(basic) multi-name credit derivativesdidattica.unibocconi.it/mypage/dwload.php?nomefile=... ·...

(Basic) Multi-Name Credit Derivatives

Paola Mosconi

Banca IMI

Bocconi University, 16/03/2015

Paola Mosconi Lecture 4 1 / 68

Disclaimer

The opinion expressed here are solely those of the author and do not represent in

any way those of her employers


Main References

Brigo, D. and Mercurio, F. Interest Rate Models – Theory and Practice. With

Smile, Inflation and Credit, Springer (2006)

Brigo, D. (2009), Essex Lecture Notes, Unit 5


Outline

1 IntroductionSecuritizationOf Models and Mathematicians

2 DependenceCredit CorrelationDependence in Reduced Form ModelsCopula Function

3 CDOs: Stylized FactsStylized Facts

4 CDO PricingMonte Carlo PricingOne Factor Gaussian CopulaLarge Homogeneous Portfolio (LHP)

5 Market Quotation StandardCompound CorrelationBase Correlation

6 Selected References


Introduction

Outline








Introduction

Introduction

Multi-name credit derivatives are characterized by payoffs which depend onmore than one underlying reference entities.

A list of them includes:

First to default;

k-th to default, last to default;

CDS indices;

CDO tranches;

CDO squared tranches;

Leveraged Super Senior (LSS) CDO tranches ...


Introduction Securitization

Asset Backed Securities and Securitization

An asset-backed security (ABS) is a security whose income payments and hencevalue is derived from and collateralized by a specified pool of underlying assets.

Securitization is the process of pooling together assets that would typically beunable to be sold individually. It allows to sell them to general investors in the formof tranches and to diversify the risk of investing.

Collateralized Debt Obligations CDOs are a particular kind of ABS, backed bya diversified pool of debt obligations, e.g.

bonds (CBOs)

loans (CLOs)

CDS (synthetic CDOs)

other structured products



Short History of Securitization: up to the Crisis(1997-2008)

In the period 1998 to 2007, the asset backed securities market increased expo-nentially both in volume and diversity.

As the crisis unfolded in 2007/2008, such market came under substantial criticismas some securitized products played a major role in the financial difficulties forvarious reasons (see Prime Collateralized Securities):

badly underwritten products

opaqueness of structures

over–leveraged issuances...



Short History of Securitization: the Role of ECB (2014)

“Despite the low issuance and the modest take-up by investors, most European structuredfinance products performed well throughout the financial crisis from a credit standpoint,

with low realized default rates.

According to Standard & Poor’s, the cumulative default rate on European consumer–

related securitizations, between the start of the financial downturn in July 2007 and Q3

2013 has been only 0.05%.”

Discussion Paper by the European Central Bank and the Bank of England (July, 2014).

According to Mario Draghi (August 2014), asset-backed securities should be“simple, transparent and real”, where:

“simple means readable”

“transparent means that you can actually go through and price them well”

“real means that they are not going to be a sausage full of derivatives”



Short History of Securitization: a Snapshot

Figure: European securitization outstanding (left) and issuance (right). Source: BOE andECB (2014).


Introduction Of Models and Mathematicians

Of Models and Mathematicians

Recipe for disaster: the formula that killed Wall Street

(Wired Magazine, 2009)

Of couples and copula: the formula that felled Wall Street

(Financial Times, 2009)

Wall Street’s math wizards forgot a few variables (New York Times, 2012)

Misplaced reliance on sophisticated math (The Turner Review, 2009)

vs

Did a mathematical formula really blow up Wall Street? (Embrechts, 2009)

Don’t blame the quants (Shreve, 2008)

Crash Sonata in D Major (Szego, 2009)

Credit Models and the Crisis, or: How I learned to stop worrying and love the

CDOs (Brigo et al, 2010)



The Formula that Killed Wall Street (Wired 2009) I

Figure: Source: Recipe for Disaster: The Formula that Killed Wall Street. WiredMagazine (2009)



The Formula that Killed Wall Street (Wired 2009) II

Undoubtedly, Li’s formula has severe flows of which mathematicians and quantswere well aware even before the crisis1.

“The most dangerous part”Mr. Li himself says of the model, “is when peoplebelieve everything coming out of it. [...] Very few people understand the essenceof the model. [...] It’s not the perfect model.”But, he adds: “There’s not a betterone yet.”

To Stanford’s Mr. Duffie, “The question is, has the market adopted the modelwholesale in a way that has overreached its appropriate use? I think it has.”

How a Formula Ignited Market That Burned Some Big Investors.The Wall Street Journal, September 2005

1We will analyze in detail the limitations of the Gaussian Copula model in this Lecture and inLecture 5.



...But, Aren’t We Missing Something?

1 Originate-to-distribute and trust in originators

2 Low interest rates

3 Increased risk appetite, excessive leverage

4 Real estate investments, and bubble

5 Equity extraction from residential properties

6 Managerial misbehavior, opaque investments and budgets

7 Systemically risky dimension

8 Adjustable rate mortgages

9 Regulatory errors

10 Herding and panic

Source: Szego, (2009)



The Heart of the Matter

CDOs tranches are difficult objects to price: tranching is a non-linear operation,which requires the knowledge of the whole loss distribution of the pool of names.

Two ways:

1 The whole distribution is simulated (Monte Carlo)

2 Single name marginal distributions+ dependence structure= copula

Where and how can we introduce dependence?


Dependence

Outline








Dependence Credit Correlation

Credit Correlation I

In literature different definitions of default correlation have been introduced(see e.g. Li (2000), Hull and White (2000), Frey et al (2001)...).

Default Correlation

Given a time horizon T (typically one year) the default correlation between twonames can be expressed in terms of:

their marginal default probabilities q1 = E[1{τ1<T}] and q2 = E[1{τ2<T}]

their joint default probability q12 = E[1{τ1<T}1{τ2<T}]

as follows:

ρ12 =q12 − q1q2

√

q1(1− q1)q2(1− q2)


Dependence Credit Correlation

Credit Correlation II

The above definition suffers from two problems:

The indicators being not elliptically distributed in general, correlation is not agood measure of dependence

It is not possible to directly estimate historical correlation (not enough data onjoint defaults). As a proxy, asset correlation has been used instead but it largelyunderestimates credit correlation.

Asset Correlation Default Correlation

10% 0.94%20% 2.41%30% 4.61%

Table: Asset correlation (estimated) vs. default correlation. Source: Frey et al (2001).


Dependence Dependence in Reduced Form Models

Single Name Framework

In the framework of reduced form (intensity) models, the default time τ is the first jumpof a Poisson process with intensity λ(t):

P(τ ∈ [t, t + dt)|τ ≥ t,market info up to t) = λ(t)dt ,

where λ is the intensity or hazard rate and represents an instantaneous credit spread.

Single name framework

Given that the cumulated intensity is distributed as an exponential random variable:

Λ(τ ) =

∫ τ

0

λ(s)ds = ξ ∼ exponential

the default time turns out to be:τ = Λ−1(ξ)



Multiple Name Framework I

Given multiple names, the dependence between default times

τ1 = Λ−11 (ξ1) , τ2 = Λ−1

2 (ξ2) , . . . , τn = Λ−1n (ξn)

can be introduced in three ways:

1 put dependence in (stochastic) intensities of the different names and keep the ξ ofdifferent names independent

2 put dependence among the ξ of different names and keep the intensities (eitherstochastic or deterministic) independent.

(This is the approach currently used for correlation products in the market.)

3 put dependence both among the ξ and the intensities of different names



Multiple Name Framework II


Dependence Copula Function

Introduction to Copula Function I

Linear correlation is not enough to express the dependence between two random variables.

Example: Correlation between X ∼ N (0, 1) and Y = X 3

X and Y have the same information content and should have maximum dependence, but

E[X 4]− E[X 3]E[X ]

Std(X 3)X=

3√15

=3

5< 1!

Correlation works well only for Gaussian variables!

In credit derivatives with intensity models dependence must be introduced the exponentialcomponents of Poisson processes for different names. This is usually done by means ofCopula functions.



Introduction to Copula Function II

DefinitionLet (U1, . . . ,Un) be a random vector with uniform margins and joint distributionC(u1, . . . , un). C(u1, . . . , un) is the copula of the random vector.

Skar’s TheoremLet H be an n-dimensional distribution function with margins F1, . . . ,Fn. Then, there existsan n-copula C such that for all x ∈ Rn,

H(x1 . . . , xn) = C(F1(x1), . . . , Fn(xn)) .

Any joint distribution function can be used to define a copula:

C(u1, . . . , un) = H(F−11 (u1), . . . ,F

−1n (un))



Gaussian Copula

The Gaussian (or Normal) copula plays a central role in the modeling of credit dependence.

Gaussian Copula

It is obtained by using a multivariate normal distribution NnR with standard Gaussian

margins and correlation matrix R as multivariate distribution H:

CN (R)(u1, . . . , un) = NnR(N

−1(u1), . . . ,N−1(un)) (1)

where N−1 is the inverse of the standard normal cumulative distribution.Notice that this formula entails a n-dimensional integral!

Properties

No closed form expression, except for n = 2.

For n names, the correlation matrix R has n(n − 1)/2 free parameters.

No tail dependence (upper/lower).

C(u, v) = C(v , u) i.e. exchangeable copula.



Gaussian Copula Simulation: Uniform R.V.

Given a random variable X , its transformation through its cumulative distributionfunction FX (X ) produces a uniform random variable U:

FU(u) = P(U ≤ u) = P(FX (X ) ≤ u) = P(X ≤ F−1X (u)) = FX (F

−1X (u)) = u

The property FU(u) = u is characteristic of a standard uniform distribution U(0, 1).

The random variable ξ = Λ(τ ) ∼ exp(1) can be expressed in terms of a uniformrandom variable U ∼ U(0, 1) as follows:

Fξ(ξ) = 1− e−ξ = U =⇒ ξ = − ln(1− U)

Goal

Model dependence across default times τ1, . . . , τn, by introducing dependence directlyamong standard uniforms!



Gaussian Copula Simulation

Uniform simulation

1 Find the Cholesky decomposition A of the correlation matrix R, such that R = AAT

2 Simulate n independent random variables Z1, . . . ,Zn from N (0, 1)

3 Set X = AZ

4 Set ui = N(xi ), i = 1, . . . , n

5 (u1, . . . , un) ∼ CN (R)

Default times simulation

1 Simulate (or calibrate to CDS market quotes) individual intensities λi (in thesimplest case, λ are independent and deterministic)

2 Simulate n uniforms according to the above copula procedure

3 Set different names default times according to:

τ1 = Λ−11 (− ln(1− U1)), . . . , τn = Λ−1

n (− ln(1− Un))

The dependency among the τ is loaded into a copula function on the U.



Example Reloaded

Dependence between X ∼ N (0, 1) and Y = X 3

X and Y have the same information content. By using the copula function we show thatthey have maximum dependence, i.e.

P(U1 ≤ u1,U2 ≤ u2) = min(u1, u2)

where U1 = FX (x) and U2 = FY (y).

Consider that:

U2 = FY (y) = FX 3(x3) = P(X 3 ≤ x3) = P(X ≤ x) = FX (x) = U1

Therefore:

P(U1 ≤ u1,U2 ≤ u2) = P(U1 ≤ u1,U1 ≤ u2) = P(U1 ≤ min(u1, u2)) = min(u1, u2)

q.e.d.


CDOs: Stylized Facts

Outline








CDOs: Stylized Facts Stylized Facts

CDOs: The Portfolio Loss Distribution

CDOs are instruments related to the loss distribution of a pool of names.

The portfolio loss distribution is not symmetric but shows the following features:

skewed bell when the correlation islow

monotonically decreasing whencorrelation increases

U–shaped when correlation is closeto 1 (either all names survive or de-fault)

Figure: Portfolio loss distribution.Source: Lehman (2003)



(Synthetic) CDOs

Synthetic CDOs with maturity T are obtained by pooling together CDSs of differentnames (up to n) with the same maturity and tranching the total loss:

Loss(T ) =n∑

i=1

LGDi 1{τi≤T} =n∑

i=1

(1− Reci)1{τi≤T} (2)

along two attachment points A and B, with A < B. The protection seller pays theprotection buyer the cumulated tranched loss that exceeds A and does not exceed B.

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Figure: Synthetic CDO.



CDOs: Tranched Loss

The (percentage) tranched loss between attachment points A and B at time t is givenby:

LosstrA,B(t) :=1

B − A

[

(Loss(t)− A)1{A<Loss(t)≤B} + (B − A)1{Loss(t)>B}

]

=1

B − A

[

(Loss(t)− A)1{A<Loss(t)} − (Loss(t)− B)1{Loss(t)>B}

]

=1

B − A

[

(Loss(t)− A)+ − (Loss(t)− B)+]

(3)

or, in a compact notation:

LosstrA,B(t) =

0 if Loss(t) < ALoss(t)−A

B−Aif A < Loss(t) ≤ B

1 if Loss(t) > B



CDOs: Equity Tranche

The tranche [0,X ] absorbs the first losses up to X% of the total portfolio loss and it iscalled equity tranche.

The corresponding tranched loss, according to (3) with A = 0 and B = X , is given by:

LosstrX (t) := Losstr0,X (t) =1

X

[

Loss(t)− (Loss(t)− X )+]

It is useful to express any tranche [A,B] in terms of equity tranches, as follows:

LosstrA,B(t) =1

B − A

[

B LosstrA (t)− A LosstrB (t)]

(4)



CDOs: Cash Flows

The contract consists of two legs: the default leg and the premium leg.Schematically the cash flows of a CDOs contract can be summarized as follows:

Protection → Prot. dLosstrA,B(t) at all t ∈ (T0,Tb ] → ProtectionSeller ← rate R at Ta+1, . . . ,Tb on the outstanding notional ← Buyer

where:

dLosstrA,B(t) is the tranched loss increment at time t

the outstanding notional is given by the survived positive (re-scaled) notional atthe relevant payment time:

OutSttrA,B(t) = 1− LosstrA,B(t)



CDOs: Premium Leg

A premium rate RA,B

0,T (0) fixed at time T0 = 0 is paid at times T1, . . . ,Tb = T from theprotection buyer to the protection seller. The rate is paid on the survived positive notionalat the relevant payment time. This notional decreases of the same amount as the tranchedloss increases, taking into account the recovery.

Discounted premium leg payoff

ΠPremLA,B (0) =b∑

i=1

D(0,Ti )RA,B

0,T (0)

∫ Ti

Ti−1

OutSttrA,B(t)dt

≈ RA,B

0,T (0)

b∑

i=1

D(0,Ti )αi [1− LosstrA,B(Ti )]

where αi = Ti − Ti−1.



CDOs: Default (Protection) Leg

Once enough names have defaulted and the loss has reached A, the counts start. Eachtime the loss increases, the corresponding loss change re-scaled by the tranche thicknessB −A (i.e. dLosstrA,B(t)) is paid to the protection buyer, until maturity arrives or until thetotal pool loss exceeds B, in which case the payments stop.

Discounted default leg payoff

ΠProtLA,B (0) =

∫ T

0

D(0, t)dLosstrA,B(t) ≈b∑

i=1

D(0,Ti )[LosstrA,B(Ti )− LosstrA,B(Ti−1)]



CDOs: Price of the Tranche

Assuming deterministic interest rates, the price of the tranche for the protection buyeris:

TrancheA,B0,T (0) = E[ΠProtLA,B (0)]− E[ΠPremLA,B (0)]

=b∑

i=1

P(0,Ti )E[(LosstrA,B(Ti )− LosstrA,B(Ti−1))]

− RA,B

0,T (0)

b∑

i=1

P(0,Ti )αi [1− E(LosstrA,B(Ti ))]

(5)

where E[.] denotes the expectation under the risk neutral measure.

The premium rate that makes the contract fair at inception is therefore given by:

RA,B0,T (0) =

∑b

i=1 P(0,Ti )[E[LosstrA,B(Ti )]− E[LosstrA,B(Ti−1)]]

∑b

i=1 P(0,Ti )αi [1− E(LosstrA,B(Ti ))](6)


CDO Pricing

Outline








CDO Pricing

CDO Pricing

Goal

The problem of pricing a CDO tranche or calculating its premium rate consists incalculating the corresponding expected tranched loss.

In the following we present two approaches:

the Monte Carlo method, which is more computational intensive,

a semi-analytical approach based on two approximations: the GaussianCopula approach and the Large Homogeneous Portfolio approximation.


CDO Pricing Monte Carlo Pricing

Monte Carlo Pricing

Pricing a CDO by means of Monte Carlo goes through the following steps:

1 Calibrate individual name intensities λi from CDS market quotes.

2 Estimate the correlation matrix R.

3 Consider N scenarios and for each scenario j , simulate the dependent defaulttimes through the copula approach outlined in slide Gaussian Copula Simulation.

4 At any given time t and for any scenario j , compute the loss and tranched lossgiven respectively by eq. (2) and eq. (3).

5 Average across all scenarios in order to find the expected tranched lossE[LosstrA,B(t)] at the desired date t or set of dates Ti .


CDO Pricing Monte Carlo Pricing

Monte Carlo Pricing: Limits and Alternative Methods

Monte Carlo pricing is conceptually straightforward, but has two main limitations:

1 An accurate estimate requires a large number of simulations and considering thatCDOs are composed of hundreds of names the process can be very timeconsuming.

2 Estimation of the correlation matrix for n names involves n(n-1)/2 estimates ofpairwise correlations.

Semi-Analytical Methods

In order to overcome the limitations of the Monte Carlo approach, alternative methodshave been proposed, which rely on the semi-analytical computation of the portfolioloss. One popular method combines:

the One Factor Gaussian Copula approach to calculate the joint default probability

the Large Homogeneous Portfolio (LHP) approach to calculate the portfolio loss


CDO Pricing One Factor Gaussian Copula

One Factor Gaussian Copula: Introduction

A one factor copula model is a way of modeling the joint defaults of n differentnames.

The structure for this model was suggested by Vasicek (1987) and it was firstimplemented by Li (2000) and Gregory and Laurent (2005).

Idea

One Factor Gaussian Copula reduces the dimensionality of the problem,increasing analytical tractability. For this reason, it has become a standard whenpricing CDOs and CDS Index tranches.



One Factor Gaussian Copula I

The One Factor Gaussian Copula approach goes through the following steps:

1 We consider n names and start from their default times, to which the copula will beapplied:

τ1 = Λ−11 (− ln(1− U1)), . . . , τn = Λ−1

n (− ln(1− Un))

2 We rewrite the definition of copula given by eq. (1) under the risk neutral measureQ:

C(u1, . . . , un) = NnR(N

−1(u1), . . . ,N−1(un))

= Q(X1 < N−1(u1), . . . ,Xn < N

−1(un))(7)

This entails the calculation of a n-dimensional integral!



One Factor Gaussian Copula II

3 Inspired by the works of Merton (1974) and Vasicek (1987), we set Ui = N(Xi ),where the standard Gaussian variables Xi are expressed in terms of:

a systematic common factor Y ∼ N (0, 1)

a idiosyncratic term, specific to each name, ǫi ∼ N (0, 1) and i.i.d.

Xi =√ρi Y +

√

1− ρi ǫi (8)

such that corr(Xi ,Xj ) =√ρiρj

Here, a first simplification occurs: the original number of free correlationparameters is reduced from n(n − 1)/2 to n!



One Factor Gaussian Copula III

4 Applying the law of iterated expectations to eq. (7), by conditioning on Y , weobtain:

C(u1, . . . , un) = E

[

Q(X1 < N−1(u1), . . . ,Xn < N

−1(un)|Y )]

(9)

5 Conditional on Y = y , the variables Xi are independent and the joint probabilityQ(.|Y ) can be written as the product of the following single name probabilities:

Q(Xi < N−1(ui)|Y = y) = Q(

√ρi Y +

√

1− ρi ǫi < N−1(ui)|Y = y)

= N

(

N−1(ui )−√ρi y√1− ρi

)

(10)



One Factor Gaussian Copula: Results

Substituting the single name probabilities (10) into eq. (9), the Gaussian CopulaC(u1, . . . , un), in the One Factor framework, can be expressed as one-dimensionalintegral:

C(u1, . . . , un) =

∫

[

n∏

i=1

N

(

N−1(ui)−√ρi y√1− ρi

)]

ϕ(y)dy (11)

where ϕ(y) is the standard Gaussian probability density.

Dimensionality Reduction

Original Problem One Factor Gaussian Copula

integral dimension n 1free correlation parameters n(n − 1)/2 n



One Factor Gaussian Copula: Deterministic Intensity

Under the assumption of deterministic intensities:

Q(τi < T |Y = y) = Q(Γi (τi) < Γi (T )|Y = y) = Q(ξi < Γi (T )|Y = y)

= Q(Ui < 1− e−Γi (T )|Y = y) = N

(

N−1(1− e−Γi (T ))−√ρi y√1− ρi

)

(12)

the (unconditional) joint default probability of n names becomes:

Q(τ1 < T , . . . , τn < T ) =

∫

[

n∏

i=1

N

(

N−1(1− e−Γi (T ))−√ρi y√1− ρi

)]

ϕ(y)dy (13)

Remark The market further reduces the dimensionality of the problem by introducing a

single value of correlation ρi = ρ for quotation reasons.


CDO Pricing Large Homogeneous Portfolio (LHP)

LHP: Goal and Assumptions

Goal

The Large Homogeneous Portfolio (LHP) approach allows to derive closed formexpressions for the (expected) portfolio loss, Loss, and the (expected) tranched loss,LosstrA,B .

Assumptions:

1 Gaussian Copula

2 homogeneity of the characteristics of names underlying the credit portfolio:

Notionali = Notional, Reci = Rec, Γi = Γ, ρi = ρ

3 large number n of names (above 100)



LHP: Homogeneity I

Homogeneity in recovery rates, default probabilities and correlations implies that:

1 the probability of a single default in the portfolio, conditional on the systematicfactor Y , as given by eq. (12) in the Gaussian Copula framework, is independent ofthe defaulting name i and therefore unique:

Q(τi < T |Y = y) = N

(

N−1(p)−√ρ y√1− ρ

)

where p = 1− e−Γ(T ) is the unconditional probability of default.

2 Exploiting independence of single names, conditionally on Y , and the commonvalue of the default probability across names, the conditional probability of havingk defaults among the n obligors is:

Q(k defaults|Y = y) =

(

n

k

)

Q(τ < T |Y = y)k [1−Q(τ < T |Y = y)]n−k (14)



LHP: Homogeneity II

3 The unconditional probability of having k defaults among the n obligors is ob-tained from eq. (14), by integrating on the common risk factor:

Q(k defaults) =

∫ +∞

−∞

Q(k defaults|Y = y)ϕ(y)dy (15)

In general, this integral has to be computed numerically. This is the reason why thiskind of approach is called semi-analytical.

4 In the homogeneous portfolio framework, under the assumption of constant recoveryrate Reci = Rec, the probability of having a portfolio loss

K = k · (1− Rec)

caused by the default of k names is equal to the probability of having k defaults:

Q(Loss = K) = Q(k defaults)



LHP: Large Pool Model (JP Morgan) I

McGinty and Ahluwalia (2004) have exploited the third assumption of the LHP approach,i.e. the infinite size of the portfolio.

We introduce the default rate (DR) of the pool, at a given time T , as the fraction ofdefaulted names w.r.t. the total pool of names. Conditional on the systematic factor Y itis given by:

DRnT (Y ) =

1

n

n∑

i=1

1{τi≤T |Y}

Conditional on Y , defaults are i .i .d . variables with mean given by the conditional proba-bility of default:

p(Y ; ρ) := E[1{τi≤T |Y=y}] = Q{τi≤T |Y=y} = N

(

N−1(p)−√ρ y√1− ρ

)

.



LHP: Large Pool Model (JP Morgan) II

Law of Large Numbers

By applying the Law of Large Numbers, the default rate DR, tends to:

DRnT (Y ) −−−→

n→∞p(Y ; ρ)

and the conditional percentage loss turns out to be:

Loss∞T (Y ; ρ) = (1− Rec) p(Y ; ρ) (16)



LHP: Tranched Loss I

GOAL: To derive a closed form expression for expected tranched losses

Consider an equity tranche with attachment points [0,X ].

1 Conditional on Y , using the large pool results, the expected tranched losscoincides with the tranched loss:

E[Losstr,∞0,X (Y ; ρ)] = Losstr,∞0,X (Y ; ρ) :=1

Xmin(Loss∞T (Y ; ρ),X )

2 The unconditional (expected) tranched loss is obtained by integrating over thecommon risk factor Y :

E[Losstr,∞0,X (ρ)] = Losstr,∞0,X (ρ) =

∫

1

Xmin(Loss∞T (Y = y ; ρ),X )ϕ(y)dy (17)



LHP: Tranched Loss II

Expected Equity Tranche Loss

The integral (17) admits a closed form solution, yielding the expected Equity Trancheloss:

Losstr,∞0,X (ρ) = N(D) +LGD

XN2

(

−D ,N−1(p),−√ρ)

(18)

where p = 1− eΓ(T ),

D :=1√ρ

[

N−1(p)−

√

1− ρN−1

(

X

LGD

)]

and N2(., ., r) is the standard normal cumulative distribution function with correlation r .

(Expected) tranched losses associated to generic tranches with attachment points[A,B] are retrieved through eq.s (18) and:

Losstr,∞A,B (ρ) =1

B − A

[

B Losstr,∞0,A (ρ)− A Losstr,∞0,B (ρ)]

(19)


Market Quotation Standard

Outline









Market Quotes

Markets quote CDO tranches only for standardized pools of CDS on different names.

The most liquid indices are the DJi-TRAXX, involving 125 European names and theDJCDX, involving 125 US names.

Premium rates RA,B

0,T (0) are quoted for the maturities

T = 3y , 5y , 7y , 10y

and standard attachment points

[0%, 3%] , [3%, 6%] , [6%, 9%] , [9%, 12%] , [12%, 22%] for DJi-TRAXX

and[0%, 3%] , [3%, 7%] , [7%, 10%] , [10%, 15%] , [15%, 30%] for DJCDX



Implied Correlation

From premium rate quotes, it is market practice to derive the default correlation, whichgoes by the name of implied correlation.

Usually implied correlation is retrieved by assuming:

a Gaussian Copula model (standard or One Factor), and deterministic spreads,for the pricing

a unique correlation parameter, which does not diversify across country, sector etc.

Implied Correlation

Two types of correlation are adopted by the market:

1 compound correlation

2 base correlation



Bootstrapping Implied Correlation

The bootstrapping procedure for a given maturity T and tranche [A,B] goes through thefollowing steps:

1 The equation used to bootstrap is given by (6), that we recall here:

RAB,mkt =

∑b

i=1 Pmkt(0,Ti )[E[Loss

trA,B(Ti )]− E[LosstrA,B(Ti−1)]]

∑b

i=1 Pmkt(0,Ti )αi [1− E(LosstrA,B(Ti ))]

2 under the Gaussian Copula assumption (e.g. in the LHP approximation) theexpected tranche loss is given by eq.s (19) and (18), i.e.:

E(LosstrA,B) = Losstr,∞A,B =1

B − A

[

B Losstr,∞0,A (ρ)− A Losstr,∞0,B (ρ)]

E(Losstr0,X ) = Losstr,∞0,X (ρ) = N(D) +LGD

XN2

(

−D ,N−1(p),−√ρ)

:= fGC0X (ρ)


Market Quotation Standard Compound Correlation

Compound Correlation I

Compound correlation is based on the assumption that each tranche [A,B] ischaracterized by a unique value of correlation ρAB .

Therefore, we solve recursively eq. (6), where the expected losses are given by:

[0,A] : E(Losstr0,A) = fGC0A (ρ0A)

[A,B] : E(LosstrA,B) =1

B − A

(

B fGC0A (ρAB)− A f

GC0B (ρAB)

)

[B,C ] : E(LosstrB,c) =1

C − B

(

C fGC0B (ρBC )− B f

GC0C (ρBC )

)

. . .

(20)

Typically, the compound correlation structure presents a smile.



Compound Correlation II

Figure: Example of compound correlation for the DJ-iTraxx. Source: Brigo and Mercurio(2006)



Compound Correlation III

The smile behavior of the compound correlation can be explained as follows:

Senior Tranches: high attachment points are reached when many defaults occur, i.e.when default correlation is large (similar to historical correlation)

Mezzanine Tranches: spreads are low given the high demand for these tranches (thisimplies low correlation)

Equity Tranche: large spreads are obtained from low correlation. This tranche isimpacted by every default and a large correlation would mean a low probability of asingle default (lower than historical correlation)



Compound Correlation: Non Invertibility I

However, for some values of the market premia, it is not guaranteed that all thebootstrapping equations (20) yield a solution. In that case, the compound correla-tion is non-invertible.

Figure: DJ-iTraxx 10 year compound correlation invertibility. Tranche Market spread(solid line) versus theoretical tranche spread obtained varying the compound correlationbetween 0 and 1 (dotted black line). Source: Brigo et al (2010)



Compound Correlation: Non Invertibility II

Figure: Compound correlation invertibility indicator (1=invertible, 0=not invertible) forthe DJ-iTraxx and CDX tranches. Source: Torresetti et al (2006).


Market Quotation Standard Base Correlation

Base Correlation I

Base correlation is based on the assumption that each equity tranche [0,X ] ischaracterized by a unique value of correlation ρ0X , implying that a tranche [A,B]depends on two values of base correlation.

Therefore, we solve recursively eq. (6), where the expected losses are given by:

[0,A] : E(Losstr0,A) = fGC0A (ρ0A)

[A,B] : E(LosstrA,B) =1

B − A

(

B fGC0A (ρ0A)− A f

GC0B (ρ0B)

)

[B,C ] : E(LosstrB,c) =1

C − B

(

C fGC0B (ρ0B)− B f

GC0C (ρ0C )

)

. . .

Typically, the base correlation structure presents a skew.



Base Correlation II

Figure: Example of base correlation for the DJ-iTraxx. Source: Brigo and Mercurio (2006)



Compound Correlation vs Base Correlation

If the Gaussian Copula assumptions were consistent with market tranche prices, therewould be a unique Gaussian Copula model (i.e. unique correlation) consistent with themarket and no distinction between compound correlation and base correlation.

Compound Correlation:

More consistent at the level ofsingle tranche (one single copulamodel).

Depends on pairs of attachmentpoints.

Cannot be easily interpolatedand/or extrapolated.

Unable to price non standard (socalled bespoke) tranches.

May not exist.

Base Correlation:

Inconsistent at the level of singletranche (different parts of the samepayoff with different models).

Depends on a single attachmentpoint.

Easy to interpolate/extrapolate.

Able to price non standard tranches.

Can be always retrieved, but mayyield negative expected tranchedlosses (very steep skew).


Selected References

Outline








Selected References

Selected References I

Bank of England and European Central Bank (2014), The Case of a BetterFunctioning Securitisation Market in the European Union, Discussion Paper

Brigo, D., Pallavicini, A., and Torresetti, R. (2010). Credit Models and the Crisis, or:How I learned to stop worrying and love the CDOs. Credit Models and the Crisis: A

journey into CDOs, Copulas, Correlations and Dynamic Models, Wiley, Chichester.

Frey, R., McNeil, A.J., and Nyfeler, M.A. (2001): Modeling Dependent Defaults:Asset Correlations Are Not Enough!.http://www.risklab.ch/ftp/papers/FreyMcNeilNyfeler.pdf

Laurent, J.P., and Gregory, J., (2005). Basket Default Swaps, CDO’s and FactorCopulas. Journal of Risk, Vol. 7, No. 4, 103-122

Hull, J., and White, A. (2000). Valuing Credit Default Swaps II: Modeling DefaultCorrelations, Journal of Derivatives, Vol. 8, No. 3

The Lehman Brothers Guide to Exotic Credit Derivatives (2003)


Selected References

Selected References II

Li, D. X., (2000), On Default Correlation: A Copula Approach, Journal of FixedIncome, 9

Merton, R. (1974), On the pricing of corporate debt: The risk structure of interestrates. J. of Finance 29, 449-470

McGinty, L., Ahluwalia, R., (2004). A Model for Base Correlation Calculation, JPMtechnical document

Prime Collateralised Securities, http://pcsmarket.org/

Szego, G. (2010). Crash 08: a regulatory debacle to be mended, Special Paper 189,LSE Financial Markets Group Paper Series

Torresetti, R., Brigo, D., and Pallavicini, A. (2006). Implied correlation in CDOtranches: a Paradigm to be handled with care. Available on ssrn

Vasicek, O. (1987). Probability of loss on a loan portfolio. Working Paper, KMVCorporation


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