bloom berg, konikov] basket default swaps

8/14/2019 Bloom Berg, Konikov] Basket Default Swaps

1/22

Basket Default Swaps

M. Konikov, M. Marinescu, and H. Stein

Bloomberg LP

August 9, 2004

Contents

1 Introduction 2

2 Pricing basket default swaps 3

3 Copula models 5

3.1 Gaussian copula . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Clayton copula . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Tail dependencies in the copula models . . . . . . . . . . . . . 8

4 Implied marginal distribution of time to default 9

5 BDS sensitivity measures 9

6 Discussions 10

1


2/22

1 Introduction

A basket default swap (BDS) is a default protection instrument written ona basket of m bonds. The buyer of the protection pays a specified rateon a specified notional principal until the n-th (n m) bond in the basketdefaults or the contract expires. If the n-th default occurs before the contractexpiration, the buyer is entitled either to exchange the bond issued by then-th defaulted entity for its face value N or to receive a cash equivalentpayment given by

(1 Rn)N, (1)

where Rn is the corresponding recovery rate.1

In general, a BDS contract is characterized by: an underlying basket ofmbonds (e.g. 5), a number of defaults that triggers the default payment (e.g.2), the maturity of the deal (e.g. 5 years), the coupon frequency (e.g. quar-terly), the notional principal (e.g. $10M), and the spread or coupon payment(e.g. 100bps). In addition, one must supply information about: the actualcoupon schedule (e.g. first/last coupon dates, the date generation method),the day count convention (e.g. ACT/360), the business day calendar, andthe payment dates offsets. We assume that the recovery rates are knownquantities supplied by the user (e.g. 0.4, 0.35, 0.2, 0.8, 0.4).2

The pricing methodology presented in this paper is based on the one-

factor mixture joint distribution model, proposed by Gregory and Laurent[1], Schonbucher [2], Hull and White [3], and Konikov, Madan, and Marinescu[4]. For a general description please see [5] and [6]. This is a non-Monte-Carlo method based on the assumption that each element of the basket iscoupled to a common single stochastic process and the joint distribution ofdefault times is constructed from the corresponding marginal distributionsvia a copula.

The CDSN Bloomberg function implements two copula models: Gaus-sian and Clayton. The implied marginal distributions of default times areparameterized by a piecewise exponential function. The piecewise exponen-tial coefficients are calibrated to the credit default swap (CDS) spreads ofdifferent maturities.

1We consider all the bond notionals in the basket to be equal to N2In general the recovery rates are known only when the default event is officially

recorded, and its value may depend on many economic factors. To express this mar-

ket reality, one can model the recovery rates as random variables. However, in this paper

we follow the common practice of assuming deterministic recovery rates.

2


3/22

2 Pricing basket default swaps

As already mentioned, under the one-factor mixture joint distribution model,each element of the basket is coupled to a single common stochastic processV called, for the purpose of this presentation, the market process. A directconsequence of this assumption is that conditional on a given market state,the probabilities of default for each entity in the basket are independent, andso the BDS evaluation problem becomes quasi-analytic.

The BDS floating leg consists of a single payment of (1 Rk)N in theevent that the n-th default in the basket is produced by the bond belongingto the k-th entity, as in Eq. (1). We denote the present value of the floating

leg by F LT.In contrast, the fixed leg of a BDS contract consists of a succession ofregular coupon payments and an accrued payment if the n-th entity defaults.The present value of the fixed leg is proportional to the BDS spread, s (thecoupon rate). Let F IX denote the present value of the fixed leg for a unitBDS spread. Then, for any given BDS spread value s, the present value ofthe fixed leg is given by s F IX. Let CP and ACC be the coupons andaccrued contributions to the F IX, respectively. Then we can write,

F IX = CP + ACC. (2)

With these notations, the BDS price (present value) is given formally by

P V = F LT s FIX. (3)

A BDS is said to be priced at par if its price is zero. In this case the BDSspread is called the BDS par-spread. The BDS par-spread is given formallyby

spar =F LT

F IX. (4)

Under the one factor mixture model one can rewrite the coupons, accrued,and floating leg contributions as

CP =Dv

(v)CP(v)dv, (5)

ACC =

Dv

(v)ACC(v)dv, (6)

F LT =

Dv

(v)FLT(v)dv, (7)

3


4/22


5/22

3 Copula models

Given the marginal distributions of time to default, the joint distribution maybe constructed using a copula function (see [7] for a detailed exposition). LetC(u1, , un) be a copula function and Fj(tj) be the marginal cumulativedistributions of default times. Then the joint distribution is given by

F(t1, , tm) = C

F1(t1), , Fn(tm)

. (13)

As we mentioned, for BDS pricing two copula models were implemented:Gaussian and Clayton.

3.1 Gaussian copula

The one factor Gaussian copula is associated with the multivariate normalrandom variables that display the correlation structure induced by linear de-pendence on a single normally distributed factor. This is the most commoncopula model used in pricing basket default swaps. It is also used by mar-ket participants as a convention to quote the BDS prices in terms of thecorrelation coefficients.

Let {Xj}mj=1 be a family of m random variables such that

Xj = jV +

1 2jWj, (14)

where V and {Wj}mj=1 are independent standard normal random variables,and (1, , m) is a correlation vector (i [1, 1]). Then, the Gaus-sian copula may be defined as

C(u1, . . . , um) = P ((X1) < u1, . . . , (Xm) < um)

=

P

W1 u) = limu0

1 2u + C(u, u)

1 u.

(30)

Then, for the Clayton copula model, we have

L = 21/, (31)

U = 0, (32)

while for the Gaussian copula model we have

L = 0, (33)

U = 0. (34)

These results show that, although both copula models capture the depen-dency among times to default, the Gaussian copula model fails to describeany tail dependency.

Figures 1, 2, and 3, 4 show the contour and 3D plots of the joint probabil-ity distribution function of two normally distributed random variable underthe Gaussian copula model for correlations = 0.15 and = 0.5, respectively.In both cases the contours are symmetric ellipsoidal curves.

Using the same type of contour and 3D plots, Figs. 5, 6, and 7, 8 show thejoint probability distribution function for the same two normally distributedrandom variables using the Clayton copula for = 0.365 and = 1, respec-tively. In both cases the contours are elongated in the region where bothrandom variables take negative values. This is a characteristic signature of

8


9/22

the high order dependency for the negative tails of the marginal distribu-

tions. We mention that for positive values the joint probability exhibits alow level of dependency, similar to the Gaussian copula model.3

We conclude that a BDS pricing model based on the the Clayton copulainfers more dependency for multiple defaults that a model based on theGaussian copula can describe.

4 Implied marginal distribution of time to de-

fault

The implied marginal distributions of default times are extracted from thequoted CDS spreads curves for each individual obligor. We choose a piecewiseexponential function to model the survival function, i.e.,

S(t) = exp

1t +

N1i=1

i+1 max(t Ti, 0)

, (35)

where i are parameters and Ti are the various CDS times to maturity. Con-sequently, the marginal cumulative distribution and the probability distribu-tion function of default times are given by,

F(t) = 1 exp

1t +

N1i=1

i+1 max(t Ti, 0)

, (36)

f(t) =

1 +

N1i=1

i+11tTi

exp

1t +

N1i=1

i+1 max(t Ti, 0)

,

(37)

respectively. By 1t


10/22

The sensitivity to the CDS spreads is measured by the Spread DV01".

The Spread DV01" is defined as the BDS price variation to a 1 bps parallelshift of each individual CDS spread curve. In addition we compute an aggre-gated Spread DV01" as the BDS price variation to a 1 bps parallel shift ofall the CDS spread curves simultaneously. Note that the aggregate SpreadDV01" is not the sum of the individual curves Spread DV01", but it is agood indicator of a global sensitivity of the BDS price to the CDS spreadcurves.

The sensitivity to the interest rates is measure by the IR DV01". TheIR DV01" is defined by the BDS price variation to a 1 bps parallel shift ofthe risk-free interest rate curve. The parallel shift is performed before the

calibration of the implied marginal distributions of times to default to theCDS spread curves.

6 Discussions

It is interesting to analyze the BDS spreads behavior as a function of thebasket dependencies, whether the dependency is represented as correlationin the Gaussian copula case or as in the Clayton copula case. We willdemonstrate this behavior in the Gaussian copula model with single correla-tion. The Clayton case can be treated in a similar fashion.

We consider two baskets of 5 names. For each basket we have computedthe BDS spread for first, second, third, fourth, and fifth to default as afunction of the single (common) correlation factor . The BDS spread isreported as a percentage of the aggregate sum of the equivalent CDS spreads.

The first basket consists of 5 names of relatively equal creditworthiness.The results are presented in Fig. 9. We note that, as a function of , thefirst to default curve is decreasing, while the higher order default curvesare increasing. This behavior can be explained by the fact that while thecorrelation is increasing the basket entities tend to behave similarly. Thus,the probability to have a first to default event is decreasing as goes to 1

since the other four entities that dominate the basket are not in default. Atthe other extreme, the probability of the fifth to default event is increasingas goes to 1 since all the basket entities will tend to default at the sametime.

A peculiar behavior is presented by the second to default curve thathas a maximum around = 0.9. For < 0.9 the curve is increasing as a

10


11/22

result of the increasing probability of a second to default event. For > 0.9

the other three basket entities that are not in default dominate the basketbehavior leading to an overall effect of decreasing the probability of a secondto default event. We mention that the actual value where the curve behavioris changing depends on the marginal probability of time to default of thebasket entities.

The second basket consist of 4 names of relatively equal creditworthinessplus the fifth name that is a lot riskier than the others. The results for thefirst, second, third, fourth, and fifth to default are presented in Fig. 10. Theoverall behavior of the BDS spread curves as a function of the correlation forthis basket is similar to the first basket. However, we note that the first to

default curve is a lot higher than the others. In fact, the first to default curveis less sensitive to the correlation. This behavior is explained by the fact thatthe riskiest basket entity tends to be almost always the first to default.

11


12/22

3 2 1 0 1 2 33

2

1

0

1

2

3

variable 1

variab

le

2

Gaussian Copula Contours for = .15

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Figure 1: Contour plot of the joint probability function of two normally

distributed random variables using a Gaussian copula for = 0.15.

12


13/22

Figure 2: 3D plot of the joint probability function of two normally distributed

random variables using a Gaussian copula for = 0.15.

13


14/22

3 2 1 0 1 2 33

2

1

0

1

2

3

variable 1

variab

le

2

Gaussian Copula Contours for = .5

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18


distributed random variables using a Gaussian copula for = 0.5.

14


15/22


random variables using a Gaussian copula for = 0.5.

15


16/22

3 2 1 0 1 2 33

2

1

0

1

2

3

variable 1

variab

le

2

Clayton Copula Contours for = .3654 (L

= .15)

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16


distributed random variables using a Clayton copula for = 0.365.

16


17/22


random variables using a Clayton copula for = 0.365.

17


18/22

3 2 1 0 1 2 33

2

1

0

1

2

3

variable 1

variab

le

2

Clayton Copula Contours for = 1 (L

= .5)

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18


distributed random variables using a Clayton copula for = 0.365.

18


19/22


random variables using a Clayton copula for = 0.365.

19


20/22

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PctofAggr

rho

1TD

2TD

3TD

4TD

5TD

Figure 9: The BDS spreads for the first, second, third, fourth, and fifth todefault as functions of correlation , for a balanced basket of5 entities of rel-ative similar creditworthiness. The basket spread is reported as a percentageof the aggregated sum of the individual CDS spreads, Pct of Aggr".

20


21/22

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PctofAggr

rho

1TD

2TD

3TD

4TD

5TD

Figure 10: The BDS spreads for the first, second, third, fourth, and fifth todefault as functions of correlation , for a unbalanced basket of 5 entitieswhere four of them are of similar creditworthiness and the fifth entity is a lotmore riskier. The basket spread is reported as a percentage of the aggregatedsum of the individual CDS spreads, Pct of Aggr".

21


22/22

References

[1] Jean-Paul Laurent, Jon Gregory (2003), Basket Default Swaps, CDOsand Factor Copulas, working paper.

[2] Philipp J. Schonbucher (2003), Credit derivatives pricing models: Mod-els, Pricing and Implementation, Wiley.

[3] John Hull and Alan White (2003), Valuation of a CDO and an nth toDefault CDS Without Monte Carlo Simulation, working paper.

[4] Dilip Madan, Michael Konikov, Mircea Marinescu (2004), Credit and

Basket Default Swaps, to be publish.[5] Christian Bluhm, Ludger Overbeck, Christoph Wagner (2002), Credit

Risk Modeling, Chapman and Hall/CRC.

[6] Wolfgang Breymann, Alexandra Dias, Paul Embrechts (2003), Depen-dence structures for multivariate high-frequency data in finance, Quan-titative Finance, 3, 1-14.

[7] Roger B. Nelsen (1998), An Introduction to Copula, Springer.

22

bloom berg, konikov] basket default swaps

Documents