m Ultin Am e an d m Ultisc Ale de Fault m o de Lin g

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MULTINAME AND MULTISCALE DEFAULT MODELING

JEAN-PIERRE FOUQUE, RONNIE SIRCAR , AND KNUT SLNA

Abstract. Joint default modeling for a set of firms is crucial in the context of pricing credit derivatives. We considerhere a model for defaults among multiple firms based on the Vasicek or Ornstein-Uhlenbeck model for the hazard rates of theunderlying companies or names. We analyze the impact of volatility time scales on the default distribution for the set offirms. We also consider the associated impact on a particular credit derivative contract, the so-called CDO. We demonstratehow correlated fluctuations in the parameters of the firm hazard rates affect the loss distribution and prices associated with theCDOs. The effect of stochastic parameter fluctuations is to change the shape of the loss distribution and cannot be captured byusing averaged parameters in the original model. Our analysis assumes a separation of time scales and leads to a singular-regularperturbation problem [9, 10]. This framework allows us to compute perturbation approximations that can be used for effectivepricing of CDOs.

1. Introduction. Credit derivatives are financial securities that pay their holders amounts that arecontingent on the occurrence (or not) of one or more default events such as the bankruptcy of a firm ornon-repayment of a loan. The dramatic losses in the credit derivatives market in 2007 and subsequentlyillustrate that the problem of appropriate modeling and pricing of large portfolios of debt obligations ischallenging and also a largely open question. The mathematical challenge is to model the default times of thefirms (also referred to as names or underlyings) and, most importantly, the correlation between them. Partof the challenge is that incorporating heterogeneity and correlations may appear as intractable due to thecurse of combinatorial complexity. Here we consider pricing of collateralized debt obligations (CDOs) usingintensity-based models with multiscale stochastic volatility. The CDO contract is explained in more detailin Section 2.1, and intensity-based models in Section 2.2. A main aspect of our approach is to make use ofapproximation methods via singular and regular perturbation expansions that make the multi-dimensionalproblems tractable.

Coupulas have been the standard tool in the industry for creating correlation structures [16] in the lastfew years. This is even the case at present after the recent credit crisis. The main drawback of this approachis the fact that these are static models which do not take into account the time evolution of joint defaultrisks. This has been recognized in the academic literature on dynamic models with recent developments in themultiname structural approach [12, 15], reduced form models [4, 19, 18, 6], and top-down models [8, 17, 21].

We shall consider the case when we use the Vasicek model in the context of (bottom-up) multiname

reduced form modeling of credit risk. With bottom-up we mean an approach where we explicitly model thedefault events of the different firms involved, rather than for instance modeling the default distribution directly(top-down). Moreover, by reduced form model we mean that the firms default is modeled as the first arrivalof a Cox process with a hazard rate that itself is modeled as a stochastic process. Sometimes these modelsare therefore called doubly stochastic. Here, we let the hazard rates of the default times of theN firms bespecified as correlated Vasicek or Ornstein-Uhlenbeck processes. Clearly, in this Gaussian model, the intensitymay become negative. But, as stated in [5], the computational advantage with explicit solutions may beworth the approximation error associated with this Gaussian formulation. This is particularly the case whenit comes to calibration, which typically involves a large number of evaluations of the default probabilities aspart of an iterative procedure. In addition to explicitly modeled correlation in between name hazard rates,implicit correlation generated by a common factor is important. This phenomenon, with a commonvolatilityfactor has been analyzed in [11, 12, 19] and we continue here this line of research. We show below how theroles of explicit and implicit correlations in joint default probabilities become transparent in the Vasicek

framework when the common factor exhibit multiscale fluctuations. This modeling builds on the modelingframework we introduced in [10].

The multiscale stochastic parameter framework has recently been used in top down modeling of the defaultdistribution in [1], in the context of the modeling approach set forth in [3]. We remark that in the top downapproach, as the default distribution is modeled directly, contagion effects corresponding to rapid transmission

Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110,[email protected]. Work partially supported by NSF grant DMS-0455982.

Department of Operations Research & Financial Engineering, Princeton University, Sherrerd Hall, Princeton, NJ 08544,[email protected]. Work partially supported by NSF grants DMS-0456195 and DMS-0807440.

Department of Mathematics, University of California, Irvine CA 92697, [email protected].

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of financial shocks and default events through large groups of names may implicitly be incorporated. Suchcontagion effects may in particular lead to multimodal default distributions with relatively heavy tails. Thebottom up framework has the advantage of modeling the different firms and their default events explicitly.Thus, information about marginal default probabilities and commonality of names in between different creditderivatives can be incorporated and the model parameters have more explicit financial interpretation. Inthis paper our focus will be on correlation effects that is caused by multiscale stochastic parameter variation.However, there may also be sector wide contagion effects and in the bottom up framework and we remark thatsuch phenomena can be modeled by a regime based framework or grouping of names [19]. In the regime basedframework all the names experience a shift in hazard level in the event of a regime shift. Such mechanismcan be related to market information like sector stratification and consequences of particular market wideperturbative events, thus reinforcing the advantage of bottom up modeling in that parameters may typicallybe given rather direct financial interpretations.

We introduce the pricing problem in Section 2.1 and discuss the doubly stochastic framework in somemore detail in Section 2.2. See for instance [5, 20] for a detailed discussion of these concepts. In Section 2.3we discuss the Vasicek credit model and pricing of CDOs. Our focus is on correlation effects between namesand the effects of stochastic multiscale parameter variations. We discuss both the symmetric case with simpleexplicit formulas for survival probabilities and the case where there is heterogeneity in between the modelsfor the name intensities. In the latter case effective computational procedures via conditioning are given andwe illustrate with numerical examples.

2. Modeling.

2.1. The CDO contract. CDOs are designed to securitise portfolios of defaultable assets. Their mainfeature is that the total nominal associated with the names or obligors is sliced into tranches. Each tranche isthen insured against default. The first default events apply to the first tranche and so on. The protection sellerfor the first tranche, the equity tranche, is therefore strongly exposed to credit risk relative to the protectionsellers for the subsequent mezzanine and senior/super-senior tranches, and the CDO provides a prioritizationof credit risk. Two credit derivative indexes are the US based CDX and the European iTraxx. Each trancheis described by a lower and an upper attachment point. In the CDX case the decomposition into tranchescorresponds to{0-3, 3-7, 7-10, 10-15, 15-30}% of the total nominal and this is the decomposition we shall usebelow in our computational examples.

Let be the yield associated with tranche , that is, the rate at which the insurance buyer pays forprotection of tranche . In the event of default in this tranche, the protection seller pays a fraction 1 R ofthe loss, with R being the recovery, to the tranche holder (the buyer). We stress here that the recovery ischosen as constant, while in general it could also be modeled as being random. We shall also assume that thepayments are made at a set of predetermined times Tk, k {1, 2, , K}.

For valuation purposes, all expectaions are taken under the pricing measure IP. We have that is determined by equating expected cashflows from the insurance buyer with expected cashflows from theinsurer. This gives

k

erTkf(Tk1)(Tk Tk1) =

k

erTk (f(Tk1) f(Tk))(1 R), (2.1)

where r is the constant short rate and f(Tk) is the expected fraction of tranche left at time Tk, given asfollows. Letpn(t) be the probability ofn names defaulting by time t, and let a, b denote respectively thelower and upper attachment points associated with tranche . We then have

f(t) =(a1)N

n=0

pn(t) +(b1)Nn=aN

pn(t)(b n/N)/(b a+ 1),

with N the number of names. We stress that the tranche yields are determined by the loss distribution,pn(t), n= 0, , N , t= T1, , TK.

This is therefore the central technical question that we address in this paper, how to consistently modeland compute the loss distribution over Nnames at a set of times Tk. The expression (2.1) is an approximationcorresponding to the defaults occurring in the time interval from Tk1 to Tk being accounted at the end ofthis time interval. This is the simple model contract we use below when the tranches are those associated

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with the CDX. Further details can be found in [7], for instance. In the following, we shall compute the trancheprices when the default of the obligors are modeled in terms of the Vasicek reduced form model, and wenext describe doubly stochastic modeling.

2.2. Doubly Stochastic Modeling. We consider N obligors or underlying names. The event that aparticular obligori defaults is modeled in terms of the first arrival i of a Cox process with stochastic intensityor hazard rate X(i), the conditional hazard rate. Conditioned on the paths of the hazard rates, the default

timesi of the firms are independent, and the probability that obligor i has survived till time T, is given byexp( T0 X(i)s ds). Therefore, the unconditional survival probability is

IP{i> T} =IE

eRT0

X(i)s ds

,

with the expectation taken with respect to the risk neutral pricing measure IP.Consider a subset{i1, , in} of obligors. The probability of the joint survival of this set till timeT is

then, under the doubly stochastic framework,

IP{i1 > T, , in > T} =IE

ePn

j=1

RT0

X(ij )s ds

.

2.3. Vasicek Intensities and Survival Probabilities. We start by considering Nnames whose in-

tensities (X(i)t ) are given by correlated Ornstein-Uhlenbeck processes 1 i N. This corresponds to theVasicek model

dX(i)t =i

i X(i)t

dt +idW

(i)t , (2.2)

where the (W(i)

t ) are correlated Brownian motions, with the correlation matrixc given by:

d

W(i), W(j)

t=cijdt . (2.3)

We denote the survival probability for name i by

Si(T; xi) = IP(i > T| X(i)0 =xi) = IE e

RT0

X(i)s ds |X(i)0 =xi .We also denote the joint survival probability of all Nnames by

S(T; x, N) = IE

eRT0

PNi=1X

(i)s ds |X(1)0 =x1, , X(N)0 =xN

,

with x= (x1, , xN)IRN. From the Feynman-Kac formula it follows that the joint survival probabilityfrom time t till time T

u(t, x) =IE

eRTt

PNi=1X

(i)s ds |Xt = x

,

solves the partial differential equation

u

t +

1

2

Ni,j=1

(ij cij )

2u

xixj +

Ni=1

i(i xi)u

xi N

i=1xi

u= 0 , (2.4)

with terminal condition u(T, x) = 1.Assume first that the covariance matrix c is the identity matrix, corresponding to the components of the

intensity process X being independent. Then, as is well known, or can be readily checked, the solution isgiven by

u(t, x) =

Ni=1

Ai(T t)eBi(Tt)xi ,

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where we introduce

Bi(s) =

s0

ei d=1 ei

i, (2.5)

Ai(s) = ei

Rs0

iBi() d+12

2i

Rs0

B2i () d =e

di(sBi(s))+ 2i4i

B2i(s)

, (2.6)

and di =i 2i

22i.

In the general correlated case, we can write

u(t, x) = Ac(T t)N

i=1

Ai(T t)eBi(Tt)xi , (2.7)

with

Ac(s) = e12

PNi=1

PNj=i=1(ijcij)

Rs0

Bi()Bj()d. (2.8)

The last integral is given explicitly by

s0

Bi()Bj () d= s

ij Bi(s)

i(i+j) Bj (s)

j (i+j ) Bi(s)Bj (s)

(i+j ) . (2.9)

3. Symmetric Name Case. In this section, we analyze the symmetric names case where the dynamicsand the starting points of the intensities are the same for all the names. This is convenient to understand theeffects of the correlation and the size of the portfolio. We return to the heterogeneous case in Section 4.

Specifically, we have

dX(i)t =

X(i)t

dt + dW

(i)t , X

(i)0 =x,

with the parameters , and x assumed constant and positive. Moreover, we assume that the correlation

matrix is defined by cij = X , for i= j, with X 0, and ones on the diagonal. We remark that such acorrelation structure can be obtained by letting

W(i)

t =

1 X W(i)t +

X W(0)

t , (3.1)

where W(i), i = 0, 1, , N, are independent standard Brownian motions.It follows from (2.7) that the joint survival probability forn given names, say the firstn names, is

S(T; (x, , x), n) = IE

eRT0 (X(1)s ++X(n)s )ds |X(1)0 =x, , X(n)0 =x

=en[(TB(T))+[1+(n1)X ]

2B2(T)/(4)+xB(T)] , (3.2)

with

B(T) = 1 eT

,

= [1 + (n 1)X ] 2

22. (3.3)

This expression shows explicitly how the joint survival probability depends on the correlation X and thebasket sizen. Note in particular how the basket size enhances the correlation effect. We consider next howa characterization of the loss distribution follows from (3.2).

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3.1. The Loss Distribution. The loss distribution at timeTof a basket of sizeNis given by its massfunction

pn= IP {(#names defaulted at timeT) = n}, n= 0, 1, , N , (3.4)

and is explicitly

pn=

Nn

nj=0

nj

SN+jn(1)j , (3.5)

using the short hand notation Sn = S(T; (x, , x), n) for the joint survival probability ofn names (for aderivation of this classical formula, see for instance [12]). This gives rise to an O(N2) procedure for calculatingthe loss distribution.

However, a direct implementation of this formula is not numerically stable due to catastrophic cancellationerrors in finite precision arithmetics. We comment therefore on an alternative implementation of (3.5). Notefirst that, from the formula (3.2) for the survival probability, we can write

Sn= ed1n+d2n2 , (3.6)

with di explicitly given as

d1 = d1(T, x) = T+ (x )B(T) 12

2(1 X )B(2)(T) , (3.7)

d2 = d2(T) = 1

22X B

(2)(T) , (3.8)

B(2)(T) =

T0

B2(s) ds= (T B(T))

2 B (T)

2

2 ,

and we assume that the model parameters are chosen so that d1 > 0. In the independent case X = 0, we getthe binomial distribution:

pn=

N

n

(1 ed1)ne(Nn)d1 =: pn(d1) .

In the general case, we can write

Sn= IE

ed1n+n2d2Z

,

forZa zero mean unit variance Gaussian random variable. Therefore, in the general case we find

pn= IE

pn(d1+

2d2 Z)

. (3.9)

Thus, we get the loss distribution stably and fast by integrating (non-negative) binomial distributions withrespect to the Gaussian density. We remark that this essentially corresponds to conditioning with respect tothe correlating Brownian motion, W(0), in (3.1). The argumentd1+

2d2 Zwill be negative for Znegative

and with large magnitude. This reflects the fact that we are using a Vasicek model where the intensity may

be negative. Below, we condition the Gaussian density toZ > d1/2d2 and choose parameters such that thecomplementary event has probability less than 103.3.2. Example with Constant Parameters and Strong Correlations. In the model (2.2) we choose

the parameters

= .02 , = .5 , = .015 , x= .02 ,

and we let time to maturity T= 5 and the number of names N= 125, corresponding to th most commonCDO contracts on the CDX and iTraxx. Here an below we use 20 equally spaced payment dates over theperiod of the contract. The loss distributions withX = 0 and X = .75 respectively are shown in Figure

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3.1 (left). Note how the strong correlation widens the loss distribution. Hence, it will strongly affect trancheprices. We consider the tranche prices for the CDX, defined in Section 2.1. The short rate is chosen to befixed at 3%, and the recovery is 40%. In Figure 3.1 (right) we show the tranche prices plotted against theupper attachment point of each tranche, that are associated with the loss distributions on the left. The topplot is on a linear scale and the bottom on log scale to visually resolve well the senior tranches. Note how thestrong correlations affects all tranches and that its relative effects are strongest for the senior tranches. Theequity tranche is also strongly affected by the correlation with a negative correction.

0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

0.1

0.12

LOSS DISTRIBUTION

INDEPENDENT

STRONG CORRELATIONS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

2000

4000

6000

8000

TRANCHE PREMIA

INDEPENDENT

STRONG CORRELATIONS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510

10

105

100

105

INDEPENDENT

STRONG CORRELATIONS

Fig. 3.1. The left figure shows the loss distribution, pn, the dashed line without correlation, X = 0, and the solid linefor strong name-name correlation, X = 0.75. The right figure shows the tranche premia, the CDX tranche prices , againstupper attachment point, the diamond-dashed line for constant parameters and the solid line for strong name-name correlation.

4. Stochastic Volatility Effects. Stochastic volatility driven by common factors varying on fast andslow time scales has been shown to be effective in the structural approach, for capturing the yield spreads ofsingle-name defaultable bonds [11], and multiname loss distributions [12]. Here we extend multiname intensitymodels, described in the previous section, to incorporate multiscale stochastic volatility. The intuitive idea is

that simultaneous high volatility in intensity will generate clustering of defaults, giving the needed flexibilityin the loss distribution. The slow factor gives additional freedom for the term structure.

Under the risk-neutral probability measure we assume the model

dX(i)t =i(i X(i)t )dt+(i)t dW(i)t , (4.1)

for 1iNwhere the W(j)s are correlated Brownian motions as in (2.3). The volatilities are stochasticand depend on a fast evolving factor Y and slowly evolving factor Z:

(i)t =i(Yt, Zt),

where the functions i(y, z) are positive, bounded and bounded away from zero, and smooth in the secondvariable.

The fast process is modeled by

dYt=1

(m Yt)dt+

2

dW

(y)t ,

with the small parameter corresponding to the short time scale of the processY. In fact, it is not importantwhich particular model we choose for the fast scale, the important aspects of the process Y are that it isergodic and that it evolves on a fast time scale. We assume the correlations

d

W(i), W(y)

t=Ydt , for 1 iN .

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The slow factor evolves as

dZt = c(Z)dt+

g(Z)dW(z)

t ,

with the large parameter 1/corresponding to the long time scale of the process Z.The functions andc and g are assumed to be smooth and we assume the correlations

dW(i), W(z)t

=Z

dt .

We denote byY Z the correlation coefficient defined by

d

W(y), W(z)

t=Y Zdt

We assume the coefficients Y, Z, Y Zand the matrix (cij ) are such that the joint covariance matrix of theBrownian motionsW(i) (i= 1, , n), W(y) andW(z) is non-negative definite.

The joint survival probabilities become now

S(T; x, y , z , N ) = IE

ePN

i=1

RT0

X(i)s ds |X0 = x, Y0 = y, Z0= z

.

In this case the joint survival probability from timet

u,

(t, x; y , z , N ) = IE

eRTt PNi=1X(i)s ds |Xt= x, Yt = y, Zt= z , (4.2)

solves the partial differential equation

L,u, = 0, (4.3)u,(T, x, y , z) = 1

with the notation

L, = t

+ L(x,y,z)

Ni=1

xi

,

whereL(x,y,z) denotes the infinitesimal generator of the Markov process ( Xt, Yt, Zt) under the risk-neutralmeasure.

We write the operatorL,

in terms of the small parameters (, ) as

L, = 1L0+ 1

L1+ L2+

M1+M2+

M3,

where the operatorsLk andMk are defined by:

L0 = 2 2

y2+ (m y)

y, (4.4)

L1 =

2Y

Ni=1

i(y, z) 2

xiy, (4.5)

L2 =

t

+1

2

N

i,j=1

cij i(y, z)j(y, z) 2

xixj

+N

i=1

i(i

xi)

xi N

i=1

xi, (4.6)

M1 = Zg(z)N

i=1

i(y, z) 2

xiz, (4.7)

M2 = 12

g2(z)2

z2+c(z)

z,

M3 =

2g(z)Y Z2

yz,

Note that

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1L0 is the infinitesimal generator of the Ornstein Uhlenbeck process Y, L1 contains the mixed derivatives due to the correlation between X and Y, L2 is the differential operator corresponding to the unperturbed problem in (2.4), but evaluated at

the volatilitiesi(y, z), M1 contains the mixed derivatives due to the correlation between X and Z, M2 is the infinitesimal generator of the process Z,

M3contains the mixed derivatives due to the correlation between Y and Z.

We will next present the results of the singular and regular perturbation techniques framework introducedin [10] and adapted to credit risk to obtain an accurate characterization of the loss distribution in the regimewhere and are small. This will enable us to describe how the fluctuations in the volatility affect the lossdistribution and tranche prices.

4.1. Time-Scale Perturbations. We expand the survival probabilityu, defined in (4.2), in the smallparameters and as

u, u, =u0+

u1,0+

u0,1, (4.8)

whereu1,0 and u0,1 are the first corrections due to fast and slow volatility scales respectively.The leading order term u0 of the joint survival probability is obtained by solving the problem (2.4) with

the effective diffusion matrix

[d(z)]ij =dij(z) :=

ciji(y, z)j (y, z)(y) dy ,

with being the invariant distribution for the Y process. Since the processY evolves on the fast scale itsleading order effect is obtained by integration with respect to . The processZevolves on a relatively slowscale and at this level of approximation its effect corresponds to just evaluating this process at its currentfrozen level z . We introduce the effective operator

Le(d(z)) = t

+1

2

Ni,j=1

dij (z) 2

xixj+

Ni=1

i(i xi) xi

Ni=1

xi

,

then we haveDefinition 4.1. The leading order termu0 is the survival probability which solves

Le(d(z))u0 = 0, u0(T, x; z) = 1.As in the constant volatility case of Section 2.3, the solution is given by

u0(t, x; z) = Ac(T t)

Ni=1

Ai(T t)eBi(Tt)xi , (4.9)

withBi defined in (2.5), and

Ac(s) =e12

PNi=1

PNj=i=1(dij(z))

Rs0

Bi()Bj() d.

The last integral is given explicitly in (2.9).Next, we obtain u1,0, the correction to the survival probability due to the fast volatility factor Y. We

start by introducing the operator

A1,0 =L1L10 (L2 L2) ,

which will be given explicitly below and where the triangular brackets represent integration with respect tothe invariant distribution for the Y process.

Definition 4.2. The functionu1,0(t, x, z) solves the inhomogeneous problem

Le(d(z))u1,0=A1,0u0,u1,0(T, x; z) = 0.

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Thus, u1,0 solves a linear equation with the effective operatorLe(d(z)), but now the problem involves asource termA1,0u0, defined in terms of the leading order survival probability u0, and with a zero terminalcondition.

We consider next the correction u0,1 due to the slow volatility factor. In this case we introduce theoperator

A0,1 =

M1

,

and obtain:Definition 4.3. The functionu0,1(t, x, z) is the solution of the problem

Le(d(z))u0,1=A0,1u0, (4.10)u0,1(T, x; z) = 0,

which is again a source problem with respect to the operator Le(d(z)) and with a zero terminal condition.The PDE problems for the approximation terms given in Definitions 4.1, 4.2 and 4.3 can be motivated by

formal multiscale asymptotics. Similar calculations in the case of equity stochastic volatility models appearin [10], and the formal asymptotics are identical albeit with different definitions of the operatorsLk andMk.In the present case,L2 is associated with a multi-dimensional diffusion process, which does not present majordifficulties compared with the single-dimensional equity case in [10]. The unboundedness of the X(i) raisessome technical issues, which were addressed in the one-dimensional case in [2]. A precise accuracy result isgiven at the end of this section.

We next obtain an expression for u, and start by introducing the symmetric matrix (y, z) satisfying

L0i1,i2 = ci1,i2i1(y, z)i2(y, z) di1,i2(z) ,

and the coefficients

V3(z, i1, i2, i3) =

Y

2

i3

i1i2y

.

Using the definitions in (4.4) - (4.6) one then obtains that the scaled operator

A1,0 can be written

A1,0 = N

i1,i2,i3=1

V3(z, i1, i2, i3) 3

xi1xi2xi3.

We make the ansatz

u1,0(t, x; z) = D(T t)u0(t, x; z), which leads to the ODE for D:

D =N

i1,i2,i3=1

V3(z, i1, i2, i3)Bi1Bi2Bi3 , D(0) = 0,

using the expressions (2.6) and (2.8) for the survival probability in the constant volatility case. Therefore, wecompute the form for the correction due to fast volatility fluctuations

u1,0= N

i1,i2,i3=1V

3(z, i1, i2, i3) T0 Bi1(s)Bi2 (s)Bi3(s) ds

u0. (4.11)

Thus, the correction depends on the underlying model structure in a complicated way, but only the effectivemarket group parametersV3() are needed to compute the fast time scale correction u1,0.

Using next the definition (4.7) we find that the (scaled) operator

A0,1 in the source term of the u0,1problem (4.10) can be written

A0,1 =

M1=

Ni=1

V1(z, i) 2

zxi,

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where we introduced the coefficients

V1(z, i) =

g(z)Zi .

It then follows, again using the expressions (2.6) and (2.8), that

u0,1= 12N

i1=1V

1(z, i1)

Ni2=1

Ni3=1

d

dz (di1i2) T0 Bi

1(v) v0 Bi

1(s)Bi2 (s) dsdv

u0 . (4.12)

Note that the effective market parameters V3 andV1 depend on the underlying model in a complicated way

as explained above. However, this particular dependence will not be needed in applying the asymptotic theorysince these market group parameters rather than the full underlying model will be calibrated to market data.

We end this section with an accuracy result for our approximation (4.8).Theorem 4.1. For any fixedt < T, x IRN andy, z IR,u,(t, x, y , z) u0(t, x, z) + u1,0(t, x, z) + u0,1(t, x, z) =O(+),

whereu, is the solution of the original problem (4.3), andu0,

u1,0 and

u0,1 are given by (4.9), (4.11)and (4.12) respectively.

Sketch of Proof: One of the difficulties is that the potential (ixi) in (4.3) is unbounded from above sincethe processesX(i) are unbounded from below. As in [2] (in the one-dimensional case), the transformation

u,(t, x, y , z) = M,(t ,y,z)

Ni=1

eBi(Tt)xi

reduces to a Feynman-Kac equation for Mwith aboundedpotential in (y, z), bounded time-dependent coeffi-cients, and smooth terminal condition M,(T , y , z) = 1. The rest of the proof follows readily from the proofof accuracy given in [9, Chapter 5] with the fast factor, and generalized to fast and slow factors in [10].

4.2. Stochastic Volatility Effects in the Symmetric Case. We consider the simplified form for theasymptotic survival probabilities in the symmetric case considered in Section 4.2, with stochastic volatilties

(i)t

(Yt, Zt). One readily computes that in this case

S(T; (x, , x), y , z , n) = u,(0;(x, , x), y , z , n) u,(0;(x, , x), z , n) (4.13)=

1 +D(T; z, n) +D(T; z, n)

en[(z)(TB(T))+[1+(n1)X ]2(z)B2(T)/(4)+xB(T)]

where

2(z) =

(, z)2, (4.14)D(T; z, n) = v3(z)n

2(1 + (n 1)X )B(3)(T) , (4.15)D(T; z, n) = v1(z)n

2(1 + (n 1)X )B(3)(T) , (4.16)

B(3)(T) =

T0

B3(s) ds ,

B(3)(T) = T0

B(s)B(2)(s) ds

v1(z) =

2 g(z)Z (, z)

z

2(, z) ,

v3(z) =

Y

2

(, z) (, z)

y

,

with here being a solution to the Poisson equation in the y -variable:

L0 = 2(y, z)

2(, z) .10


11/20

Note that is computed as in (3.3), but evaluated with = (z):

(z) = [1 + (n 1)X ] 2(z)

22 .

Therefore, using the notation introduced in (3.6), we have

S(T; x,z,n) 1 +n3X v3(z)B(3)(T) +v1(z)B(3)(T) end1(T,x,z)+n2d2(T,z) , (4.17)with

d2(T , x , z) = d2(T , x , z) + (1 X )

v3(z)B(3)(T) +v1(z)B

(3)(T)

, (4.18)

whered1 andd2 are computed as in (3.7) and (3.8):

d1(T , x , z) = T+ (x )B(T) 12

(1 X )(z)2B(2)(T) ,

d2(T, z) =1

2X (z)

2B(2)(T) ,

B(2)(T) = T

0

B2(s) ds .

By the remarks following equation (3.6), we easily compute the loss distribution that follows from (4.17)in the caseX = 0. That is, we compute it by (3.9) with

pn(x) =

N

n

nj=0

nj

ex

(N+jn)(1)j (4.19)

=

N

n

(1 ex)ne(Nn)x , (4.20)

and d2 replaced by d2. In the general case with X= 0 we obtain the loss distribution by the generalizationof (3.9):

pn= IE

pn(d1+

2d2 Z) +

X (v3B(3) +v1B

(3))

p

n(d1+

2d2 Z)

, (4.21)

whereZis anN(0, 1) random variable. Note thatN

n=0

p

n(x) = d3

dx3

Nn=0

pn(x) = 0 ,

so that indeedN

n=0pn= 1. We remark however that outside of the domain of validity of the approximationwe may have pn < 0. Thus, when applying the approximation the vis must be chosen small enough so thatindeed the computedpns define a distribution. In the modeling above the vis are O(

,

) and are therefore

small. From the representation (4.21) we see that the effect of the stochastic volatility in the uncorrelatedcase with X = 0 is a modification of the hazard rate, to the order we consider. While the combined effect

of correlation and stochastic volatility is qualitatively different and gives a correction to the binomial shape.We can also observe that the effects of the slow and fast volatility scales are qualitatively similar, givingthe computed correction a canonical character, it gives the structure of the correction under a large class ofunderlying models. This will be further reinforced by our analysis below.

We next continue the numerical example introduced in Section 3.2. We choose the parameters as

= .03 , = .5 , = .02 , x= .03 ,

and we let the time to maturity T = 5 and the number of names N= 125. The short rate is chosen to befixed at 3% and the recovery is 40% as before. Here and below, when we show numerical examples they are

11


12/20

based on the asymptotic approximations of the type (4.21). Our analysis has shown that name correlationcan be generated in various ways. Either by directly correlating the innovations or Brownian motions drivingthe hazard rates of the names or alternatively by introducing time scale effects in the volatility. We remarkthough that the time evolution of the loss distribution depends somewhat on how the correlation is generated.In Figure 4.1, we illustrate the relative strong effect that combined name correlation and stochastic volatilityhas, here with X =.01 andv3= 3 104. We letv1 = 0, as the influence of this parameter is similar to thatofv

3. Note in particular the relative strong effect on the senior tranches and that in this case the shape of

the loss distribution is affected, however, the equity tranches are relatively less affected.

0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

0.1

0.12

LOSS DISTRIBUTION

CONSTANT PARAMETERS

STOCHASTIC VOLATILITY

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

2000

4000

6000

8000

10000

12000

TRANCHE PREMIA

CONSTANT PARAMETERS


0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510

2

100

102

104

106

CONSTANT PARAMETERS


Fig. 4.1. The left figure shows the loss distribution, pn, the dashed line with constant parameters, v3 = 0, and the solid linewith stochastic volatility, v3 = 3104. The right figure shows the tranche premia, the CDX tranche prices , against upperattachment point, t he diamond-dashed line for constant parameters and the solid line for stochastic volatility.

4.3. Name Correlation via Fluctuations in Hazard Rate Level. In the modeling above we as-sumed that the mean reversion level of the name hazard rates, the parameter, was constant. We nextexamine the effects of time variations in this parameter by generalizing the model in (4.1) as

dX(i)t =

(Y(2)t ) X(i)t

dt+(Y

(1)t ) dW

(i)t ,

for 1 iNwhere theW(j)s are correlated Brownian motions as in (2.3). Note that here we consider onlythe case with a symmetric model and fast scale time fluctuations in the parameters. The general case can beanalyzed with a similar approach. The fast processes are modeled by

dY(j)t =

1

(mj Y(j)t )dt+

j

2

dW(j,y)

t , for j {1, 2} .

and we assume the correlations

d

W(i), W(j,y)

t=Yjdt , for 1i N and j {1, 2} , (4.22)

which can be decomposed in terms of independent Brownian motions ( W(0)t , W(j,y)t ) as follows:

W(j,y)

t =

1 yj W(j,y)t + yj W(0)t ,yj =Yj/

X.

Thus, we assume that there is one short time scale associated with the market, the time scale characterizedby.

In (4.8) we now only have the correction due to the fast scale variation

u u =u0+

u1,0.

12


13/20

The functions u0 and u1,0 solves again the problems in Definition 1 and Definition 2 respectively, upon thereplacements

L02

j=1

2j

2

yj2+ (mj yj )

yj

+

y1y212

2

y1y2,

L12

j=12j Yj(y1)

Ni=1

2

xiyj ,

L2 t

+2(y1)

2

Ni,j=1

cij2

xixj+

Ni=1

((y2) xi) xi

Ni=1

xi

.

It then follows that the expression (4.13) for the asymptotic loss distribution becomes:

S(T; x, n) = u(0;(x, , x), y1, y2, n) u(0; x, n)= (1 +D(T; n) +D(T; n)) e

n[(TB(T))+[1+(n1)X ]2B2(T)/(4)+xB(T)] ,

with now

2 = 2()1 ,

= ()2 ,(z) = [1 + (n 1)X]

2

22,

wherei denotes integration with respect to the invariant distribution for Y(i), i {1, 2}. Moreover,D isdefined as in (4.15), but without the z dependence:

D(T; n) = v3n2(1 + (n 1)X )B(3)(T) ,

v3 =

Y1

2

() ()

y

1

,

L0,1 = 2(y)

2()1

,

with

L0,1 being the infinitesimal generator for Y1. The correction,D, due to fluctuations in the parameter

is given by:

D(T; n) = v2n2B(2)(T) ,

v2 =

2Y22

(y1)

2(y2)

y

1,2

,

L0,22 = (y) ()2 ,withL0,2 being the infinitesimal generator for Y(2) and1,2 denoting integration with respect to the jointinvariant distribution for Y(1) andY(2) .

Observe that the form of the joint survival probability is as in (4.13), which confirms the canonicalstructure of the correction to the survival probability due to multiscale parameter fluctuations. It also followsthat we can use the same procedure as the one described in Section 4.2 for computing the loss distributionand associated tranche prices. In the present case we can write

S(T; x, n) u(0; x, n)=

1 +c1n2B(2)(T) + (c2n

2 +c3n3)B(3)(T)

S0(T; x, n) ,

S0(T; x, n) = en[(TB(T))+[1+(n1)X ]2B2(T)/(4)+xB(T)] ,

with c2 and c3 calibration parameters of magnitudeO(). Note that the form of the correction in (4.13)is identical with respect to the form in n, but slightly different in the temporal dependence. We continue inthe next section by briefly discussing the case with a stochastic short rate and show that then we also getcorrections terms that areO(1) andO(n).

13


14/20

4.4. Short Rate Term Structure Effects . Consider the case were the short rate is stochastic, alsomodeled as a Vasicek process under the risk neutral measure:

dX(i)t =

(Y(2)) X(i)t

dt +(Y(1)) dW

(i)t ,

drt =

r(Y

(4)) rt

dt +r(Y(3)) dWrt .

We again assume a symmetric model with the time scales defined by 1/ and 1/ so that the fast scales areall modeled by:

dY(j)t =

1

(mj Y(j)t )dt+

j

2

dW(j,y)t

with the symmetric correlations of the same form as above:

d

W(i), W(j,y)

t=Yjdt , for 1 i N , 1 j 4 ,

d

Wr, W(j,y)

t=r,Yjdt , for 1j 4 ,

dW(i), W(j)t =Xdt , for i =j , dW

(i), Wrt = rdt , for 1 iN .In this case, the quantity of interest that we need to compute in order to price the CDO is:

q(r)(T; x, n) = IE

eRT0 (r(s)+X(1)s ++X(n)s ) ds |X(1)0 =x, , X(n)0 =x, r(0) =r0

the expectation under the risk neutral measure of the discounted joint survival probabilities. This expressionis of the same form as in (2.7). By generalization of the multiscale analysis we then easily find the asymptoticapproximation

q(r)(T; x, n)

1 + (c0+c1n+c2n2 +c3n

3)B(3)(T)

S(r)0 (T; x, n) ,

S(r)0 (T; x, n) = e

n[(TB(T))+[1+(n1)X ]2B2(T)/(4)+xB(T)]

e[r,(TB(T))+[1+nr ]

2rB

2(T)/(4)+r0B(T)] ,

with ci, i {0, , 3}calibration parameters of magnitudeO() and2r =2r ()3,

r,=r()4 [1 +nr] 2r

22.

Therefore, when we introduce a multiscale short rate we find that the calculation of the CDO prices is modifiedin two ways: i) the correction for the survival probabilities involve nowO(1) andO(n) terms as announcedabove, ii) the discounting factor is modified as:

erT e[r,(TB(T))+[1+nr ]2rB2(T)/(4)+r0B(T)] .We finish this section with a numerical example illustrating short scale fluctuation effects, shown in Figure4.2. The parameters are chosen as in the constant parameter case of Figure 4.1. In addition we choose r = .1,c0 =.05 and c1 =.0007 to illustrate how multiscale short rate fluctuations may affect the tranches. Notethat with these parameters the loss distribution is not much affected. However, term structure effects influencerelatively strongly the mezzanine tranches. This mechanism is somewhat complementary to the modificationseen in the previous examples.

5. Name Heterogeneity. We return now to a non-symmetric model and describe how we can effectivelycompute the CDO price in this situation. We consider the case with one volatility factor:

dX(i)t =(i X(i)t )dt+if(Yt) dW(i)t ,14


15/20

0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

0.1

0.12

LOSS DISTRIBUTION

CONSTANT PARAMETERS


0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

2000

4000

6000

8000

10000

12000

TRANCHE PREMIA

CONSTANT PARAMETERS


0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510

2

100

102

104

106

CONSTANT PARAMETERS


Fig. 4.2. Loss distribution and tranche prices without and with short rate stochastic volatility effects. The left figure showsthe loss distribution, pn, t he dashed line with constant parameters, c0 = 0, c1 = 0, and the solid line with stochastic volatility,c0 = .05, c1 = .0007. The right figure shows the tranche premia, the CDX tranche prices , against upper attachment point,the diamond-dashed line for constant parameters and the solid line for stochastic volatility.

for 1 iNand with vi adi constants. Moreover, the W(j)s are correlated Brownian motions as in (2.3)and

dYt=1

(m Yt)dt+

2

dW

(y)t ,

with again the symmetric correlations:

d

W(i), W(y)

t=Ydt , for 1 iN ,

d

W(i), W(j)

t=Xdt , for i=j .

Thus, we assume that the market is characterized by the two time constants 1 / and 1/, moreover, thatthe volatilities dependence on the fast factor is via the common term f(Y). Therefore, in terms of theirtime scale contents the names are symmetric. We shall see below that this means that we can compute theCDO prices effectively via conditioning on, and subsequent integration with respect to, one Gaussian randomvariable. In the case that there are several time constants, several s characterizing the market, the CDOcan be computed via conditioning with respect to several Gaussian random variables as we describe below.

The approximation for the joint survival probability can now be expressed by:

S(T; x, n) 1 +v3

(1 X ) n

i=1

i

ni=1

2i +X

ni=1

i

3B(3)(T)

Ac(T)n

i=1

Ai(T)eB(T)xi , (5.1)

where

Ai(T) = eiB(1)(s)+ 122i (1X)B(2)(T) ,

Ac(T) = e(Pn

i=1i)2(XB(2)(T)/2) ,

2i =2i f2() ,

B(k)(T) =

T0

Bk(s) ds .

15


16/20

In order to compute the CDO price, we need the loss distribution pn(T; x, N):

pn(T; x, N) = IP {(#names defaulted at time T) = n|X0 = xi} .

Consider first the case in which the names are independent:

S(T; x, n) =

ni=1

Si(T; xi) .

In this case we can compute the loss distribution via the recursive algorithm described in [14]. In thisprocedure the loss distribution forn names is easily computed from the loss distribution ofn 1 names dueto independence, and so on. The procedure isO(n2).

We now comment on how this procedure can be used to compute the loss distribution associated with thesurvival probabilities in (2.7). First, we discuss the case without stochastic volatility, but with correlation.Then we can write

S(T; x, n) = IE

ni=1

eZPn

i=1i

XB(2)(T)Ai(T)eB(T)xi

=IE n

i=1

qi(T; xi)

,

where the expectation is with respect to the standard Gaussian random variable Zand we assume X 0.The integration is therefore over cases corresponding to independent names, the situation in which the CDOprice can be computed effectively by the iterative algorithm in [14]. Note that, as above, we constrain theGaussian random variables so that 0qi1.

In Figure 5.1, we illustrate the effects of correlation in between the names and of hazard rate heterogeneity.We choose here X = .3, and we weight the first ten names intensities by a factor of five, we keep the nextfifty with with weight one, and the last 65 with weights 0 .2. The parameters are otherwise chosen as above.The result is shown by the solid line. The case without name correlation is shown by the dash-crossed line.The dashed line corresponds to replacing the heterogeneous hazard rate by its average. Note that this givesa very different loss distribution in this case with groupingof the hazard rate.

0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

LOSS DISTRIBUTION

HAZARD HETEROGENEITY

CORRELATION

MEAN PARAMETERS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

2000

4000

6000

8000

10000

TRANCHE PREMIA


CORRELATION

MEAN PARAMETERS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510

10

105

100

105


CORRELATION

MEAN PARAMETERS

Fig. 5.1. Loss distribution and tranche prices without and with correlation and heterogeneity effects. The left figure showsthe loss distribution, pn, the dashed line with mean parameters, the solid line with hazard heterogeneity and strong name-name

correlation, X = .3, and the diamond-dashed line the case of hazard heterogeneity and X = 0. The right figure shows thetranche premia, the CDX tranche prices , against upper attachment point, with the lines defined as in the left plot

Now consider the case with both correlation and stochastic volatility. In order to extend the calculation

16


17/20

of the loss distribution to this case we use expansions of the form:

ni=1

i

ni=1

2i

=

e3/8(

Pni=1

2i ) 1

e1/8(

Pni=1i) (5.2)

e1/4/2(Pn

i=1 i)2

e1/4(

Pni=1 i)

2 1

e1/8(

Pni=1i) e1/4/2(

Pni=1 i)

2

+ O(7/8)

ni=1

i

3

=

e3/8(

Pni=1i)

2 1

e1/8(

Pni=1i) (5.3)

e1/4/2(Pn

i=1 i)2

e1/4(

Pni=1 i)

2 1

e1/8(

Pni=1i) e1/4/2(

Pni=1 i)

2

+ O(7/8) .

If we substitute the approximations in (5.2) and (5.3) in (5.1) we find that S(T; x, n) can be expressed in terms

of a linear combination of exponentials, whose exponents are of the form ni=1c1,i,k (T)+c22,k(T) ( i)2 /2ni=1B(T)xiwithk = 1, , KforKthe number of different exponentials. It then follows that we can express

the loss distribution as a sumof terms in the form:

S(T; x, n) =K

k=1akIE

n

i=1ec1,i,k(T)+Zc2,k(T)iB(T)xi

, (5.4)

with c1 > 0 and c2 beingO(1) and havingO() corrections due to stochastic volatility. Note that we havethe normalization

Kk=1

ak = 1,

in view of the first factor in the right hand side of (5.2) and (5.3). We can therefore again obtain the lossdistribution via integration with respect tooneGaussian random variable over independent cases. This followsexplicitly since using (5.4) we can write

pn(T; x, N) = IP {(#names defaulted at time T) = n}

=IEsSn

isc

eRT0

X(i)t dtis

1 e

RT0

X(i)t dt

|X0 = x

=IE

sS

cn(s)is

eRT0

X(i)t dt |X0 = x

sS

cn(s)K

k=1

akIE

is

ec1,i,k(T)+Zc2,k(T)iB(T)xi

=

Kk=1

akIE

sS

cn(s)is

ec1,i,k(T)+Zc2,k(T)iB(T)xi

=

Kk=1

akIE{pn(T; Z, x, N)}

=

pn(T; Z, x, N) dK(Z) ,

with Sn being the collection of distinct subsets of size n of{1, , N}, S being all the distinct subsets orS = S1 SN and with sc being the complement set of s. Observe that pn again can be computedeffectively by the algorithm of the independent case as described in [14]. Moreover,K is a signedmeasurewith unit total mass. Indeed the loss distribution calculated by the above algorithm will therefore also haveunit total mass. We remark that the possibility that the distribution can go negative can be dealt with viathe framework set forth in [13].

We finally remark on the case with many time scales, that is with many different is. Note that we then

17


18/20

can write

log

Ac(T)

=X

2

ni,j=1

j i

T0

B(s; i)B(s; j ) ds

=X

2

n

i,j=1

j i T

ij B(T; i) B(T; j ) +B(T; i+j ) , , (5.5)

with B (t; ) = (1 et)/. Assume that there are at mostMdistinct time scales 1, , M. We denotebyAij the summands in (5.5). Then the matrix A = (Aij ) is symmetric non-negative definite and has rankat mostM so that in terms of the corresponding eigenvalue factorization we have

A=M

i=1

iv(i)

v(i)T

.

Using this fact we express

S(T; x, n) = IE

n

i=1

M

j=1

e2jZj

Pni=1v

(j)i Si(T; xi)

=IE

ni=1

Mj=1

e2jZjv

(j)i Si(T; xi)

,

where the expectation is with respect to the independent standard Gaussian random variables Zi. We remarkthat the spectrum A decays as{(i+j )1}, which typically has extremely fast decay.

We can decompose the terms in D similarly so that the CDO price in the case with many time scales,differentis, can be obtained via integration with respect to a set of Gaussian random variables both in thecase with and without stochastic volatility. In the case with stochastic volatility we integrate with respect toat mostM2 Gaussian random variables. This follows since we can write

i

2

j T0 B(s; i)B(s; j )

2

ds

N

i,j=1 =

Mi=1 iui(vi)

T

,ij k

T0

B(s; i)B(s; j )B(s; k) ds

Ni,j=1

=M2i=1

i,kvi(vi)T ,

and then use decompositions as in (5.2) and (5.3).The integration over the Gaussian random variables is again over cases corresponding to independent

names, the situation in which the CDO price can be computed effectively by the iterative algorithm in [14].We finish this section with a numerical example. In Figures 5.2, we show the case when the stochastic

volatility parameter v3 = .001 and i = .02 + .02 exp(i/120). Note how the heterogeneity affects the lossdistribution tail as shown in Figure 5.2. The parameters are chosen so that the loss distribution is almostzero before a small tail mode generated by implicit correlations. This gives a relatively large correction of themost senior tranche as shown in the the bottom right graph. We remark that if we replace i by its simpleaverage then the correction is relatively smaller.

6. Conclusions. The results show that much progress can be made with multiscale stochastic volatilityasymptotic approximations, even when built around a simple Vasicek-based model for stochastic intensities,by providing additional correlation through potentially large and rapid volatility excursions. Generalizingthis approach for other popular intensity models such as CIR or expOU, while keeping the computationaltractability, is an important challenge, which is work in progress. However, the Vasicek analysis providesinsights into the relative roles of the model parameters, and their effects on loss distributions and CDOtranche spreads. For example, the name-name correlation affects strongly the equity tranche. Uncertainty

18


19/20

0 10 20 30 40 50 600.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

LOSS DISTRIBUTION

CONSTANT PARAMETERS


MEAN PARAMETERS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.5

1

1.5

2x 10

4 TRANCHE PREMIA

CONSTANT PARAMETERS


MEAN PARAMETERS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510

2

100

102

104

106

CONSTANT PARAMETERS


MEAN PARAMETERS

Fig. 5.2. Stochastic volatility and name heterogeneity effects. The left figure shows the loss distribution, pn, the dashedline with mean parameters, the solid line with stochastic and level volatility heterogeneity, and the diamond-dashed line is

with volatility level heterogeneity only. The right figure shows the tranche premia, the CDX tranche prices , against upper

attachment point, with the lines defined as in the left plot

in the short rate affects relatively strongest the mezzanine tranches, while stochastic volatility in hazardgives a strong correction to senior tranches. Calibration to real data is an essential next step, which shouldbe facilitated by explicit approximations computed here. In the context of calibration we mention that thecalibrated parameters also provide a linkage to other credit derivatives, like option on tranches, via theircorrected prices.

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[2] P. Cotton, J.-P. Fouque, G. Papanicolaou, and K. R. Sircar. Stochastic volatility corrections for interest rate models.Mathematical Finance

, 14(2):173200, 2004.[3] X. Ding, K. Giesecke, and P.I. Tomecek. Time-changed birth processes and multi-name credit derivatives.Preprint, availableat www.defaultrisk.com, 2009.

[4] D. Duffie and N. Garleanu. Risk and valuation of collateralized debt obligations. Financial Analysts Journal, 57(1):4159,2001.

[5] D. Duffie and K. Singleton. Credit Risk. Princeton University Press, 2003.

[6] A. Eckner. Computational techniques for basic affine models of portfolio credit risk. Journal of Computational Finance,2007. To appear.

[7] A. Elizalde. Credit risk models IV: Understanding and pricing CDOs. www.abelelizalde.com, 2005.

[8] E. Errais, K. Giesecke, and L. Goldberg. Pricing credit from the top down with affine point processes. Working paper,Stanford University, February 2007.

[9] J.-P. Fouque, G. Papanicolaou, and R. Sircar. Derivatives in Financial Markets with Stochastic Volatility. CambridgeUniversity Press, 2000.

[10] J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. Solna. Multiscale stochastic volatility asymptotics. SIAM J. MultiscaleModeling & Simulation, 2(1):2242, 2003.

[11] J.-P. Fouque, R. Sircar, and K. Slna. Stochastic volatility effects on defaultable bonds. Applied Mathematical Finance,13(3):215244, 2006.

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[13] J.-P. Fouque and X. Zhou. Perturbed Gaussian copula. In T. Fomby, J.-P. Fouque, and K. Solna, editors, Econometricsand Risk Management, volume 22. Emerald Group, 2008.

[14] J. Hull and A. White. Valuation of a CDO and an n-th to default CDS without Monte Carlo simulation. Journal ofDerivatives, 12(2):823, 2004.

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[17] A. Lopatin and T. Misirpashaev. Two-dimensional Markovian model for dynamics of aggregate credit loss. Technical report,Numerix, 2007.

[18] A. Mortensen. Semi-analytical valuation of basket credit derivatives in intensity-based models. Journal of Derivatives,13(4):826, 2006.

[19] E. Papageorgiou and R. Sircar. Multiscale intensity models and name grouping for valuation of multi-name credit derivatives.Applied Mathematical Finance, 2009. To appear.

[20] P. Schonbucher. Credit Derivatives Pricing Models. Wiley, 2003.

[21] P. Schonbucher. Portfolio losses and the term structure of loss transition rates: A new methodology for the pricing ofportfolio credit derivatives. Technical report, ETH Zurich, 2006.

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