is the time t‘ 0 9 The market may not be complete in the presence of default and we assume that a recovery process and a probability measure exist such that the Eq. (8.2) is satisfied. If theprobability measure is not unique (i.e., not all the risks, such as the default recovery, arehedgeable or diversifiable), then the price of the defaultable instrument is not unique,which is a main reason for the (relatively) large bidlask spreads of some of the defaultable instruments as the dealers need to make conservative assumptions on both the bidand the offer (ask) side. For pricing the default contingent instruments (such as CDS)used in the construction of the credit curve, the recovery rate values are typically notcritical.
Credit Risk Modeling and Pricing
330
Chapter 8
value of the corresponding default-free instrument that, in general,of the corresponding deshould not be equal to the time t valuefaultable instrument.It is interesting for the reader to think how to modify the above equation so that we can obtain a similar martingale relationship. We shallprovide the answers in Section 8.7.Let's first consider a simple case whereby the non-default (or sur2') (if no default occurs at or before time r ) and the devival) payofffault time process z are completely independent of each other. Thiscovers the case of a defaultable zero-coupon bond (where v ( T ) is aconstant) and some other cases. In this simple case, the expected value
v(t)
v(
[
of the non-default payoff, E: ?(T)lP,,}]
, can be simplified and Eq.
(8.2) becomes
(OltlT)Thus, the time 0 value or PV v ( 0 ) of the simple defaultable instrument is given by
(8.5)
or, in a more familiar form,
v ( 0 )= v( o)(1- Po (0,T ) )+ R (0,2') v(0)PD(0,2') ,where
Sec. 8.1
Pricing Simple Defaultab[eInstruments
33 1
V ( 0 )= P ( 0 , T ) E i [ V ( T ) ]
The reader can recognize that V (0) is the time 0 value of the corresponding default-free instrument (e.g., an OTC derivatives instrumentwith non-defaultable counterparties), Po (0, T ) is the time 0 value of thecumulative default probability in the time interval of (O,T] under thecorresponding forward measure PT, and R (0, T ) is the time 0 value ofthe effective recovery rate in the time interval of (0, T ].The expected default loss is thus given by
V(O)-?(O)=V(O)P,(O,T)(I-R(O,T)).
(8.8)
We can also use various recovery assumptions in the above equation.The simplest case is zero recovery, in which case, the defaultable zerocoupon bond reduces to a simple (extinguishing) default contingent instrument with payment only if there is no default.In practice, the prices of liquid defaultable bonds (such those givenby the above equation) or the par coupons of CDS or asset swaps areused to compute or imply the default probabilities (such as P,(O,T))under arbitrage-free probability measures (such as the forward measure
lb 1.For instance, the price of the defaultable bond price is often quotedby a credit spread (per annum) S given byV ( 0 )= V(O)exp(-SD) ,
(8.9)
where D is the duration in years (which is the same at T for a zerocoupon bond) and, for simplicity, we have assumed that the credit spreadAS is continuously compounded. Thus, implied default probability is
332
Credit Risk Modeling and Pricing
Chapter 8
given by
Po (0, T ) =
1- exp (-SO)
1-R(0,T) .
(8.10)
As an example, if S =lOObp p.a. (per annum), D =lOYears , and
R ( 0 , T ) = 50% , then Po ( 0 , T ) = 19% .The implied default probabilities, in general, can be very differentfrom (and are typically much higher than) the actual default probabilitiesestimated by the actual historical defaults. Trading the implied defaultprobabilities against the actual default probability is a (speculation) strategy for structured credit derivatives trading.Even if the non-default payoff f ( T ) and the default time process zare correlated, we can still simplify the expected value of the non-default
payoff, E:
[
[f ( T )l { T < r l ] ,using the conditional default probability
1
E,? liT<r) ( T ) ]. In this case, Eq. (8.2) can be rewritten as
(8.11)( 0 I t IT )
However, it is more subtle to handle the default payoffs, which we shalldiscuss in the next section.One interesting aspect to look at is the default probability P$ ( t , T )under the spot arbitrage-free measure Q (with respect to the default-freemoney market account numeraire p ( t ) ). Though under the equivalentprobability measures PT and Q , the default events are the same, theprobabilities of their occurrence can be different.Since
333
Pricing Simple DefaulfableInstruments
Sec. 8.1
thus, through a change of probability measure,
‘2 (tYT>= E? [l{,<rsT}]
(0 I t I T )The above equation can be further rewritten with the following convexity adjustment,
‘2
=P(~~T)ET[ ~ ( ~ ) / ~ ( t ) l { t < r _ < T } ]
(‘3‘)
= ‘(tJPT
T )covlr ( P ( T ) / P ( t ) l{r<rsT})
+‘(t,=0‘
[ P ( T ) / P ( t ) ] E T [l{t<r_<T}]
(8.14)
Y
(‘9
T , i-(‘ t ,T ,
covT (P( T ) / P (
t),l{l<r<T})
(0 I t I T )
(
)
where CovT P ( T ) / P (t ) ,l{r<rsT} is the covariance of P ( T ) / P( t ) andl{r<rsT}. We have used the following martingale relationship in deriving
the above equation
P( I > / P ( t , T )= E1‘[P( T ) / P (T J ) ]
(0 I t I T ) .
(8.15)
Thus, if the interest rate process p ( T ) / p ( t )and the default indicator or the default event process l{t<rsTjis completely independent, then(8.16)
How to handle various correlation in general shall be presented in thefuture sections.
334
Credit Risk Modeling and Pricing
Chapter 8
The interested reader may also want to consult http://defaultrisk.comfor a large amount of interesting downloadable literature on credit riskmodeling.
8.2
Default Contingent Instruments
Default contingent instruments are those whose payoffs are continent onthe default of one or more reference credit names. Some are in the formof cash instruments and others are in the form of credit or credit hybridderivatives. Examples of these instruments include single-name instruments, such as defaultable bonds, CDS, Quanto CDS, default swaptions,credit hybrid instruments, and defaultable OTC derivatives, and multiname instruments, such as credit indices,"' nth-to-default baskets,CDO/CSO, and single tranche CSO.As we discussed before, examples of credit hybrid instruments include CCDS and credit extinguishers. A CCDS is a contingent CDSwhose notional is contingent on or linked to the PV of an OTC derivatives instrument. It provides default protection (to the protection buyer)on the OTC derivatives instrument by providing the replacement of thesame instrument upon the default of the reference credit. In other words,it is an OTC derivatives instrument with a knock-in feature upon the default of the reference credit.A credit extinguisher, on the other hand, is an OTC derivatives instrument with a knock-out feature upon the default of the reference creditwith a pre-specified rebate (that can be zero or non-zero). The rebate canalso be linked to the face value of the defaulted bonds of the referencecredit, which is similar to the ISDA set-off provision whereby, underproper circumstances, one is allowed to offset one's payables to a counterparty by delivering defaulted bonds of the counterparty with the facevalue equal to the PV of the payables.The typical motivation of a CCDS is to trade the credit risk in anOTC derivatives instrument and provide a static hedge of the counterparty credit risk (as apposed to a dynamic hedge with CDS). The typicalmotivation of a credit extinguisher is to achieve trade or funding cheapening for the counterparty. It is important to point out that the credit extinguishers are not legal in all jurisdictions.
'
l o For more information, please refer to http://www.Markit.com/marketing/indices.php,http://www.dowjones.com/Pressroom/PressReleases/Other/Europe/2005/0315-Europe-DowJonesIndexes-7695.htm.
Sec. 8.3
A Simple Markov Chain Model
335
Some brokers are working on sourcing liquidity for the market forsome of these instruments. For more information, the reader may wantto refer to, e.g., http://www.SwapMeCredit.com.There are also credit rating contingent instruments, such as bondswith rating based variable coupons, which can be priced by a MarkovChain Model, such as the JLT model (Jarrow, Lando, and Turnbull,1997) as we discussed in Section 4.6.2, but we shall not go into detailshere.
8.3
A Simple Markov Chain Model
In the previous section, we discussed the pricing of a simple (extinguishing) default contingent instrument with the focus on pricing the nondefault payment. Here we shall focus on pricing the default payment of ageneric single-name default contingent instrument.In this section, we shall present a simple Markov chain model. In thenext section, we shall discuss the modeling of the default processes correlated with market processes and the default processes of multiplenames/obligors.More specifically, let us take a look at a generic single-name defaultcontingent instrument with non-default payoff of 0 and default payoff ofC, (z) at default time z of the reference credit. C, (z) can be thoughtof as the default protection payoff or the loss given default (LGD) of,e.g., a defaultable OTC derivatives instrument, and the default time z isdefined in the same way as in the previous section. For pricing the counterparty credit risk or a defaultable OTC derivatives instrument, the reference credit is actually the counterparty.Under a discrete time setting, the time t value V ( t ) of this genericdefault contingent instrument is given by
(8.17)(0 I t I < T2...< T, )
where Q is the spot arbitrage-free measure with respect to the defaultfree money market account numeraire p ( t ) and T, is the final maturityor the risk end date of the generic default contingent instrument.
's are
336
Credit Risk Modeling and Pricing
Chapter 8
chosen such that there are no non-default cashflows in the time intervalof ( q,q+j) . This is an approximation if the default payoff of C, (z) is,e.g., the default loss on an American option or a continuous barrier option.More specifically, this default contingent instrument is really a forward starting default contingent instrument with an extinguishing featuresuch that it is knocked out with zero rebate if the reference credit defaultsat or before time q . Other default contingent instruments can be represented as a linear combination of the extinguishing forward starting default contingent instruments.The summation in the above equation is to handle the intermediatecashflows due to default, as pointed out in the previous section.Define the time t value 5 ( t ) for each time period as
0
then, we can rewrite V ( t ) as
gK
v( t ) = i=l
(t)
(0 I t ) .
(8.19)
With the above equations, we can perform a straight forward MonteCarlo simulation on X, already to price a generic single-name defaultcontingent instrument. However, such straight forward Monte Carlosimulation is typically time-consuming and not accurate due to insufficient sampling of the default states or insufficient number of MonteCarlo simulation paths reaching the default states (which, in turn, is dueto the default probability being low). Importance sampling (Section2.5.7), stratified sampling, and Brownian bridge simulation are goodcandidates for enhancing the default sampling and, thus, improving thesimulation accuracy and speed. A simple Markov chain model (whichcan be combined with importance sampling) that we shall discuss provides additional enhancement in the default sampling.How to specify and calibrate the process of X,shall be discussed inthe next section.
Sec. 8.3
A Simple Markov Chain Model
337
Under various conditions, the price of a generic single-name defaultcontingent instrument can be expressed using the default probabilitiesrather than completely using the default events or states as shown in theabove equations. This can often simplify the numerical implementationsand help improving the computation accuracy and speed.In order to proceed further along this direction, we define a conditional default time z ( T ) given by the following stopping or first exittime, i.e.,Z ( T ) = inf(t]X, I H , )
(0 I T < t ) .
(8.20)
The distinctive feature of this conditional default time (as comparedto the unconditional default time z) is that it is the default time conditioned on no default at or before time T.It is easy to verify that
(8.21)
( O < T l T , <q+JIn order to handle the correlation that we discussed earlier, we canrewrite V, ( t ) with the following iterated expectation,
338
Credit Risk Modeling and Pricing
Chapter 8
( 0 5 t q <T+J
Define(8.23)which is the time t marginal default probability in the time intervalconditioned on no default at or before time t, then, we can rewrite v;. ( t ) as
( o s t s q<?+,)where ED (Ti ,q+l) is the (discounted) expected default payoff conditioned on all the market and all credit information up to time and further conditioned on the default of the reference credit in the time intervalof ( T , T + I ] .The above equations can be solved numerically by a simple Markov
Sec. 8.3
A Simple Markov Chain Model
339
chain model whereby on any simulation path of all the market and allother credit variables, the reference credit can have a default state (0)and a non-default state (ND) at every time step as shown in the figurebelow.
D
D
D
Figure 8.1. A simple Markov Chain Model whereby on any simulation path ofall the market and all other credit variables, the reference credit has a default state(D) and a non-default state (NO).
This can be achieved by drawing two samples of XT+]- X T on eachsimulation path and at each time step such that one sample correspondsto a default state (D) or T < z(T,) I q+,and the other to a non-defaultstate (ND) or q+l< z(T,). Even though, this approach doubles the pathsamples of XT+,- X q , but it ensures the sampling of a default state atevery time step on every simulation path. So, in general, this approachsamples many more default states per simulation path than other methodsthat have multiple time steps, but only ensure one default state per simulation path.The reader may want to think about how to draw simulation samplesso as to ensure a default sampling and a non-default sampling.It is important to point out that the following martingale relationshipfor the default probabilities
=P$(t,TyT+1)
( o a q <q+,)is often not satisfied in numerical procedures and can be used in the mar-
340
Credit Risk Modeling and Pricing
Chapter 8
tingale resampling for improving the numerical accuracy. The essenceof this martingale resampling is actually to numerically calibrate to thetime t default probability term structure Pj ( t , I;. ,7;+, ) . This is particularly usehl when t = 0 .One way to perform the martingale resampling on the default probabilities is by enforcing the above martingale relationship numericallyand rewriting Eq. (8.24) as follows,
( o a q <q+l)The simple Markov chain model allows the application of the martingale resampling on the default payoff is also desirable, which, in turn,can provide additional accuracy and computation speed improvement.Other advantages of the simple Markov chain model include allowingjumps in the payoff, allowing sub-sampling (with martingale interpolation and resampling) for significant computation speed improvement,and, under various conditions, allowing a smooth transition to defaultprobability based models.Part of the correlation among the reference credit and the marketvariables (including the credit spread and default processes of othercredit names) comes from the diffusion (term) correlation of the expecteddefault payoff( q,q+,) and the default probability
c,
l{T<rlP;
( q,I;:, q+,) when there are no jumps, and part comes from the
jump correlation of the jump to default of the reference credit and thejump of the market variables including the credit spread and defaultprocesses of other credit names in computing( q ,q+l) or C, ( r ).An example of the jump correlation is the situation of a Quanto CDSwhereby the reference credit obligation is an external USD denominateddebt of an emerging market sovereign entity and the default payoffC, ( r ) is a fixed amount to be paid in the local currency of the sovereign entity (in exchange of the defaulted external debt). Since the sovereign's default can cause a significant jump devaluation of its local CUT-
c,
Sec. 8.4 Modeling Correlated Default Event Processes with a Factor Model
341
rency, thus, the default payoff C , (z) can also experience a significantjump devaluation in its value in USD. This situation can also occur inthe counterparty default loss in cross-currency IR swaps and in the counterparty default gain in cross-currency IR swap extinguishers.Another example is the credit or default contagion whereby onecredit name’s jump to default can cause other credit names’ creditspreads to jump widen or even cause other credit names to jump to default. This situation can occur if the reference credit sells credit protection on other credit names through CDS or CDO/CSO, in which case, theLGD (loss given default) with respect to the reference credit will alsoexperience a jump.Yet another example is an entity selling put options on its own stock(as a funding cost saving or funding cost arbitrage strategy in buyingback its own stocks) and, when the entity defaults, its equity price willexperience a significant jump devaluation (and vice versa), in whichcase, the LGD (loss given default) with respect to the entity will also experience a jump.We shall discuss how to handle such correlation in the next section.
8.4
Modeling Correlated Default Event Processes with aFactor Model
A popular approach in handling the correlation is to describe or modelthe entire market (including the market risk processes and the defaultprocesses of multiple names/obligors) by a factor model.Let’s assume that the entire market can be described by rn variableswith X j ( t ) ( j = O,l, ...,rn -1) denoting the j f h variable that can be theasset process of a credit name or a market risk variable, such as an interest rate, an FX rate, or an equity price, etc.More specifically, X ( t ) is modeled through the following factormodel
dX j ( t )= oj( t )(PjdW, ( t )+ ajdWi ( t ) ),
(8.27)
where W j ( t ), a one-dimensional Brownian motion for the j r hvariable, isthe idiosyncratic driving factor that is independent of any other drivingfactors, W, ( t ), an n-dimensional independent Brownian motion, is a
Credit Risk Modeling and Pricing
342
Chapter 8
nxl vector of common driving factors for the entire market including themarket risk processes and the default processes.It is important to point out that conditioned on Fp" (the filtrationgenerated by W, ( t ) up to time T), all Xi( t ) ( t I T ) are independent ofone another. This conditional independence is a foundation for the application of various efficient numerical algorithms, such as FFT (FastFourier Transformation), for modeling correlated defaults.The common driving factors for the default processes can be categorized as one global economy driving factor and one driving factor foreach of the geographical regions and industry sectors. Additional common driving factors can be added, if needed, for additional degrees offreedom for specifying the correlation. The common driving factors alsoinclude the driving factors for the market risk variables, such as interestrates, FX rates, equities, etc.a j , a scalar, and pj , a l x n vector, are the coefficients that deter-
mine the variance and correlation of dYj ( t). Typically, we choose
J
aJ. = l - p j p j T .
(8.28)
Thus,f l j ( t )= aj ( t ) ( p j d W , ( t )+ , , / m d W ,
V..(dYi
( t ) ),
(8.29)
(8.30)
( t ) )= aj ( t ) 2dt ,
andcorr(dYi ( t ) , d Y j( t ) )= pjpjT
(if j).
(8.31)
The correlation among the default processes (and the market processes) is specified through the correlation among the hypothetical underlying asset value processes (and the processes of the market variables), orCor+Xi(t),dYj(t)).
In the case of pair-wise constant correlation, the pi can be easilysolved given the correlation Corr (dYj ( t ) ,d Y j ( t ) ).
Sec. 8.4
Modeling Correlated Default Event Processes with a Factor Model
343
t
If pi is state-dependent, e.g., dependent on jpjdWM(s),for in0
stance, then we have a simple model for handling the correlation skew,or the state-dependent correlation (similar to the volatility skew).The above equations can also be rewritten as
dXj(t)=oj(t)dW,'(t),
(8.32)
where Wit( t ) is a standard (one-dimensional) Brownian motion given by
dWj'(t)=pjdWM(t)+a;dWj(t),
(8.33)
The factor model provides a framework of simulating correlated default event processes that can be correlated among themselves and withmarket variables.More specifically, let Xi( t ) and H j ( t ) be the hypothetical underlying asset value process and the default bamer (or the hypothetic liability level) for t h e f hcredit name, and its default time zj process is givenbyz j = inf ( t l X j ( t )I Hi( t ) )
(0 < t ) .
(8.34)
So far, we have not specified the probability measure for theBrownian motions. One way to specify the probability measure is basedon how the model is calibrated or whether the model is calibrated to theimplied default probability (e.g., under the spot arbitrage-free measureQ ) and the historical default probability.Hull and White (2001) deals with the situation where o,( t ) is con-
stant and Hi( t ) is time dependent, and Hi( t ) can be used to calibrateto today's implied default probability term structure p,4"0 (O,O,t) of thejfhcredit name through a bootstrap procedure with numerical integrations.Brigo and Tarenghi (2005) ' I ' assumes a constant default barrier Hiand calibrates the time-dependent volatility oj( t ) to today's implied~
'I'
~
~~~~
This paper can be downloaded from http://www.damianobrigo.it/cdsstructural.pdf.
344
Credit Risk Modeling and Pricing
Chapter 8
default probability term structure PD: (O,O,t) of the jth credit namethrough a bootstrap process with closed-form solutions.Finkelstein, Lardy, et al (2O02)ll2 provides an uncertain default barrier model where the barrier H is random.
,
Here we start with Brigo and Tarenghi (2005). More specifically, westart with'FD
('9
' 9
q = Ef ['{O<.r<T ] ]
(8.35)
7
which, with a constant default barrier H , , can be reduced to
~2~( ~ 0q ,) = Pr (0 < z5 q )(8.36)
where @ (.) is the cumulative (one-dimensional) normal distribution, ormore specifically, the probability for an N (0,l) variable to be less than xand the factor 2 comes from the reflection principle of Brownian motion.The above equation can be used, through a bootstrap procedure, todetermine or calibrate a, ( t ), if it is piecewise constant.Similarly, the conditional default probability is given by
~j490(T,<,<+~)= Pr( q < z(q) IT + ~ )(q)).
= 2 ~ rX(, (<+,) I H , ~ x ,
(8.37)
(
H , -xj( q )
For a review of the correlated default modeling in CDO pricing, the
'''
This paper can be downloaded fromhttp://www.creditgrades.com/resources/pdf/CGtechdoc.pdf.
Sec. 8.4
Modeling Correlated Default Event Processes with a Factor Model
345
reader can refer to, e.g., Burtschelll, Gregory, and Laurent (20O5).lI3For an application of the correlated default modeling in CD02 pricing,the reader can refer to Li and Liang (2005)."4Here we focus on the discussion of the default process of a singlecredit name. This includes defaultable OTC derivatives, CCDS, extinguishers, and counterparty credit risk.Let's first handle the diffusion correlation presented in the previoussection. In other words, in this case, the default payoff C, (z) does nothave any jump component upon the default of the reference credit.For simplicity, we let
x,= x,( t ) .
(8.38)
It is important to point out that, conditioned on F/ (the filtrationgenerated by W, ( t ) up to time T), all X j ( t ) are independent. Thisconditional independence is a foundation of various efficient numericalalgorithms, particularly for modeling correlated defaults.Eq. (8.18) can be rewritten in a similar form to Eq. (8.24), i.e.,
,
(8.39)
where
It is downloadable fromhttp://~~~.mathematik.uni-ulm.de/finmath/ss_05/fe/jplaurentcomparative.pdf.' I 4 It is downloadable fromhttp://finmath.stanford.edu/seminars/documents/GMCDOsquared.pdf.
346
Credit Risk Modeling and Pricing
Chapter 8
is the time t conditional default probability in the time interval (q,q+l]conditioned on that no default has happened at or before time t, and further conditioned on the market filtration 3,. It is generally state dependent (i.e., dependent on the market states).Under proper technical conditions and in the continuous time limit,Eq. (8.17) and the above equation can be reduced to
(8.41)( O I t I T , <T, = T )
This is also a generic formula for pricing the default leg of a CDSwhen the default payoff does not have any jump. Stochastic recoveryrates can also be incorporated in the above equations.With a convexity adjustment, the above equation can be further rewritten as
, (8.42)
( O r t I q <T, = T )and p are the covariance and
See. 8.4
Modeling Correlated Default Event Processes with a Factor Model
347
correlation between c D ( u ) and -dPu $ ( t , t , u , 3 ~ ) , respectively, q
P(4
the percentage volatility of
du
P(4
, and o2 the percentage volatility of
In deriving the above equation, the following iterated expectationrelationship was applied,
(8.43)
= duP' ( t , t , u )Another way of evaluating Eq. (8.41) is through a change of probability measure, which can often yield more accurate approximations.Now let's handle the jump correlation. In other words, in this case,the default payoff C, (z) has a jump component upon the default of thereference credit.The examples of the jumps that we discussed before can be specifiedas a dependency of C, (z) on the market variables A? (*, z) that, in turn,can experience jumps upon the default of the reference credit.In other words,(8.44)
where z- denotes the pre-default time (and the state) immediately priorto the default of the reference credit.While theoretically one can use very sophisticated jump models,practically, simple and parsimonious models are typically used due to thelack of reliable market data needed for the model calibration, which, inturn,is due to that these jumps are rare events.A simple approach is to exogenously specifj the post-default valuesof the market variables a ( * , z ) (immediately after the default jump)
348
Credit Risk Modeling and Pricing
Chapter 8
being a deterministic (often times linear) function of the pre-default Val(immediately prior to the default jump), i.e.,ues(
0
,
~
~
)
Iii(*,z)=f(Iii(*,c)).
(8.45)
To proceed further, let's first define the market variables M ( t )without jumps following Eq. (8.27), i.e.,(8.46)We can then define the pre-default market variables as
n;r (q-)= M (z) ,
(8.47)
and Eq. (8.45) for the post-default market variables can be rewritten as
A2 (v)= f ( M (z)).
(8.48)
Typically, the following linear function can be used,M(*,t)=lI,<s]M(t)+l~s<t)(a(t)M(t)+b(t)).
(8.49)
Thus, Eq. (8.44) can be rewritten asCD (z)= cD (z,f(
('))?.).
(8.50)
The simple Markov chain model can naturally allow jumps in thedefault payoff. Particularly, under the above simplified assumptions forthe jumps, Eq. (8.41) still holds.In terms of the counterparty credit risk pricing, the diffusion andjump correlation effects are the main cause of the wrong-way and rightway exposures.
8.5
Modeling Correlated Default Time Processes with theCopula Approach
A generic way of specifying the joint (default) probability distributionfrom the marginal (default) probability distributions is through the Copula approach (Li, 1999).
Sec. 8.5
Modeling Correlated Default Time Processes with the Copula Approach 349
This also provides a simple framework for simulating default times(as opposed to the default events presented in the previous sections).Such default times can be correlated among themselves and with marketvariables.As compared to the default event simulation, the default time simulation is simpler and has a better convergence, as one simulation sample ofthe default time (within a certain time interval) corresponds to many default event simulation samples (within the same time interval). However,default event simulation is more general, as the default time simulationcannot easily handle instruments whose values explicitly depend on thedefault events, such as certain default options including the counterpartydefault options.More specifically, letpi ( t , ) = Pr( z, st,)
(8.51)
denote the default probability of thej'' credit name at or before time ti.Then, the joint default probability or the joint default time distribution ofN credit names is given by
where C ( p ,( t l ) , p Z( t 2 ) ., . . , p N( t N ) ) is a Copula function.A popular choice of the Copula function is an N