SPONSOR’S STATEMENT
Breaking through basket
credit default swaps pricing
First and second-to-default credit default swaps are becoming popular hedging instruments. In this paper,
Jean-Noël Dordain and Stéphane Mayrargue outline a simple pricing approach based on a first-order
dynamic replication and an ‘asset’ estimation of the credit risk correlation
hen hedging credit risk on a global position of credit risky financial assets – fixed income products, hybrid derivatives as convertible bonds – a credit risk analysis per issuer may be irrelevant because of cost and liquidity issues. Hence, a new range of credit
risk securities has been developed to hedge, not a single issuer default
event, but the first or second default event observed during a certain lifetime on a set containing many issuers. These underlying credit structures
are referred to below as first- or second-to-default credit baskets.
We present hereafter the pricing of credit default swaps written on
credit baskets. This pricing gives rise to several difficulties:
W
■ decomposing of a basket credit default swap as a series of plain
vanilla credit default swaps carrying on a single issuer;
■ understanding and estimating of the credit correlation between the
issuers contained in a credit basket and;
Definition of credit default swaps
Credit default swaps (CDS) involve two counterparts: the protection
buyer and the protection seller and carries on a credit basket. A credit basket is set up as a series of issuers and reference assets (Ij,Vj). For
each issuer Ij, the reference asset Vj is an asset issued by Ij. If the
default event observed carries on Ij, the credit risk protection is calculated on the instrument V taking into account its recovery rate δj. The
credit event for the credit basket may be the first or the second issuer
to default during the CDS lifetime.
While no credit event is observed, the protection buyer pays to the
protection seller a fixed-rate leg and the protection seller pays nothing
to the protection buyer.
Figure 2. Price with respect to simulation runs.
Second to default – 10 issuers
1.6
1.8
1.4
1.6
Price (% of the notional)
Price (% of the notional)
Figure 1. Price with respect to simulation runs.
First to default – 10 issuers
■ building an effective method to value the basket credit default swap
based on a Monte Carlo simulation.
1.2
1
0.8
0.6
0.4
0.2
1.2
1
0.8
0.6
0.4
0.2
0
–0.2
1.4
0
5,000
10,000
15,000
20,000
Number of simulations
25,000
30,000
0
5,000
10,000
15,000
20,000
Number of simulations
25,000
30,000
Figure 3. Price with respect to the correlation.
First to default – two issuers – 5000
simulation runs
Figure 4. Price with respect to the correlation.
Second to default – two issuers – 5000
simulation runs
1.6
Price (% of the notional)
1.4
–100 –80
–60
–40
1.2
0
–20 –0.2
0.6
0.4
0.2
–60
–40
0
–20
–0.2
60
80
100
20
40
60
80
100
–0.6
Price (% of the notional)
0.8
–80
40
–0.4
1
–100
20
–0.8
–1
–1.2
–1.4
–1.6
–1.8
–2
–0.4
–2.2
Correlation (%)
Correlation (%)
If the credit event occurs before the CDS maturity, the swap is cancelled out as follows. Denote by I the issuer involved in this credit event
and V its reference asset. The protection buyer pays the accrued
amount and gives back to the protection seller a quantity Q of the reference asset V such that the complete value paid back is equal to
where δN is the recovery rate of I. In return, the protection seller pays
the protection buyer a fixed cash amount equal to N.
■ its value if no default occurs: FixedLeg, (the present value of the
fixed leg paid by the protection buyer);
■ its value if a default occurs CDSj(default).
Hence,
CDS j (default) =
CDS j − (1 − Pj (T))FixedLeg
Pj (T)
Basket credit default swaps decomposition
Similarly, we decompose the probability space into n+1 disjoint events:
In this section, we show how to decompose the pricing of the basket
CDS into the pricing of plain vanilla CDS. This decomposition is theoretically justified as a first order approximation. A generic Monte Carlo
■ no default occurs;
■ the default is observed on Ij.
Credit default swap baskets
can be dynamically replicated
by a portfolio of plain vanilla
credit default swaps
method similar to the one explained below may be applied when relaxing the first order approximation assumption. But, this approach is
rather time consuming and the precision gain is really tiny.
In the first order approximation framework, some semi-closed formulas are available for plain vanilla CDS as explained in our previous
article ‘Practical Credit Risk Hedging’ (Dordain et al, 2001). Hence, in
this, the sequel, we only focus on the pricing of basket CDS.
CDSj denotes the plain vanilla CDS carrying on Ij with reference asset
with a notional equal to N. Tj denotes the stopping time corresponding to the first default time of Ij, Pj(t) is the cumulative probability law
of Tj. A marked-to-market calibration of Pj(t) is also detailed by
Dordain et al (2001).
Because the two events are mutually exclusive, CDSj may be
decomposed as the sum of two terms:
In a first order approximation, if we neglect correlation terms in the
computation of conditional expectations, we get


CDSBasket =  1 − ∑ ω j  FixedLeg + ∑ ω jCDS j (default)


j
j
where ωj is the probability that the default event occurs before T and
carries on Ij. More precisely, for a first to default credit basket, we have
ω i = Prob { Ti < T and Ti = Tfirst }
where Tfirst = inf(Ti) is the first default event observed on the set of
issuers, and for a second to default credit basket, we have
ω i = Prob { Ti < T and Ti = Tsecond }
where Tsecond = inf(Ti>Tfirst) is the second default event observed on
the set of issuers.
Definition of a ‘practical’ credit risk correlation
To value the weights ωj’s, we need to rebuild the joint cumulative probability of the vector (Tj)j. Using results about Copula functions (see
Sklar, 1959), we can see that, if f is a joint cumulative probability law
and fj is its 1-dimensional projection on its jth variable; the function P
defined by
P (x1,..., x n ) = f (f1−1 ( P1 (x1 ))),..., fn−1 ( Pn (x n ))
SPONSOR’S STATEMENT
is a joint cumulative probability law whose 1-dimensional projections
are exactly the cumulative default probabilities for each issuer.
Conversely, any joint probability law can be written in such a manner.
The main assumption is to take f as a standard correlated n-dimensional normal law. The correlation matrix M defining the normal distribution f is called the asset correlation between the issuers.
Indeed, if the credit default event of an issuer depends only on the
market price of its share, it can be shown (see Merton, 1974) that,
under a certain trigger on the Brownian motion driving the share price,
the default occurs. If ρ is the correlation between the share of Ii and the
share of Ij, it is easily seen that the (i,j)th correlation of M is equal to ρ.
Figures 3 and 4 give the dependence of a basket CDS on two issuers
with the same credit risk data with respect to the asset correlation
ranging from –100% to 100%. When the correlation nears 100%, the
two issuers tend to default at the same time. This minimises the price
of a first-to-default swap and maximises the price of a second-todefault swap.
Moreover, when the correlation is worth 100%, the credit risk parameters being the same, we retrieve, for both cases, the price of the
simple CDS: CDSplain vanilla with same credit parameters (here –0,2643).
As the data are the same for both issuers in the basket, we have, independently of the asset correlation, the following parity relationship:
CDS1 = CDS2 = CDSplain vanilla
Effective pricing of credit default swaps
We need to compute the weights ωi’s. This is done using a Monte
Carlo simulator to simulate the samples (Tj)j. To do this, we compute
n-dimensional normal samples with the correlation matrix M to obtain
a simulation sample (ui)i. The simulated Tj is then obtained by
Ti = Pi−1 ( N (u i ))
where N is the 1-dimensional standard cumulative normal law.
Numerical results
We consider a single set of issuers. It is composed of 10 issuers with
different credit risk parameters. The CDS rate curves are quite flat and
low (about 75bp) in order to respect the first order approximation. The
two basket CDS are two-year maturity swaps with different fixed rates
(7% for the first-to-default and 2% for the second-to-default), so the
prices of the swaps are relevant for a convergence test. Finally, the
recovery rates are taken within the range [50%, 90%].
The asset correlation is
directly used to estimate
the credit risk correlation
The number of simulation runs needed to obtain a good accuracy
depends on the regularity of the term structure of CDS rate identifying
the credit risk of each issuer especially because we need to invert it
numerically to obtain the first or second time to default for each issuer.
The numerical results are presented in the figures 1 and 2. Of course,
first- and second-to-default swaps are less expensive than the equivalent plain vanilla credit default swaps as they offer a poorer hedge.
We examine below the correlation sensitivity in the simplest case of
a credit basket containing two issuers with the same credit risk data
(ie, have the same market prices for plain vanilla CDS). In this case,
we have:
CDS1 = CDS2 = CDSplain vanilla
Conclusion
We have proposed, in this paper, a practical approach for the pricing
of basket CDS. This marked-to-market pricing is based on term structure of plain vanilla CDS rates, and the correlation between the share
prices of the issuers. Using the asset correlation as a credit correlation
estimation, is fully significant in the Merton asset-liability Model (see
Merton, 1974). This pricing method is valid if the default probability of
each issuer remains small. This pricing checks the arbitrage parity relationships and matches the prices of vanilla instruments. ■
We thank Patrick Loiseleur for his helpful comments and Arnaud Vinciguerra for his
contribution to this research project. The charts were produced using the version
4.1.3 of SOPHIS Risque software.
Bibliography
Dordain J N, Loiseleur P and J M Mercier, 2001. “Practical
Credit Risk Hedging”, Risk December 2001
Merton R C, 1974. “On the Pricing of Corporate Debt, the Risk
Structure of Interest Rates”, J Finance 29, pp 449–470
Sklar A, 1959. “Fonctions de répartition à n dimensions et leurs
marges”, Publications de l’Institut de Statistiques de
l’Université Paris 8, pp 229–231
CONTACTS
Jean-Noël Dordain and Stéphane Mayrargue
Quantitative Analysts
SOPHIS
Tel: +33 1 4455 3773
Fax: +33 1 4260 2006
e-mail: info@sophis.net
Website: www.sophis.net