-dimensional Normalor Gaussian Copula function that is given by
where O N(.) is the N-dimensional cumulative normal distribution function, C is the correlation matrix, and a(.) is the one-dimensional cumulative normal distribution function. The student t Copula is anotherpopular choice for better handling the default contagion. Different Copula models give rise to different default contagion effects.The default time from a Gaussian Copula can be easily simulated bythe following procedure. First, one needs to simulate a set of properly
350
Credit Risk Modeling and Pricing
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correlated standard Gaussian ( N (0,l)) variables, which can be achievedby using the factor model in Eq. (8.29). Let Z j denote such correlatedstandard Gaussian variable corresponding to the jthcredit name, then thesimulated cumulative default time ti of thejthcredit name is given byt j = p;l ( @ ( Z j ) ) .
(8.54)
Similar to the volatility skew/smile of the Black volatilities in theoptions market, there is significant correlation skew effect whereby different base correlation is needed in a (Gaussian) Copula model in orderto price to the market different tranches of a CDOKSO. This means thata (Gaussian) Copula model cannot consistently model all the tranches ina CDOKSO, which is similar to the situation whereby a Black-Scholesmodel cannot consistently model options with different strikes. Moredetails can be found in, e.g., Hull and White (2006).There is also significant model risk in terms of the correlation valueand the dynamics of the correlation. It has been reported that variousfunds have incurred significant losses due to the inaccurate or incorrectassumptions of the correlation.
8.6
Recovery Rate Modeling
The stochastic recovery rate modeling is typically achieved by using aBeta distribution whose probability density function is defined in the interval of [0,1] and is given by'15
!up-1 (1 - u)"-' du
(8.55)
0
The corresponding cumulative probability density function is given by(8.56)
'I5
For more details, see, e.g., Wikipedia at:
http://en. wikipedia.org/wiki/Beta-distribution.
Sec. 8.7
Risky Market Model for Credit Spread Modeling
351
The parameters p , q of the Beta distribution are typically obtained bythe mean E p [ x ] and the variance Varp[ x ] of the Beta distribution asfollows,
(8.57)
In order to simulate correlated recovery rate, first, one needs to simulate a set of properly correlated standard Gaussian ( N(0,l) ) variables,which can be achieved by using the factor model in Eq. (8.29). Let Zpdenote such correlated standard Gaussian variable corresponding to recovery rate process of t h e f h credit name, then the simulated recoveryrate R, is given by
R j = Fi:. (@(Z;))
,
(8.58)
where Fp, ( x ) cumulative probability density function for the recoveryrate process of t h e f hcredit name.
8.7
Risky Market Model for Credit Spread Modeling
As we can see, the simplified Merton or structural model in the previoussections actually predicts the future volatilities of the credit spreads (forgiven recovery rates) and such volatilities are likely to be inconsistentwith the volatilities implied from the default swaption market or by othermeans. Regardless of the liquidity of the default swaption market, it isuseful to have some freedoms so that we specify our view on volatilitiesof the credit spreads.This is not really an issue for pricing instruments that are insensitiveto the credit spread volatilities, such as CDS and CDO/CSO. However,it is an important issue for pricing default options, such as options onCDS and CDO/CSO, and the credit charge or CVA.The volatilities of the credit spread for both the counterparty and the
352
Credit Risk Modeling and Pricing
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underlying reference credit names are important for the credit chargecalculation, but the volatilities of the credit spread for the underlying reference credit names are more important, as the credit charge of a CDS isessentially a series of default swaptions.In this section, we shall present the martingale modeling of the creditspread processes, which is one step towards jointly modeling the creditspread processes and the default processes that we shall discuss in thenext section.While, in general, we can use the credit spread models based on theshort hazard rate or default intensity, here we choose the martingalemodeling approach, such as the credit spread market model (see, e.g.,Schonbucher, 2000 and 2003'16). The credit spread market model (or themartingale credit spread model), similar to the LIBOR or swap marketmodel, directly models the market observed credit spreads and provides aconvenient simulation framework for modeling the dynamics of thecredit spread and specifying or calibrating the credit spread volatilities.In order to proceed further, we first need to define the non-default orsurvival value and the extinguishing value of a default contingent instrument.The time t non-default or survival value, r? ( t ), of a default contingent instrument is its time t value conditioned on no default of the reference credit occurring at or before time t. As we discussed before, in general, the following martingale relationships do not hold, ie.,
(8.59)
(OItIT)This is due to that r? ( f ) or r? ( T ) do not fully capture the cashflowsof the instruments. More specifically, they only capture the non-defaultor survival payoffs, but do not capture the default payoffs. As a matterof fact, P ( t , T ) E T [ r ? ( T ) ]or p ( t ) E ? [ v ( T ) / p ( T ) ] is the time t value' I 6 These papers are downloadable from http://ssm.com/abstract=261O51 andhttp://www.schonbucher.de/papers/cdsoptions.pdf.
Sec. 8.7
Risky Market Model for Credit Spread Modeling
353
of the corresponding instrument that is not default contingent.On the other hand, the time t extinguishing value, V (t ) , of a defaultcontingent instrument is the time t value of the same default contingentinstrument but with an extinguishing feature in that the instrument isknocked out with zero rebate if the reference credit defaults at or beforetime t. More specifically,
(8.60)where, z, as usual, is the default time of the reference credit and l{t<r)represents the extinguishing feature. In other words, V ( t ) is the time tvalue of an extinguisher or the non-default or survival leg of the samedefault contingent instrument which is a traded asset.Equivalently,
V ( t ) = V ( t ) l z< t ,
(8.61)
Thus, if there are no intermediate non-default cashflows before timeT and at or after time t, then, by applying the Harrison-Pliska martingaleno-arbitrage theorem, we have the following martingale relationships onthe extinguishing values, i.e.,
(8.62)L
.
.
(0I t I T )
Let's further proceed to define the survival probability measure Pfiwith respect to a traded default contingent numeraire asset with time tpositive survival value of f i ( t ) and with no intermediate non-defaultcashflows before time T and at or after time t.So far, we have assumed and will continue to assume that the defaultcontingency of the instrument, the default contingency of the numeraire,and the extinguishing feature refer to the same single reference credit.But, similar arguments can be applied to the cases involving multiple
Credit Risk Modeling andpricing
354
Chapter 8
reference credit names.One of the goals to define the survival probability measure is to answer the question what are the probability measure and the numerairet ) normalized by the numeraire survival value is a martinsuch that
v(
gale. As we shall see that v ( t ) / f i ( t )is a Pfi-martingale.Following Schonbucher (2000 and 2003), we define the survival arbitrage-free probability measure Pfi through the following RadonNikodym derivative
(8.63)(OSt)
where we have assumed that no default has occurred at time 0.It is easy to verify that since
(8.64)
(OIt IT)
thus,
(8.65)(OIt)
Note, the same as before,(8.66)( O I t IT )
The term survival probability measure comes from the fact that Pfi
Sec. 8.7
355
Risky Market Model for Credit Spread Modeling
only describes the non-default or survival states and assigns zero probability or measure to the default states or events IlrSt) , i.e.,
(8.67)
=O(OIt)
But, typically,(8.68)(0 5 t )
These indicate that the probabilities measures Pfi and Q are notequivalent. Rather, PR is absolutely continuous with respect to Q .Furthermore, we can see that
(t)/I? ( t ) is a Pfi-martingale as fol-
lows,
(OIt5T)
In deriving the above equation, we have used
Credit Risk Modeling and Pricing
356
Chapter 8
(8.70)(OSt)
To proceed further on the martingale modeling of the credit spreadprocess, we start by identifying the martingale measure of a creditspread.Let j j (t,q,q+l)denote the time t par coupon of the CDS of thejthcredit name for the accrual and default protection period from I;: to I;:+l
Aj ( t ,q,q+l) the time t survival value of a defaultable annuity of the jthcredit name for the accrual and default protection period from I;: to I;:+l,and P- the annuity survival measure with respect to the defaultable anA1
nuity Aj ( t ,q,q+l) as the numeraire.Since 1 f < r , ij( t ,q ,q+l) 2,( t ,I;:, q+l) is the time t extinguishing
{
I1
value of the protection or survival leg of the CDS for thejth credit name(for the accrual and default protection period from I;: to I;.+1where z j isthe default time of thejth credit name) which is a traded asset, thus wehave the following martingale relationship
( O l t l T 5 q <I;:+J
If we restrict ourselves only to the non-default or survival states, thenthrough a change of probability measure, we obtain
ij(t,q,q+l)= Epi [s, (T,I;:,I;:+,)]>
(8.72)
where the expectation is taken under the annuity survival measure P-
Aj
Sec. 8.7
Risky Market Model for Credit Spread Modeling
357
Similarly, the time t value of an extinguishing forward starting CDSon thejthcredit name (with reset at time T and the accrual and protectionperiod from q to) is given by
=1{l<rj
1 i j(t,q,q+]) w ( K - 3, (t,q,q+]))(OStlTIT
where K is the fixed coupon rate or premium of the CDS, w = 1 is forselling the credit protection (or being short in the credit spread or long inthe credit market), and w = -1 for buying the credit protection (or beinglong in the credit spread or short in the credit market).In deriving the above equation, a change of probability measure fromthe spot arbitrage-free measure Q to the annuity survival measure PiJ
and the martingale relationship in Eq. (8.72) were applied.Furthermore, the time t value of an extinguishing default swaption(DSO) on thejthcredit name (with expiry at time T for optionally entering a CDS with the accrual and protection period from q to q+,)isgiven by
Chapter 8
Credit Risk Modeling and Pricing
358
1
L
( O < t < T < T <q+l)where K is the strike and the option holder has an option to enter a CDSwith coupon rate of K for selling ( w = 1 ) or buying ( w = -1 ) the creditprotection.Similar to before, in deriving the above equation, a change of probability measure from the spot arbitrage-free measure Q to the annuitysurvival measure P- was applied. If j j (t,qyq+l)follows a simpleA,
lognormal, normal, or shifted lognormal process, then the above equationreduces to a simple (lognormal, normal, or shifted lognormal) Blackformula. The market typically quotes the option premium in terms of theimplied Black lognormal volatilities.Based on the martingale relationship in Eq. (8.72), the diffusionprocess of the credit spread j j (tyq,q+l)can be specified by the martingale representation theorem (in the absence of jumps and under the usualtechnical conditions) as
(8.75)
s. 2 .
where 6 - is the instantaneous credit spread volatility and W,sione-dimensional P- -Brownian motion.
is a
Ai
6 - can be determined by calibrating to the initial volatilities of thei'
See. 8.8
Joint Credit Spread and Default Modeling
359
credit spreads implied from the default swaption market or by othermeans through Eq. (8.74) and W,''"jother driving factors.
8.8
needs to be properly correlated to
Joint Credit Spread and Default Modeling
In this section, we shall present an approximate methodology for jointlymodeling the credit spread processes and the default processes. The goalis to price instruments sensitive to both default and credit spread volatility and, particularly, the credit charge or CVA.The general idea is to introduce additional degrees of freedom to thedynamics of the credit spread process in the simplified structural modelso that it can have enough parameters to calibrate to the credit spreadvolatilities.Conceptually, we can use the uncertain default barrier model ofFinkelstein, Lardy, et al (2002) or the like as the additional degrees offreedom. Stochastic future recovery rates can also be considered.Here we present a simple approximation. The basic idea is to introduce an additional exogenous diffusion component to the credit spreadprocess by combining the simplified structural model and the martingalecredit spread model presented in the previous sections.More specifically, we decompose the credit spread ( t ,q ,q+l) asfollows
sj
s j ( l , q , T . + l =s~l(~,I;.,q.+l)+sj"(t,q,q+l))(OltSTSq
q+,)
Y
(8.76)
is the credit spread implied by the simplified strucwhere ,?j'(t,q,q+l)tural model and j j "( t ,I;. ,q+,) is an additional credit spread as the cor-
sj
rection term.(t,T,q+l) is solved by the martingale model in Eq.(8.75) and is calibrated to the initial credit spread volatilities.Since both
sj( t ,q ,q+l) and sjl( t ,q
) are solved by their pro-
spective models, on the surface, it looks trivial to obtain ;3from Eq. (8.76).
However, the subtlety is that
(l,q.,q+l)
sj (t,q,q+l)and
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Credit Risk Modeling and Pricing
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3; ( t ,q.,I;.+1) are typically solved under different probability measures.If we use Eq. (8.75), then j j (t,I;:,q+,)is solved under the survival annuity probability measure Pij , whereas j j " (t,q,q+,)is typicallysolved under the spot arbitrage-free measure Q . So, we will need toperform a change of measure to a common probability measure.One way to approach this is to rewrite Eq. (8.75) in the spot arbitrage-free measure Q , which can be achieved through the application ofGirsanov's change of measure theorem.Alternatively, we can derive the change of measure ourselves. If weonly consider the non-default or survival states, then
Since
-dsj (t,q,q+Jdij( t ) / i j(t)+ L i p j
( t , q , ~ + , ) ~ j ( t ) ),/ i j ( t )(8.77)
(O<t<T<I;:<I;:+,)then, based on the martingale representation theorem (in the absence ofjumps and under the usual technical conditions), we obtain
where 6 - is the instantaneous credit spread volatility,W,s'.Q is a onesi
Sec. 8.8
Joint Credit Spread and Default Modeling
dimensional Q -Brownian motion, 6
-
361
is the instantaneous percentage
A,
volatility of 2,( t ), and pSJ
-
is the instantaneous correlation between2’4,
j j (t,?,T+]) and j j ( t ).Essentially, we have derived the risky market model for creditspreads similar to the interest rate market model.under the spotWith the above equation, we can solve Sj (t,q,q+l)arbitrage-free measure Q on the survival or non-default states. 6 - cans;
be determined by calibrating to the initial volatilities of the credit spreadsimplied from the -defaultswaption market through Eq. (8.74) or can bespecified, and FfJ’AJ needs to be properly correlated to other drivingcan be approximated with methodologies simifactors. 6 = and p A,
’J
lar to the approximations of the BGM model. A simple approximationunder the 1-factor setting is given by
(8.79)
An additional drift correction Ap, is needed in Eq. (8.78) so that wecan perform martingale resampling such that the martingale relationshipsin Eq. (8.72) and Eq. (8.73) are satisfied. In other words,
The initial value of j j ( t ,q ,
) is the extinguishing forward CDS
par rate that can be derived from the term CDS par rates as
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Credit Risk Modeling and Pricing
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If there are jumps in the credit spread processes due to credit contagion, we can superimpose exogenous jump components in the creditspread processes similar to the previous section.We can also incorporate (correlated) stochastic recovery rate processinto the risky market model. If we adopt the common approach and usea Beta distribution for the recovery rate process, then Eq. (8.78) becomesan approximation.
8.9
Counterparty Credit Risk Pricing in OTC Derivatives
As we discussed before, pricing counterparty credit risk is the same aspricing two default options or expected default lossedgains on a basketof OTC derivatives or a counterparty portfolio.One is the option for one’s counterparty to default first and the otheris the option for one to default first oneself. The first to default nature istypically a second order effect, unless the default probability of one orboth parties is very high. Thus, we generally ignore the first-to-defaultnature, and, consequently, pricing these default options is the same aspricing the generic single-name default contingent instruments (or, morespecifically, options) presented in the previous sections.The PV of such default options or expected default losses/gains areintegral parts of the market values of the OTC derivatives, and, thus, theyshould be priced in and, for market making purposes, they should also behedged and risk managed like other trades.As we discussed in Chapter 1, there are many subtleties in the pricing of the counterparty credit risk in OTC derivatives as compared to,e.g., the pricing of the credit risk in a bond. The following are some examples.1. In addition to the volatility in the default recovery rate, the default or credit exposure (or the replacement value of the counterparty portfolio at the time of the counterparty’s default) is alsohighly volatile. Accurately modeling the credit exposure is one
Sec. 8.9
Counterparry Credit Risk Pricing in OTC Derivatives
363
of the key components and challenges of the counterparty creditrisk pricing.2. The counterparty credit risk is typically on a netted counterpartyportfolio, not on a single trade. Thus, the counterparty defaultoptions are really options on a basket of derivatives.3. One needs to handle the diffusion and jump correlation betweenthe market and counterparty default processes in order to handlethe wrong-way and the right-way exposures.4. One also needs to model various credit enhancement or contingency features, such as rating based collaterals and rating basedbreaks.5. Furthermore, the trades in a counterparty portfolio can be modeled by different pricers, booked in different systems, from different desks, or even from different product areas. This presentssignificant challenges to the counterparty credit risk modelingand the system.In general, we can use Eq. (8.41) and Eq. (8.42) to price the PV ofany one of the default options. More specifically, we can rewrite Eq.(8.41) and Eq. (8.42) as
P$ ( f ) = P$ (o,o, t ,F?)
(8.82)
(O<T),
P$ ( 0,t)= P$ (o,o,t)and
(8.83)
The default cashflow can be expressed with the default exposure
Credit Risk Modeling and Pricing
364
Chapter 8
V,, ( t ) (or potential value at default risk) and the default recovery rateR ( t ) as follows(8.84)
Thus,
where VDE‘( t ) is the effective default exposure given by
5; (f ) = VE, ( t) (1 + W P t )
*
(8.86)
If there is a jump component in the default exposure V,, ( t ) upondefault, then Eq. (8.49) and Eq. (8.50) need to be considered.The above equations are typically solved by (multi-dimensional)Monte Carlo simulations whereby all the underlying market variablesand the default processes and the credit spread processes of the counterparties and the underlying obligations are collectively simulated.The market variables, such as the interest rates, equity prices, FXrates, are simulated by their perspective models. The major differencesbetween the counterparty credit risk pricing model and the trade-level (ormicro) model for the underlying derivatives pricing, as we discussed earlier, are that the former being a portfolio-based or macro model, requires,consistent modeling of all the variables involved with, e.g., global calibrations of volatilities and proper correlation among all the variables.The correlation is typically specified through the instantaneous correlation among the incremental driving factors (e.g., Brownian motions),which can be achieved through Choleski decomposition or a factormodel.The joint default and credit spread processes for the underlying obligations and the counterparties will also need to be simulated, e.g., withthe methodologies presented in the previous sections. The incrementaldriving factors for these processes should also be properly correlated
Sec. 8.9
Counterparty Credit Risk Pricing in OTC Derivatives
365
with the incremental driving factors.In some cases, the recovery rate processes will need to be simulatedand jump components in the processes of the market and credit variableswill need to be considered.Once all the market and credit variables are simulated, a major (probably the most difficult) task to compute the cash or dirty price’I7 of eachtrade in the counterparty portfolio on each simulation path as of eachsimulation date. This can be achieved by directly calling the trade-levelpricers or by simulating through the grids or simulation paths of thetrade-level pricers.This is also a generic scenario analysis engine that has many additional applications.8.9.7
Credit Charge Calculation
The credit charge (or credit premium) Vcc is the PV of the default option due to one’s counterparty’s default. More specifically, it is given by
where V,,( t ) is the asset-side default exposure or positive exposurewhen one has net receivables or the default exposure with respect to thecounterparty’s default, VAsset’( t ) is the effective default exposure,P$(t) is the counterparty default probability, RCP(t) is the counterparty default recovery rate, and T is the risk end date of the counterpartyportfolio (beyond which there is no risk at all).Since one has a short position in the option due to the counterparty’sdefault, Vcc, representing the PV of such short position, is no greaterthan zero. V,, is the same as the asset side of CVA.Since, at any given time t, one will have counterparty default risk
’”
The reason that we use the cash or dirty price is due to that the coupon or accruedinterest is subject to the counterparty credit risk.
366
Credit Risk Modeling mdPricing
Chapter 8
only when one has net receivables from the counterparty (or one is net( t ) isowed money by the counterparty), thus, the default exposure V,,given by(8.88)with N trades where V;: ( t ) is
for an all-netted counterparty
the time t cash or dirty price of the ithtrade and Vco,( t ) is the amount ofcollaterals to be received (a positive value) or posted (with a negativevalue) based on the collateral agreement or a CSA (Credit Support Annex). We further denote Y ( t ) the time t cash or dirty price of the coun-
c ).N
terparty portfolio, i.e., Y ( t ) =
(t
i=l
We shall discuss again on the collaterals in Section 8.9.4. Roughlyspeaking, the effect of the collaterals is to put a cap on the counterpartydefault exposures.In terms of the risks, a credit charge is short in the counterpartycredit spread, short in the Vega and Gamma of a call option on the counterparty portfolio, long in Theta, and short in the correlation between thecounterparty credit spread and the market value of the counterparty portfolio. If such correlation is positive (for the case of wrong-way exposures), then the credit charge is short in the counterparty credit spreadvolatility. If such correlation is negative (for the case of right-way exposures), then the credit charge is long in the counterparty the credit spreadvolatility.8.9.2
Expected and Potential Exposures and Expected Shortfall
Other counterparty credit risk measures include the expected and potential exposures and shortfall.The positive or asset side expected default exposure curve or profile(with respect to the counterparty's default) is given by'I8 An all-netted counterparty portfolio means that the counterparty credit risk is a function of the net value (or cash or dirty price) of the portfolio. It is common that a counterparty portfolio is partially netted. This is almost the same as multiple counterparty portfolios except that the collateral amount can still be determined by the all-netted portfoliovalue.
Sec. 8.9
367
Counterparty Credit Risk Pricing in OTC Derivatives
(8.89)The potential exposure (PE) curve or profile PE ( t ) at the percentileof PpE (due to the counterparty's default) is given by(8.90)For instance, a 95'h percentile PE (i.e., PpE =95%) means that thePV of the default exposure or VAsse'(z) is less the PE P E ( t ) with theP(t>
probability of 0.95.The maximum PE MPE ( t ,,t2) within a certain time band or interval
[t, ,t, ) is defined asMPE ( t ,,tz ) = max( PE( t)[tlI t < t 2 ) .
(8.91)
A better credit risk measure than PE or MPE is the expected shortfallor expected tail loss ETL defined asr
1
In general, ETL can be defined as
where f(*)is a certain tail distribution function with 1a special case.The expected tail loss allows more meaningful comparison of thecounterparty credit risk among counterparties with different creditspreads and recoveries, and different trades.
368
8.9.3
Credit Risk Modeling and Pricing
Chapter 8
Credit Benefit Calculation
The credit benefit Vcs is the PV of the default option due to the one’sown default. More specifically, it is given by
(O<T)
where VLiab( t ) is the liability-side default exposure or negative exposurewhen one has net payables or the default exposure with respect to one’sown default, VLiab’( t ) is the effective default exposure, P& ( t ) is one’sown default probability, RSey( t ) is one’s own default recovery rate, and
T is the risk end date of the portfolio (beyond which there is no creditrisk).Since one has a long position in the option due to one’s own default,Vcs ,representing the PV of such long position, is no less than zero. VcBis the same as the liability side of CVA.Since, at any given time t, one will have counterparty default riskonly when one has net receivables from the counterparty (or one is netowed money by the counterparty), thus, the default exposure VLiab( t ) isgiven by
yLiab( t )= min
“C
1
5 ( t )- vc0,( t ) , o
i= I
for an all-netted counterparty portfolio with N trades where
(8.95)
( t ) is the
time t value of the ith trade and Vcor( t ) is the amount of net collateralsreceived (a positive value) or posted (with a negative value) based on thecollateral agreement or a CSA.The negative expected default exposure curve or profile (with respectto one’s own default) is given by
Sec. 8.9 Counterparty Credit Risk Pricing in OTC Derivatives
369
(8.96)In terms of the risks, mostly opposite to the credit charge, a creditbenefit is long in one’s own credit spread, long in the Vega and Gammaof a put option on the counterparty portfolio, short in Theta, and short inthe correlation between one’s own credit spread and the market value ofthe counterparty portfolio. If such correlation is positive (for the case ofright-way exposures to the counterparty), then the credit benefit is shortin one’s own credit spread volatility. If such correlation is negative (forthe case of wrong-way exposures to the counterparty), then the creditbenefit is long in one’s own credit spread volatility.
8.9.4
Collateral or Margin Agreement
A CSA typically specifies a bilateral collateral or margin agreement with) and a minimum transfer amount (MTA, VMTA ). Thea threshold ( VThreshthreshold determines the potential or contractual collateral amount andthe MTA determines whether or an actual transfer of collaterals willhappen. For instance, the CSA with one counterparty can specifyVThresh= $lOMM and VM, = $1MM , and the CSA with another counterparty can specify VThresh= 0 and VM, = $0.2MM .In addition, the threshold is typically dependent on the credit ratingof the parties involved. Thus, the threshold (and possibly the MTA) foreach party can be different. Thus, for generality, we denote vizh andV i g to be the asset side of the threshold and MTA (when one has receivables) that governs the collaterals posted by the counterparty, andLiabVThreshand V!,$ito be the liability side of the threshold and MTA (whenone has payables) that governs the collaterals posted by oneself.Roughly speaking, the goal of a collateral agreement is to limit thepotential default loss by imposing a cap (roughly VEGi + V,&i‘ ) for thepositive or asset side exposure and a cap (roughly Vzzth + V , $ ) for theabsolute value of the negative or liability side exposure.Here we present a simple example for the collateral calculation.Let’s assume that there is no initial margin, i.e., the initial actualBefore the first time whencollateral amount V,,, (0) = 0 .
Cred