the latter ones.

Unfortunately, the integrals in the pricing expressions of the LHP model with double tfactor copula cannot be computed analytically. The major problem is the instability of theStudent t distribution under convolution. The calculation of default thresholds (that arequantiles of the distribution of asset returns) requires a numerical root search procedureinvolving numerical integration that increases the computation time dramatically. Thus,using a different heavy tailed distribution that is similar to Student t but stable underconvolution would help to decrease the computation time tremendously. As computationtime is an important issue for a large range of applications such as the determination ofan optimal portfolio asset allocation (including CDO tranches) where CDO tranches haveto be repriced in each scenario path at each time step in the future, the usage of such adistribution is crucial.

In our opinion, the Normal Inverse Gaussian (NIG) distribution is an appropriate dis-tribution to solve the problem. The family of NIG distributions is a special case of thegeneralised hyperbolic distributions (Barndorff-Nielsen [3]). Due to their specific charac-teristics, NIG distributions are very interesting for applications in finance - they are a gen-erally flexible four parameter distribution family that can produce fat tails and skewness,the class is convolution stable under certain conditions and the cumulative distributionfunction, density and inverse distribution functions can still be computed sufficiently fast([12] and chapter four). The distribution has been employed, e.g., for stochastic volatilitymodelling (Barndorff-Nielsen [4]).

This paper is organised as follows: In the first section we recall the general pricing approachfor synthetic CDOs. The one factor Gaussian copula model and the LHP approach arediscussed in the second section. In the third section we present the extension of LHP to adouble t factor model. The fourth section contains a brief review of the properties of theNIG distribution including a short description of its efficient implementation. In the fifthsection we present the one factor NIG copula model and the CDO pricing formulas forour NIG LHP model. Finally, the last two sections provide empirical investigations andcomparative analyses of the pricing abilities of the three models and their tail dependences.

1 General semi-analytic approach for pricing synthetic CDOs

We consider a CDO with a reference portfolio consisting of credit default swaps. SuchCDOs that use credit derivatives rather than bonds or loans to get an exposure to creditrisk are call synthetic CDOs. A protection seller of a synthetic CDO tranche receivesspread payments on the outstanding notional at regular payment dates (usually quarterly).If the total loss of the reference credit portfolio exceeds the notionals of the subordinatedtranches, the protection seller has to make compensation payments for these losses to theprotection buyer.

Basically, the pricing of a synthetic CDO tranche that takes losses from K1% to K2% of

3

the reference portfolio can be performed in the same way as the pricing of a credit defaultswap. Let’s assume that

0 <= t0 < ... < tn−1 (1)

denote the spread payment dates.

Then, the value of the premium leg of the tranche is the present value of all expectedspread payments:

Premium Leg =

n∑

i=1

∆ti · spread ·(

1 − EL(K1,K2) (ti−1))

· B(t0, ti−1), (2)

where ∆ti = ti − ti−1, B(t0, ti) is the discount factor and EL(K1,K2) (ti) is the expectedpercentage loss of the (K1 − K2) CDO tranche. The value of the protection leg can becalculated according to:

Protection Leg =

tn∫

t0

B(t0, s)dEL(K1,K2) (s) (3)

≈n∑

i=1

(

EL(K1,K2) (ti) − EL(K1,K2) (ti−1))

· B(t0, ti).

At issuance of the CDO tranche the tranche spread is determined so that the values ofpremium leg and protection leg are equal. Note, that all expectations are calculated underthe risk neutral measure.

Assume, we had already computed the time t discrete loss distribution of the referenceportfolio (for simplicity we omit the index t):

{Lk with probability pk}k=1,...,m.

Then the (K1 − K2) CDO tranche suffers a loss of (min(Lk,K2) − K1)+ with probability

pk and the percentage expected loss of the tranche can be easily calculated:

EL(K1,K2) =1

K2 − K1

m∑

k=1

(min(Lk,K2) − K1)+ · pk. (4)

Given a continuous portfolio loss distribution function F (x), the percentage expected lossof the (K1 − K2) CDO tranche can be computed as:

EL(K1,K2) =1

K2 − K1

1∫

K1

(x − K1)dF (x) −1∫

K2

(x − K2)dF (x)

. (5)

Thus, the central problem in the pricing of a CDO tranche is to derive the loss distributionof the reference portfolio. In the next sections we present the factor copula model ofcorrelated defaults as well as semi-analytical and analytical approximation methods tocompute the portfolio loss distribution and the expected loss of a tranche.

4

2 One factor Gaussian model and LHP approximation

The LHP approach is based on a one factor Gaussian copula model of correlated defaults.Assume that the portfolio of reference assets consists of m financial instruments and theasset return of the i-th issuer in the portfolio, Ai, is assumed to be of the form:

Ai = aiM +√

1 − a2i Xi, (6)

where M,Xi, i = 1, ...,m are independent standard normally distributed random variables.Then, conditionally on the common market factor M , the asset returns of the differentissuers are independent. Note, that due to the stability of normal distributions underconvolution the asset return Ai follows a standard normal distribution as well.

According to Merton’s approach we assume that default occurs when the asset return ofobligor i crosses the threshold Ci, which is implied by the obligor’s default probability pi :

pi = P [Ai ≤ Ci] = Φ (Ci) . (7)

The model is calibrated to observable market prices of credit default swaps, i.e. the defaultthresholds are chosen so that they produce default probabilities implied by quoted creditdefault swap spreads:

Ci = Φ−1(pi).

According to equation (6), the i-th issuer defaults if

Xi ≤Ci − aiM√

1 − a2i

.

Then the probability that the i-th issuer defaults conditional on the factor M is

pi(M) = Φ

Ci − aiM√

1 − a2i

. (8)

Now, we assume that the portfolio is homogeneous, i.e. the correlations of the asset returnsto the market ai and the default thresholds Ci are the same for all i. Then, the defaultprobability of all issuers in the portfolio conditional on M is given by

p(M) = Φ

(

C − aM√1 − a2

)

. (9)

Furthermore, we assume that the notional amounts of all issuers in the portfolio are thesame and denote the notional of the portfolio by N . Finally, the recovery rate R is as-sumed to be constant and the same for all issuers.

Then, the loss distribution of the portfolio can be computed as follows – the probabilityof the loss L being Lk = k

mN(1 − R) is equal to the probability that exactly k out of missuers default:

P [L = Lk|M ] =

(

m

k

)

Φ

(

C − aM√1 − a2

)k (

1 − Φ

(

C − aM√1 − a2

))m−k

. (10)

5

Hence, due to conditional independence and two possible states the conditional loss dis-tribution is binomial. The unconditional loss distribution P [L = Lk] can be obtained byintegrating (10) with the distribution of the factor M :

P [L = Lk] =

(

m

k

)

∞∫

−∞

Φ

(

C − au√1 − a2

)k (

1 − Φ

(

C − au√1 − a2

))m−k

dΦ (u) . (11)

Since the calculation of the loss distribution in (10) is quite computationally intensive forlarge m, it is desirable to use some approximation. The large portfolio limit approximationproposed by Vasicek [15], [14] is a very simple but powerful method.

For simplicity let us first assume a zero recovery rate.

We consider the cumulative probability of the percentage portfolio loss not exceeding θ

Fm(θ) =

[mθ]∑

k=1

P [L = Lk] .

With the substitution s = Φ(

C−au√1−a2

)

and from (11) we get the following expression for

Fm(θ):

Fm(θ) =

[mθ]∑

k=1

(

m

k

)

1∫

0

sk(1 − s)m−kdΦ

(√1 − a2Φ−1(s) − C

a

)

. (12)

Since

limm→∞

[mθ]∑

k=1

(

m

k

)

sk(1 − s)m−k =

{

0, if θ < s

1, if θ > s

the cumulative distribution of losses of a large portfolio equals

F∞(θ) = Φ

(√1 − a2Φ−1(θ) − C

a

)

. (13)

In case of the large homogeneous portfolio it is possible to compute the integrals in (5)analytically and the expected loss of the mezzanine tranche is given by:

EL(K1,K2) =Φ2

(

−Φ−1 (K1) , C,−√

1 − a2)

− Φ2

(

−Φ−1 (K2) , C,−√

1 − a2)

K2 − K1,

where Φ2 is the bivariate normal distribution function.

6

Now, let us assume that assets have the same (maybe non-zero) recovery R. The total lossof the equity tranche of K% will occur only when assets of the total amount of K%

1−R havedefaulted. Thus, the expected loss of the tranche between K% and 100% is given by

ELR(K,1) =

1∫

K

1−R

(1 − R)

(

x − K

1 − R

)

f(x)dx

= (1 − R)EL( K

1−R,1),

where ELR denotes the loss under recovery R. Finally, it is easy to see that the expectedpercentage loss of the mezzanine tranche taking losses from K1 to K2 under the assumptionof constant recovery R is

ELR(K1,K2)

= EL� K1

1−R,

K2

1−R

�. (14)

3 LHP with double t copula

One natural extension of the LHP approach could be an assumption of some heavy taildistribution. The double t one factor model proposed by Hull and White [7] assumesStudent t distribution for the common market factor M as well as for the individualfactors Xi. Then the loss distribution F∞ in (13) is modified in the following way:

F∞(x) = T

(√1 − a2T−1 (x) − C

a

)

,

where T denotes the Student t distribution function. Unfortunately, it is not possible tosolve the integral in (5) analytically and one has to use some numerical integration method.

The asset returns Ai do not follow necessarily Student t distributions since the Studentt distribution is not stable under convolution. The distribution function Hi of Ai mustbe computed numerically. Afterwards, it is possible to find the default thresholds Ci =H−1

i (pi). This procedure is quite time consuming and it makes the double t model tooslow for Monte Carlo based risk management applications.

4 NIG distribution and its main properties

4.1 Definition and properties of the NIG distribution

The normal inverse Gaussian distribution is a mixture of normal and inverse Gaussiandistributions.

A non-negative random variable Y has inverse Gaussian distribution with parametersα > 0 and β > 0 if its density function is of the form:

fIG(y; α, β) =

α√2πβ

y−3/2 exp

(

−(α − βy)2

2βy

)

, if y > 0

0 , if y ≤ 0.

7

A random variable X follows a Normal Inverse Gaussian (NIG) distribution with parame-ters α, β, µ and δ if:

X | Y = y ∼ N (µ + βy, y)

Y ∼ IG(δγ, γ2) with γ :=√

α2 − β2,

with parameters satisfying the following conditions: 0 ≤ |β| < α and δ > 0. Wethen write X ∼ NIG(α, β, µ, δ) and denote the density and probability functions byfNIG(x; α, β, µ, δ) and FNIG(x; α, β, µ, δ) correspondingly.

The density of a random variable X ∼ NIG(α, β, µ, δ) is quite complicated:

fNIG(x; α, β, µ, δ) =δα · exp (δγ + β(x − µ))

π ·√

δ2 + (x − µ)2K1

(

α√

δ2 + (x − µ)2)

,

where K1(w) := 12

∫∞0 exp

(

−12w(t + t−1)

)

dt is the modified Bessel function of the thirdkind. The NIG moment generating function M(t) = E[exp (tx)] has a quite simple form:

MNIG(x; α, β, µ, δ) = exp (µt)exp

(

δ√

α2 − β2)

exp(

δ√

α2 − (β + t)2) .

The main properties of the NIG distribution class are the scaling property

X ∼ NIG(α, β, µ, δ) ⇒ cX ∼ NIG(

α

c,

β

c, cµ, cδ

)

, (15)

and the closure under convolution for independent random variables X and Y

X ∼ NIG(α, β, µ1, δ1), Y ∼ NIG(α, β, µ2, δ2)

⇒ X + Y ∼ NIG (α, β, µ1 + µ2, δ1 + δ2) . (16)

The moments (mean, variance, skewness and kurtosis) of a random variable X ∼NIG(α, β, µ, δ) are:

E(X) = µ + δβ

γV(X) = δ

α2

γ3

S(X) = 3β

α · √δγK(X) = 3 + 3

(

1 + 4

(

β

α

)2)

1

δγ.

8

4.2 Efficient implementation of the NIG distribution

So far we have seen that, theoretically, it is much more convenient to work with theNIG distribution than with the Student t distribution. Still, it is not obvious if the NIGdistribution, especially due to its rather complicated density function involving a Besselfunction, can be implemented efficiently. As the main driver of our considerations has beenspeed of computation, an efficient implementation is crucial. In the following paragraphswe briefly describe our implementation in Matlab.

• The implementation of the NIG distribution can be done straightforward using thepretty efficient Matlab version of the Bessel function.

• Instead of only using the obvious representation of

FNIG(x) =

x∫

−∞

fNIG(t)dt

we additionally use the fact that the NIG distribution stems from a convolution ofthe normal and the inverse Gaussian distribution:

FNIG(x) =

∞∫

0

Φ(x − (µ + βy)√

y

)

fIG(y, δγ, γ2)dy

Depending on the number of points where the same distribution function is evalu-ated, we either use the first or the second representation. If only evaluation of a fewpoints is necessary, the second version is more efficient. If more points should be cal-culated, the first representation can be used and the integral is split into appropriatesmall integrals.

Both integrals do not live on a compact interval, hence we further split up theintegral in a compact part and the rest. For this remainder, we use a logarithmictransformation. If we use for example t = exp(−y) the latter integral is transformedto the compact interval

FNIG(x) =

1∫

0

Φ(x − (µ − β log(t))

√

log(t)

)

fIG(− log(t), δγ, γ2)1

tdt

We approximate all occuring integrals by a Gauss cubature formula using four Gaussnodes. Depending on the required accuracy we recursively integrate on subdividedintegration domains.

• For the calculation of the inverse distribution function – even if it is only evaluatedat one point y – we table the distribution function FNIG on a fine grid around themean (with size equal to several standard deviations). This table is then used to findan x with FNIG(x) = y by using linear interpolation inside the grid and logarithmicinterpolation outside the grid. This can still be done much faster than using forexample Newton’s method for root finding for FNIG .

As we can see from table 1 the computation time is a factor 6 faster than for the Studentt example, where we have used similar tricks for the speed-up of the computation, whileworking with a much more flexible distribution.

9

5 LHP with NIG copula

We define the one factor NIG copula model in the following way. The asset return of thei-th issuer in a portfolio Ai is assumed to be of the form:

Ai = aM +√

1 − a2Xi,

where M,Xi, i = 1, ...,m are independent NIG random variables with the following para-meters:

M ∼ NIG(

α, β, − αβ√

α2 − β2, α

)

Xi ∼ NIG(√

1 − a2

aα,

√1 − a2

aβ, −

√1 − a2

a

αβ√

α2 − β2,

√1 − a2

aα

)

.

Then, using the scaling property and stability under convolution of NIG distribution in(15) and (16) we get NIG distributed asset returns:

Ai ∼ NIG(

α

a,

β

a, −1

a

αβ√

α2 − β2,

α

a

)

.

The second and the third parameters are chosen to get zero expected value. Fornon-zero β we will get a skewed distribution. To simplify notations we denote

FNIG

(

x; sα, sβ, −s αβ√α2−β2

, sα

)

with FNIG(s)(x).

One big advantage of the NIG copula is that the default thresholds are easy and fast tocompute:

C = F−1NIG( 1

a)(p).

The default probability of each instrument in the portfolio is given by

p(M) = FNIG

�√1−a2

a

�(C − aM√1 − a2

)

,

and the loss distribution function of the large homogeneous portfolio is

F∞(x) = FNIG(1)

√1 − a2F−1

NIG�√

1−a2

a

�(x) − C

a

.

6 Comparison of the models: pricing DJ iTraxx

In order to compare the properties of the LHP model with Gaussian, double t and NIGcopulas, we consider the example of the tranched Dow Jones iTraxx Europe with 5 years

10

Table 1: Pricing DJ iTraxx tranches with LHP model with different distributions

Market Gaussian t(4)-t(4) t(5)- t(5) t(6)-t(6) NIG(1) NIG(2)

0-3% 24,70% 24,70% 24,70% 24,70% 24,70% 24,70% 24,70%3-6% 160 bp 243 bp 135 bp 155 bp 171 bp 157 bp 159 bp6-9% 49 bp 73 bp 56 bp 60 bp 64 bp 63 bp 51 bp9-12% 22,5 bp 25 bp 33 bp 33 bp 34 bp 36 bp 23,5 bp12-22% 13,75 bp 4 bp 18 bp 16 bp 14 bp 16 bp 8 bp

absolute error 120 bp 47 bp 28 bp 39 bp 33 bp 10 bp

correlation 19,47% 26,26% 25,16% 24,79% 22,62% 25,59%α 0,8833 1,2558β 0 -0,2231

comp. time 0,3 s 11 s 9 s 8,5 s 1,7 s 1,5 s

maturity. The reference portfolio consists of 125 credit default swap names. The stan-dard tranches have attachment/detachment points at 3%, 6%, 9%, 12% and 22%. Theinvestors of the tranches receive quarterly spread payments on the outstanding notionaland compensate for losses when they hit the tranche they are invested in. The investorof the equity tranche receives an up-front fee that is quoted in the market and an an-nual spread of 500 bp quarterly. The settlement date of the third series of this index is20-September-2005 and matures on 20-September-2010. We consider the market quotesof iTraxx tranches at 11-April-2005. The average CDS spread of the corresponding CDSportfolio with 5 years maturity is 37,5 bp at this day. We use the constant default intensitymodel to derive the marginal default distributions, and estimate the default intensity ofthe large homogeneous portfolio from the average portfolio CDS spread. The constantrecovery rate is assumed to be 40%.

Gaussian and double t factor copulas have only one parameter: correlation. We estimatethis parameter so that the price of the equity tranche fits the market quote, i.e. impliedequity correlation. The same correlation is used to price the other tranches. The versionsof the NIG factor copula we consider have one or two parameters besides the correlation.We use least squares estimation to estimate these parameters such that they fit all theobserved tranche spreads.

Table 1 presents the market quotes of the iTraxx tranches as well as the prices of the LHPmodel with Gaussian one factor copula, double t distribution with 4, 5 and 6 degrees offreedom and NIG factor copula with one and two free parameters. In the one parameterNIG copula, the parameter β is set to zero which makes the distribution symmetric. Thedouble t factor copula models with 4, 5, and 6 degrees of freedom overprice the three mostsenior tranches. The one with 5 degrees of freedom provides the best result for the 3-6%tranche. The results of the NIG copula with one parameter are similar to the results ofdouble t copula with 5 degrees of freedom. The additional free parameter in the NIGcopula makes it more flexible: all tranches besides the most senior one can be fitted by

11

the NIG(2) model very well.

We plot density and cumulative distribution functions of portfolio loss from the LHPmodel with Gaussian, double t(5) and NIG factor copulas in the Figure 1. From the Fig-ure 2 that shows the differences between modified (t or NIG) and Gaussian densities it isespecially easy to see that all modified models redistribute risk out of the lower end of theequity tranche to its higher end. Note that total risk difference within the equity trancheis zero since we have calibrated all models to fit the equity tranche. Gaussian model al-locates more risk to mezzanine tranches than the modified models. Figure 3 shows thatNIG(2) copula allocates more risk at the 3-6% tranche than the double t copula. Figure4 presents the density function of the asset returns. The density of the NIG(2) model isslightly skewed to the right.

We have reformulated the prices produced by all models into base correlations (Figure 5).Base correlations are the implied Gaussian model correlations of the corresponding equitytranches, i.e. of 0-3%, 0-6%, 0-9%, 0-12% and 0-22% tranches. The NIG model with twoparameters fits all market base correlations besides the last one very well.

Base correlation is very popular among the market participants since due to its monotonic-ity it can be easily interpolated and used rather than the compound correlation to price theoff-market tranches with the Gaussian model. However, the big problem of this method isthat it can produce arbitrage prices. We have computed the prices of different off-markettranches for our test sample using the Gaussian model with base correlations. We couldobserve the problem of arbitrage prices for some senior tranches. Figure 6 shows thespreads for 15-16%, 15-17%, ..., 15-22% tranches of iTraxx. The spreads start growingfrom the detachment point of 18%.

The big advantage of our NIG model is that we can compute the price of any off-markettranche with the same correlation value that guarantees arbitrage-free pricing. We havedone this and compared the results with the base correlation Gaussian model. Figure 7shows that the off-market prices of the NIG model are very close to those of the Gaussianmodel in its arbitrage-free part. In the lower right part of the figure the Gaussian modelproduces arbitrage prices for the most senior tranches while the prices of the NIG modelcontinue decreasing monotonically.

7 Comparison of the tail dependence

In this section we investigate the tail dependence of the one factor copulas under study. Inparticular, the amount of dependence in the lower-left-quadrant is relevant for modellingcredit portfolios.

Let X1 and X2 be continuous random variables. We consider the coefficient of lower taildependence:

λL(x) = P{X2 ≤ x|X1 ≤ x}.Random variables X1 and X2 are said to be asymptotically dependent in the lower tail if

12

λL > 0 and asymptotically independent in the lower tail if λL = 0, where

λL = limx→−∞

λL(x).

The coefficient of upper tail dependence is defined as

λU (x) = P{X2 > x|X1 > x},

and X1 and X2 are asymptotically dependent in the upper tail if λU > 0, where

λU = limx→∞

λU (x).

We plot the tail dependence coefficients of Gaussian, double t(5) and NIG factor copulasin Figure 8. The Gaussian copula shows no tail dependence. The tail dependence coeffi-cient of double t(5) copula is significantly larger than that of the Gaussian copula. Thetail dependence of the NIG copula with one parameter is similar to the tail dependenceof the double t(5) copula. Note, that the upper and the lower tail dependence structuresof Gaussian, double t(5) and NIG(1) copulas are symmetric. This is not the case for twoparameter NIG copula. This copula has a higher lower tail dependence coefficient lyingbetween those of Gaussian and double t copulas, and a very small upper tail dependencecoefficient.

These properties of the copulas under study can be observed in Figure 9 as well. Wehave simulated pairs of correlated asset returns with Gaussian, double t with 5 degreesof freedom, NIG with one and two parameters factor copulas and plotted the contours oftheir joint densities. The double t factor copula produces more extreme values than theother copulas. The NIG copula with two parameters has more extreme values in the lowerleft tail.

Conclusion

This paper has introduced the NIG factor copula as an extension of the LHP model forpricing synthetic CDOs. We have presented an analytical formula for the distributionfunction of the portfolio loss. In general, the NIG distribution has four free parameters.The standardised symmetric NIG distribution with zero mean and unit standard devia-tion still has one free real parameter. The ability of this structure to fit CDO tranchesis similar to that of the double t copula that has one free integer parameter. Further-more, we have studied the properties of the one factor copula model with a skewed NIGdistribution having two free parameters. This copula has non symmetric upper and lowertail dependence. The two parameter NIG factor copula model is more flexible to achievea good fit of all iTraxx tranches and thus providing a more realistic distribution of theportfolio loss.

The main purpose of developing the NIG factor copula was simplification and speedingup computation of the default thresholds. Incorporating the effect of tail dependence intothis simple one factor credit portfolio model by means of replacing Gaussian distributionwith Student t distribution for the factors has the following disadvantage: it is not possible

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to compute the distribution function of asset returns analytically because of the lack ofstability under convolution. This leads to a dramatic increase of computation time andmakes it impossible to use this model for some important applications in practice. TheNIG distribution still has a higher tail dependence than the Gaussian distribution. Inaddition, it is stable under convolution under certain conditions. The employment of theNIG distribution does not only bring significant improvement with respect to computationtimes but also more flexibility in the modelling of the dependence structure of a creditportfolio.

References

[1] Leif Andersen and Jakob Sidenius. Extensions to the Gaussian copula: randomrecovery and random factor loadings. Journal of credit risk, 1(1), 2005.

[2] Leif Andersen, Jakob Sidenius, and Susanta Basu. All your hedges in one basket.Risk, November 2003.

[3] O.E. Barndorff-Nielsen. Hyperbolic distributions and distributions on hyperbolae.Scandinavian Journal of statistics, 5:151–157, 1978.

[4] O.E. Barndorff-Nielsen. Normal inverse gaussian distributions and stochastic volatil-ity modelling. Scandinavian Journal of statistics, 24:1–13, 1997.

[5] X. Burtschell, J. Gregory, and L.-P. Laurent. A comparative analysis of CDO pricingmodels. Working paper, April 2005.

[6] P. Embrechts, F. Lindskog, and A. McNeil. Modelling Dependence with Copulas and

Applications to Risk Management, 2001.

[7] John Hull and Alan White. Valuation of a CDO and an n-th to default CDS withouta Monte Carlo simulation. Journal of Derivatives, 2004.

[8] Lean-Paul Laurent and Jon Gregory. Basket default swaps, CDO’s and factor copulas.Working paper, September 2003.

[9] D. O’Kane and L. Schlogl. Modelling credit: Theory and practice. Lehman Brothers

Analytical Research Series, February 2001.

[10] Dominic O’Kane and Lutz Schloegl. An analytical portfolio credit model with taildependence. Quantitative credit research, January 2003. Lehman Brothers.

[11] Lutz Schloegl and Andrei Greenberg. The shape of implied loss distribution. Quan-

titative credit research, March 2005. Lehman Brothers.

[12] Michaela Tempes. Implementierung einer copula-toolbox unter matlab. Master’sthesis, TU Munich, 2005.

[13] Minh Trinh, Robin Thompson, and Mukundan Devarajan. Relative value of CDOtranches: a view through ASTERION. Quantitative credit research, March 2005.Lehman Brothers.

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[14] Oldrich Vasicek. Limiting loan loss probability distribution. Memo, KMV corpora-tion, August 1987.

[15] Oldrich Vasicek. Probability of loss n loan portflio. Memo, KMV corporation, Feb-ruary 1987.

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0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Loss (%)

Gaussian copulat(5)−t(5)NIG(1) copulaNIG(2) copula

(a) Cumulative distribution function

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

Loss (%)

Gaussian copulat(5)−t(5)NIG(1) copulaNIG(2) copula

(b) Density function

Figure 1: Portfolio loss distribution from LHP model

16

0 1 2 3 4 5 6 7 8 9 10−25

−20

−15

−10

−5

0

5

10

15

Loss (%)

t(5)−t(5) − GaussianNIG(1) − GaussianNIG(2) − Gaussian

Figure 2: Difference between modified and Gaussian loss density

0 1 2 3 4 5 6 7 8 9 10−10

−5

0

5

10

15

Loss (%)

NIG(1) − t(5)−t(5)NIG(2) − t(5)−t(5)

Figure 3: Difference between double t and NIG loss density

17

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45Gaussian copulat(5)−t(5)NIG(1) copulaNIG(2) copula

(a) Density function

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −10

0.05

0.1

0.15

0.2

0.25

Gaussian copulat(5)−t(5)NIG(1) copulaNIG(2) copula

(b) Density function: left tail

Figure 4: Distribution of asset returns

18

3 6 9 12 2215

20

25

30

35

40

45

50

55

60

tranche (%)

base

cor

rela

tion

(%)

MarketGaussian Copulat(5)−t(5)NIG(1)NIG(2)

Figure 5: Implied base correlation

16 17 18 19 20 21 2212

12.2

12.4

12.6

12.8

13

13.2

13.4

13.6

13.8

14

detachment point (x%)

tran

che

spre

ad (

bp)

Figure 6: Spreads for the 15-x% tranches calculated with the Gaussian model with basecorrelations

19

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220

200

400

600

800

1000

1200

x(%)

(bp)

Gaussian CopulaNIG(2)

0−x% tranche

3−x% tranche

4−x% tranche

5−x% tranche

(a)

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220

10

20

30

40

50

60

70

80

90

100

x(%)

(bp)

Gaussian CopulaNIG(2)

(b)

Figure 7: Spreads for off-market tranches

20

−7 −6 −5 −4 −3 −2 −10

0.05

0.1

0.15

0.2

0.25

0.3

x

lam

bda(

x)

Gaussian copulat(5)−t(5)NIG(1) copulaNIG(2) copula

(a) Lower tail dependence

1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

0.3

x

lam

bda(

x)

Gaussian copulat(5)−t(5)NIG(1) copulaNIG(2) copula

(b) Upper tail dependence

Figure 8: Tail dependence coefficients

21

Gaussian

−2 0 2

−3

−2

−1

0

1

2

3

t(5)−t(5)

−2 0 2

−3

−2

−1

0

1

2

3

NIG(1)

−2 0 2

−3

−2

−1

0

1

2

3

NIG(2)

−2 0 2

−3

−2

−1

0

1

2

3

Figure 9: Density contours of one factor copula

22