Multi-Name Credit Derivatives - unibocconi.itdidattica.unibocconi.it/mypage/dwload.php?nomefile=... · Large Homogeneous Portfolio (LHP) 5 Implied Correlation Market Quotation Standard

Multi-Name Credit Derivatives

Paola Mosconi

Banca IMI

Bocconi University, 15/04/2019

Paola Mosconi 20541 – Lecture 8-9 1 / 80

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 Introduction2 Dependence

Credit CorrelationDependence in Reduced Form ModelsCopula Function

3 CDOs: Stylized Facts4 CDO Pricing

Monte Carlo PricingOne Factor Gaussian CopulaLarge Homogeneous Portfolio (LHP)

5 Implied CorrelationMarket Quotation StandardCompound CorrelationBase CorrelationOpen Problems

6 AppendixLHP

7 Selected ReferencesPaola Mosconi 20541 – Lecture 8-9 4 / 80

Introduction

Outline






6 AppendixLHP


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

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


Introduction

Short History of Securitization: before 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

...


Introduction

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”


Introduction

Short History of Securitization: the Capital Market Union(2015-2019)

On 2 December 2015, the Permanent Representatives Committee (Coreper) of the EUCouncil of Ministers approved a negotiating stance on proposals aimed at facilitating thedevelopment of a securitization market in Europe.

“These proposals aim to relaunch the securitisation market, by promoting simple,transparent and standardised (STS) securitisations. The objective is to con-

tribute to the financing of the economy and hence to the creation of jobs and

growth”

Pierre Gramegna, minister of finance of Luxembourg and president of the Council.

A framework for securitization is the first major building block of the EU’s plan, launchedduring 2015, to develop a fully functioning capital markets union by the end of 2019.

Developing a securitization market will help create new investment possibilities and providean additional source of finance, particularly for SMEs and start-ups.


Introduction

Short History of Securitization: Securitization Package(2015-2017)

“Securitisation can allow diversification of funding sources and a broader distribution of risk by

allowing banks to transfer the risk of some exposures to other institutions or long-term investors,such as insurance companies and asset managers. This allows banks to free the capital they set

aside to cover for risks of those exposures, allowing them to generate new lending to households

and SMEs. STS securitisations will also provide new investment opportunities for institutionalinvestors such as pension funds and insurance companies.”

European Commission - Press release, May 30, 2017

September 2015: proposal by the EU Commission as part of the Capital MarketsUnion (CMU) action plan. According to the Commission’s estimates, securitizationwould generate up to EUR 150bn in additional funding for the economy

October 2015: EU Parliament appraisal

May 30, 2017: EU reaches agreement on the Securitization Package


Introduction

Short History of Securitization: a Snapshot

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


Introduction

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)


Introduction

The Formula that Killed Wall Street (Wired 2009) I

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


Introduction

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 crisis.

“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


Introduction

...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)


Introduction

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






6 AppendixLHP


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 a goodmeasure of dependence

It is not possible to directly estimate historical correlation (not enough data onjoint defaults). Default correlation can be deduced implicitly from asset correlation.However, the resulting default correlation is much lower than the correspondingasset correlation in all cases.

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 forcorrelation products in the market

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



Multiple Name Framework II

Dependenceacross:

Pros Cons

Intensities

• Tractability• Correlation can be estimated

historically from time seriesof credit spreads

• Default of one name does notaffect intensity of the othernames

• Unrealistically low dependenceacrossdefaults

• Deterministic intensitiespossible

• Sufficient levelsof dependenceacrossdefault times(throughthe copula function)

• No natural source for historically estimating the copula

• It ignores credit spread volatilities (large) and correlations

Intensities +

• Credit spread volatility• Sufficient levelsof dependence

acrossdefault times(throughthe copula function)

• No natural source for historically estimating the copula

• Default of one name affectsthe intensity of other names(untractable)


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 ]

Stdev(X 3)Stdev(X )=

3√15

=

√

3

5< 1!

Correlation works well only for Gaussian variables!

In credit derivatives with intensity models, dependence must be introduced in the expo-nential components of the Poisson processes for different names.

This is usually done by means of Copula 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.

Sklar’s TheoremLet H be an n-dimensional distribution function with margins F1, . . . ,Fn. Then, there existsan n-copula C (i.e. a joint distribution function on n uniforms) 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 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.

Idea

To model dependence across default times τ1, . . . , τn, by introducing dependencedirectly among standard uniforms!

Probability Integral Transform

Given a random variable X , its transformation through its cumulative distributionfunction FX (X ) produces a uniform random variable U, i.e. FX (X ) = U ∼ U(0, 1).Proof:

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). ✷

Exponential Random Variable

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)



Gaussian Copula Simulation

Uniform simulation from Multivariate Gaussian distribution

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 ≤ x

3) = 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)

✷


CDOs: Stylized Facts

Outline






6 AppendixLHP



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 LosstrB (t)− A LosstrA (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,B0,T (0) fixed at time T0 = 0 is paid at times T1, . . . ,Tb = T from the

protection 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,B0,T (0)

∫ Ti

Ti−1

OutSttrA,B(t)dt

≈ RA,B0,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,B0,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






6 AppendixLHP


CDO Pricing

CDO Pricing

CDOs look like contracts selling (or buying) insurance on portions of the loss ofa portfolio.

The valuation problem is trying to determine the fair price of this insurance.

Pricing (marking to market) a tranche: taking expectations of the futuretranche losses under the risk neutral measure.


CDO Pricing

Tranching I

Tranching is a non-linear operation, which requires the knowledge of the wholeloss distribution of the pool of names. The expectation will depend on all momentsof the loss and not just the expected loss.

Figure: Source: Brigo (2010)


CDO Pricing

Tranching II

The complete description of the portfolio loss is obtained in two alternative ways,through:

the knowledge of the whole distribution (e.g. Monte Carlo simulation)

Single name marginal distributions+ dependence structure= copula

Dependency is commonly called“correlation”

(abuse of language).

The dependence of the tranche on correlation is crucial. The market assumes aGaussian Copula connecting the defaults of the n names belonging to the portfolio.


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 28.

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

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

The One Factor Copula model provides a way of modeling the joint defaults of ndifferent names and it allows to derive their joint default probability.

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

Idea

The 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.



Joint Default Probability: Derivation I

The One Factor Gaussian Copula allows to derive the joint default probability of n names,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 represents the (unconditional) joint default probability of n names and entailsthe calculation of a n-dimensional integral.



Joint Default Probability: Derivation 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!



Joint Default Probability: Derivation 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)



Joint Default Probability: Results

(Unconditional) Joint Default Probability

Substituting the single name probabilities (10) into eq. (9), the Gaussian CopulaC(u1, . . . , un), which represents the unconditional joint default probability of n names,in the One Factor framework, can be expressed as a one-dimensional integral:

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



Joint Default Probability: 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 builds on the One Factor GaussianCopula and allows to derive closed form expressions for:

the (expected) portfolio loss, Loss

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)

[Large Pool Model (JP Morgan), McGinty and Ahluwalia (2004)]


CDO Pricing Large Homogeneous Portfolio (LHP)

LHP: Tranched Losses

Expected Equity Tranche Loss:

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

XN2

(

−D ,N−1(p),−√ρ)

(14)

where p = 1−eΓ(T ), N2(., ., r) is the standard normal cumulative distribution functionwith correlation r and

D :=1√ρ

[

N−1(p)−

√

1− ρN−1

(

X

LGD

)]

(Expected) Tranched Loss with attachment points [A,B] is retrieved through eq.(14) and:

E[LosstrA,B ] ≡ Losstr,∞A,B (ρ) =1

B − A

[

B Losstr,∞0,B (ρ)− A Losstr,∞0,A (ρ)]

(15)

(See the Appendix for the proofs.)


Implied Correlation

Outline






6 AppendixLHP


Implied Correlation Market Quotation Standard

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,B0,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.

A standard (liquid) index composed of n = 125 names (e.g. DJ-iTraxx Index) is associateda copula, parameterized by a matrix of 125× 124/2 = 7750 pairwise correlation values.

Implied Correlation

However, when looking at a given tranche:

7750 parameters −→ 1 parameter

The unique correlation parameter is reverse-engineered to reproduce the price of theliquid tranche under consideration. This is the implied correlation and once obtained itis used to value related products.

Two types of correlation can be implied from the market: compound correlation and basecorrelation (the market has chosen this one as a quotation standard).



Bootstrapping Implied Correlation from the Market

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 (15) and (14), i.e.:

E(LosstrA,B) ≡ Losstr,∞A,B (ρ) =1

B − A

[

B Losstr,∞0,B (ρ)− A Losstr,∞0,A (ρ)]

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

XN2

(

−D ,N−1(p),−√ρ)

:= fGC0X (ρ)


Implied Correlation Compound Correlation

Compound Correlation: Definition

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 fGC0B (ρAB)− A f

GC0A (ρAB)

)

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

C − B

(

C fGC0C (ρBC )− B f

GC0B (ρBC )

)

. . .

(16)

Typically, the compound correlation structure presents a smile.



Compound Correlation: Smile I

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



Compound Correlation: Smile II

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: this tranche is impacted by every default and a large correlationwould mean a low probability of a single default (lower than historical correlation)



Compound Correlation: Tranches

Two tranches on the same pool (same maturity) yield different values of compound cor-relation. This implies that the two tranches are priced with two models having differentand inconsistent loss distributions.




Compound Correlation: Non Invertibility I

For some values of the market premia, it is not guaranteed that all the bootstrappingequations (16) yield a solution. In that case, the compound correlation 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).



Compound Correlation: Multiple Solutions

Figure: Upper charts: DJi-Traxx 5 (left charts) and 10 (right charts) year CompoundCorrelation uniqueness. Lower charts: CDX. Blue dots highlight the dates where morethan one compound correlation could reprice the tranche market spread. Source: Brigo et

al (2010)


Implied Correlation Base Correlation

Base Correlation: Definition

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 fGC0B (ρ0B)− A f

GC0A (ρ0A)

)

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

C − B

(

C fGC0C (ρ0C )− B f

GC0B (ρ0B)

)

. . .

Typically, the base correlation structure presents a skew.



Base Correlation: Skew

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



Base Correlation: Tranches

The market prefers base correlation as a quotation standard. Base correlation decomposese.g. the 3%−6% tranche in terms of the 0%−3% and the 0%−6% equity tranches, usingtwo different correlations (and hence distributions) for those. Therefore, base correlation,though allowing for an easier interpolation, is inconsistent even at the single tranchelevel.




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).


Implied Correlation Open Problems

Open Problems

The One Factor Gaussian Copula model and implied base correlation havebecome the market standard for valuing CDOs and similar instruments.

However, such model presents the following issues:

inconsistency across the capital structure

inconsistency across maturities

difficult pricing of bespoke1 portfolios and tranches.

1Portfolios constructed specifically for one structured credit derivative, for which there is noliquid information on implied correlation.



Inconsistency Across the Capital Structure

The phenomenon of correlation skew means that, in order to match the observedmarket prices, the correlation must depend on the position in the capital structureof the particular tranche being priced.

Inconsistency across the capital structure means that there exist different modelsassociated to different tranches (compound correlation) or even to the sametranche (base correlation).



Inconsistency Across Maturities

The expected [0%−3%] tranche loss calibrated to the 3y, 5y and 10y [0%−3%] trancheson April, 26th 2006 (in the One-Factor Homogeneous Finite Pool Gaussian Copula model)do not overlap. Source: Brigo et al (2010)

When valuing the same expected tranche loss E0[Losstr0%,3%(Ti )] for Ti <3y, we are using

three different numbers depending on the tranche maturity even though the pool ofunderlying credit references is the same!


Appendix

Outline






6 AppendixLHP


Appendix LHP

LHP: Proof of Eq. (14)

In this Appendix we show how to derive the closed form expression of the ExpectedEquity Tranched Loss given by eq. (14), based on the following assumptions:

One Factor Gaussian Copula

Homogeneity of notional, recovery and correlation

Large number n of names (Large Pool Model – JP Morgan)


Appendix LHP

LHP: Homogeneity Assumption 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 (17)


Appendix LHP

LHP: Homogeneity Assumption II

3 The unconditional probability of having k defaults among the n obligors is ob-tained from eq. (17), by integrating on the common risk factor:

Q(k defaults) =

∫ +∞

−∞

Q(k defaults |Y = y)ϕ(y)dy (18)

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)


Appendix LHP

LHP: Large Pool Model Assumption 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

)

.


Appendix LHP

LHP: Large Pool Model Assumption 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) (19)


Appendix LHP

LHP: Expected Equity Tranched Loss Eq. (14)

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 (20)

The integral can be computed analytically yielding eq. (14).

✷


Selected References

Outline






6 AppendixLHP


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, hiip://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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