Directory UMM :Data Elmu:jurnal:J-a:Journal Of Banking And Finance:Vol25.Issue2.2001:

Journal of Banking & Finance 25 (2001) 295±316
www.elsevier.com/locate/econbase

The analytic pricing of asymmetric defaultable
swaps
Georges H
ubner
a

a,b,*

Department of Management, University of Liege, Bld. du Rectorat 7 - B31, B-4000 Liege, Belgium
b
Limburg Institute of Financial Economics, Faculty of Economics and Business Administration,
Maastricht University, Postbus 616, NL-6200 MD Maastricht, Netherlands
Received 17 November 1998; accepted 23 September 1999

Abstract
Swaps where both parties are exposed to credit risk still lack convincing pricing
mechanisms. This article presents a reduced-form model where the event of default is
related to structural characteristics of each party. The cash ¯ows submitted to credit risk

are identi®ed before the swap is priced. Analytical pricing formulas for interest rate and
currency swaps are computed using a Gaussian model for risky bonds. Currency swaps
exhibit additional correlation risk. The bene®ts from netting depend on the balance
between exposures and market conditions in valuation. We show that sources of credit
risk asymmetries are also likely to impact on credit spreads. Ó 2001 Elsevier Science
B.V. All rights reserved.
JEL classi®cation: G13; G15; G33
Keywords: Derivatives; Interest rate swap; Currency swap; Default risk

The valuation of undefaultable interest-rate swaps and swaptions under a
speci®ed term structure of interest rates is a classical exercise. However, the risk
that either party defaults on its ¯ow of payments should entail the addition of a

*

Tel.: +32-4-366-27-65; fax: +32-4-366-28-21.
E-mail address: G.Hubner@ulg.ac.be (G. HuÈbner).

0378-4266/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 1 2 2 - 3

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G. H
ubner / Journal of Banking & Finance 25 (2001) 295±316

spread to the coupon charged as a reward for the corresponding expected loss.
Some authors have simpli®ed the problem by considering the interest rate swap
as a simple exchange of loans, as did Sundaresan (1991). Another simpli®cation consists of taking into account the presence of one risky counterparty
only, as in Baz (1995), Li (1998) and Cooper and Mello (1991).
These approaches overlook the presence of bilateral default risk. Whether its
value is positive or negative for each party (i.e., corresponding to an asset or a
liability), does not add much complexity in itself. However, the usual settlement rule in case of default creates asymmetry: for the defaulting party, the
swap is terminated through payment in full of its market value if it is positive,
but only through fractional payment if it is negative.
In order to deal with this issue, two solutions have been proposed. First, as
in Due and Singleton (1997) and Nielsen and Ronn (1996), it is possible to
deliberately restrict investigations to cases involving perfectly symmetric credit
exposures. Under these conditions, the empirical use of market-wide data is
possible. Second, allowing asymmetric default risk often leads to numerical

approximations, as in Due and Huang (1996).
This paper uses the approach of default risk developed by Due and Singleton (1999) for the valuation of credit risky securities, as Due and Singleton
(1999) do. Contrarily to these authors, who seek to simplify the swap valuation
formula by directly modeling its market value, the model proposed here focuses on the interpretation of default risk for each party to endogenously
provide the swap value. This more rigorous approach leads to analytic pricing
equations. Thanks to this new valuation principle, the model yields pricing
formulas for defaultable interest rate swaps and currency swaps with reasonable assumptions.
Obtaining closed-form solutions for swaps with bilateral credit risk allows us
to isolate the various components of default spreads and to analyze the possible
sources of asymmetry in credit qualities. In particular, the paper generalizes the
results of Due and Huang (1996) about the use of a netting master agreement
to reduce credit exposures by performing a systematic analysis of the impact of
netting for any initial swap value with any additional notional.
The paper is organized as follows. Section 1 presents the general valuation
framework. In Section 2, it is applied to interest rate and currency swaps, including the impact of netting. Section 3 concludes the article.

1. Valuation of swaps subject to bilateral credit risk
1.1. Principles
The starting point of the analysis is to identify how a defaultable swap
di€ers from its default-free version. Obviously, any party can only be re-

G. H
ubner / Journal of Banking & Finance 25 (2001) 295±316

297

sponsible for default on her own payments, as for any other obligation. A
credit risk model must ultimately recognize this fact. But the swap contract
involves bilateral risks which interact with each other, and the settlement rule
in case of default may be asymmetric.
The swap contract is priced in two stages. First, the possible future occurrence of default of each party is explicitly integrated before the price is computed. Second, the pre-default contract value is obtained from the comparison
of these defaultable components with the immunization of the one which is not
subject to immediate default risk. This procedure substantially di€ers from the
Due and Huang (1996) approach who directly model the pricing consequence
of immediate default risk: they implicitly assume that default is tied to the
contract value instead of underlying net payments.
The approach proposed here takes into account the contract settlement rule
in case of default, which is a key factor of pricing di€erences. The most
commonly used one is called the ``full two-way payment rule'' and obeys the
International Swap Dealers Association (1992) Master Agreement: under this

rule, the defaulting party receives a closed-out payment in full if the predefault
value of the swap is positive for her, and pays a fractional amount otherwise.
Due and Huang (1996) integrate this directly in the value by considering that
default risk depends on its sign. However, each party also exhibits credit risk as
she may not service the swap in the future. Only the party whose swap value is
positive bears immediate default risk, but the other party is at risk too: if the
swap remains alive, the sign of the price may reverse and she would then su€er
from immediate default. This potential risk is introduced in the valuation and
re¯ected in swap prices.
The pricing procedure also takes easily into account the ``limited two-way
payment rule'', stipulating that the defaulting party receives a fractional termination payment if the swap value is positive for her at default time. Thus, the
consequence of default is not tied to the sign of the swap price, and the
probability of defaulting on one's obligations is also contingent on the other
party's credit conditions.

1.2. Valuation model
This subsection proposes a two-stage valuation formula, with an emphasis
of how it economically and technically di€ers from the results proposed by
Due and Huang (1996).
The core di€erence is the way default risk is taken into account. This paper's

approach focuses on assessing credit qualities prior to valuing the swap. First,
one must identify the ¯ows subject to credit risk and determine their present
value. Secondly, the compensation for the undefaultable fraction of future
payments is computed in order to derive the swap pricing equation.

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G. H
ubner / Journal of Banking & Finance 25 (2001) 295±316

1.2.1. Identi®cation and valuation of cash ¯ows subject to credit risk
The payments that are a€ected by credit risk considerations must logically
be the ones that can actually occur. Therefore, the approach of identifying a
swap to an exchange of loans is not sustainable in the case of interest-rate
swaps, since principals are never exchanged. Besides, the settlement rule in case
of early termination has a clear in¯uence on who may default on which payments. Under the full two-way payment rule, a party can only default on her
own obligations. Conversely, under the limited two-way payment rule, the
consequence of the event of default is independent of the sign of the ¯ow.
Once the relevant ¯ows are identi®ed, default risk has to be explicitly integrated in order to reach a present value for expected dividends. The proper
method is to compute the expectation of all discounted payments under a riskadjusted probability measure.

To consider default risk in valuation, two competing approaches can be
distinguished. The ``structural form approach'' rests on the assumption that
default is triggered when the value process of the ®rm gets lower than or equal
to a speci®c value. It provides an economic interpretation of the event of
default, but fails to explain the nature and magnitude of the associated loss.
More recently, the ``reduced-form approach'' has been exploited by Due and
Singleton (1999), Jarrow and Turnbull (1995) and Jarrow et al. (1997). It has a
more descriptive point of view, and focuses on the spread added to the discount factor used in risk-neutral valuation. This spread turns out to be a
function of the probability of default and of the loss experienced when default
occurs.
Three di€erent formulations of the loss function currently exist in the literature: the defaulted pay-o€ is either a fraction of par (Madan and Unal,
1998), of a risk-free bond with the same notional and coupons (Jarrow and
Turnbull, 1995) or of the market value of the security just prior to default
(Due and Singleton, 1999). The latter is especially suited to the case of swaps:
in case of default by a party, the swap is unfolded but its price re¯ects credit
conditions prevailing if it were renegotiated among similar parties, with a
possibly reduced notional. This actually corresponds to the line followed in
Due and Huang (1996).
The random variable representing the time elapsed before the occurrence of
default is assumed to follow an exponential distribution with a stochastic timedependent parameter whose cumulative density function is

F …t† ˆ Pr‰0 < s 6 tŠ ˆ
Q

Z

0

t

h…X…s†; s† eÿh…X…s†;s†s ds;

…1†

where s is the timing of default, X…s† the vector of stochastic state variables
determining default risk, Q a risk-adjusted probability measure and h…X…s†; s†
is the associated risk-neutral arrival rate of default or hazard rate.

G. H
ubner / Journal of Banking & Finance 25 (2001) 295±316

299

When a party defaults, the current market value of the claim considered is
instantaneously reduced by a fraction l…X…t†; t†, which represents magnitude
risk.
In a risk-neutral world, the discount rate adjusted for default risk R…X…t†; t†
of the promised pay-o€ should leave the same present value as does the expected pay-o€ discounted at the riskless rate. This characterization corresponds to the framework developed by Due and Singleton (1999), who show
that
R…X…t†; t† ˆ r…t† ‡ h…X…t†; t†l…X…t†; t† ˆ r…t† ‡ s…X…t†; t†:

…2†

The discount factor for a risky cash-¯ow is just the spot rate plus a default
spread, which is itself the product of the arrival rate and the loss rate.
This approach can be used to di€erentiate the settlement rules in case of
default. If the arrival and loss rates for the parties A and B of a swap are
hA …XA …t†; t†, lA …XA …t†; t†, hB …XB …t†; t† and lB …XB …t†; t†, respectively, where XA …t†
and XB …t† can be di€erent or identical, and considering as negligible the
probability of simultaneous default, the spread associated to payments by A is:
8

sA …XA …t†; t† ˆ hA …XA …t†; t†lA …XA …t†; t†
>
>
>
>
< for full two-way;
…3†
sA …X…t†; t† ˆ sA …XA …t†; XB …t†; t† ˆ hA …XA …t†; t†lA …XA …t†; t†
>
>
…X
…t†;
t†l
…X
…t†;
t†
‡h
>
B
B

B
B
>
:
for limited two-way:
Under the full two-way payment rule, the spread for a party is only in¯uenced
by the state variables determining her own credit risk, while in the limited twoway payment rule, the spread is a€ected by both parties' default risk. For the
latter settlement rule, the sign of the ¯ow actually does not matter, because
then: sA …XA …t†; XB …t†; t† ˆ sB …XA …t†; XB …t†; t†.
Introducing credit risk at the level of defaultable ¯ows yields the following
standard expression for the present value equation:
 
Z T
Rs

ÿ
R …X…u†;u† du
e t j
dDj …s†  Ft ;
…4†
Vj …t† ˆ EQ
t

where Dj is the nonnegative process of all promised payments which are defaultable to j, j ˆ A; B, and Rj …X…t†; t† is the default-adjusted discount rate for
these ¯ows at time t.
1.2.2. Swap pricing equation
The original feature involved in the second stage is the use of a compensation principle. Since the contract is priced after credit risk has been taken into
account, part of it cancels out from comparing exposures and compensating
them.

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G. H
ubner / Journal of Banking & Finance 25 (2001) 295±316

It is important to note that the swap value V …t† is a market value that results
from discounting cash ¯ows subject to default risk, but it is still defaultable by
both parties. As cash ¯ows have already been identi®ed, it does not make sense
anymore to expose them fully to credit risk because they can be partly hedged
by netting the ¯ows. Only the unnetted part of the discounted cash ¯ows
should be ultimately vulnerable to default risk.
Denote VA …t† and VB …t† the present value of all payments to parties A and B
of a swap. Credit risk has already been accounted for in each expression. The
lower of these two value is actually compensated by a fraction of the higher,
because valuation takes place before the possible occurrence of default is
considered. This netted part is
I…t† ˆ min‰VA …t†; VB …t†Š

…5†

and the vulnerable part, which remains subject to credit risk at t is equal to
max‰VA …t†; VB …t†Š ÿ I…t† ˆ jVA …t† ÿ VB …t†j:

…6†

The swap value for party A is the vulnerable part if VA P VB and the opposite of
the vulnerable part otherwise. It gives her exposure to B's immediate default.
The swap pricing formula simpli®es to
V …t† ˆ VA …t† ÿ VB …t†:

…7†

Vj …t† is continuous when Dj …t† is continuous, and so is V …t†. It is crucial to stress
that V …t† also encloses a latent component of default risk: through the netted
part, default risks of both parties have been introduced in the swap price. Even
when V …t† < 0, A bears some credit risk under the limited two-way payment
rule: the higher B's credit risk, the lower A's compensated part, and the more
negative the vulnerable part. In terms of replicating securities, this swap is
equivalent to a portfolio of a long position on a defaultable cap and a short
position on a defaultable ¯oor (or conversely) with the same party, where the
lower of the values of these positions is compensated through netting with a
fraction of the higher one.
As RA …X…t†; t† ˆ RB …X…t†; t† ˆ R…X…t†; t† in the limited two-way payment
rule, and since dDA …t† ‡ dDB …t† ˆ dD…t† 8t , V …t† can immediately be simpli®ed
to
 
Z T
Rs

ÿ
R…X…u†;u† du
e t
dD…s†  Ft
V …t† ˆ EQ
t

under limited two-way payment rule;

…8†

which can easily be solved by the classical martingale approach for swap
pricing under a multi-factor interest rate model. Due to its limited interest and
use in practice, this type of settlement need not be examined further in the
paper.

G. H
ubner / Journal of Banking & Finance 25 (2001) 295±316

301

This reduced-form modeling of the swap is easy to compare with Proposition 1 in Due and Huang (1996). It presents the price as the unique solution
of a recursive integral equation:
 
Z T
R s DH

ÿ
R …X…u†;u† du
e t
dD…s†  Ft ;
…9†
V DH …t† ˆ EQ
t

where RDH …X…t†; t† ˆ r…t† ‡ sA …X…t†; t†1fV DH …t†