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(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
Paola Mosconi Lecture 4 2 / 68
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
Paola Mosconi Lecture 4 3 / 68
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
Paola Mosconi Lecture 4 4 / 68
Introduction
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
Paola Mosconi Lecture 4 5 / 68
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 ...
Paola Mosconi Lecture 4 6 / 68
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
Paola Mosconi Lecture 4 7 / 68
Introduction Securitization
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...
Paola Mosconi Lecture 4 8 / 68
Introduction Securitization
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”
Paola Mosconi Lecture 4 9 / 68
Introduction Securitization
Short History of Securitization: a Snapshot
Figure: European securitization outstanding (left) and issuance (right). Source: BOE andECB (2014).
Paola Mosconi Lecture 4 10 / 68
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)
Paola Mosconi Lecture 4 11 / 68
Introduction Of Models and Mathematicians
The Formula that Killed Wall Street (Wired 2009) I
Figure: Source: Recipe for Disaster: The Formula that Killed Wall Street. WiredMagazine (2009)
Paola Mosconi Lecture 4 12 / 68
Introduction Of Models and Mathematicians
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.
Paola Mosconi Lecture 4 13 / 68
Introduction Of Models and Mathematicians
...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)
Paola Mosconi Lecture 4 14 / 68
Introduction Of Models and Mathematicians
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?
Paola Mosconi Lecture 4 15 / 68
Dependence
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
Paola Mosconi Lecture 4 16 / 68
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)
Paola Mosconi Lecture 4 17 / 68
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).
Paola Mosconi Lecture 4 18 / 68
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)
Paola Mosconi Lecture 4 19 / 68
Dependence Dependence in Reduced Form Models
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
Paola Mosconi Lecture 4 20 / 68
Dependence Dependence in Reduced Form Models
Multiple Name Framework II
Paola Mosconi Lecture 4 21 / 68
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.
Paola Mosconi Lecture 4 22 / 68
Dependence Copula Function
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))
Paola Mosconi Lecture 4 23 / 68
Dependence Copula Function
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.
Paola Mosconi Lecture 4 24 / 68
Dependence Copula Function
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!
Paola Mosconi Lecture 4 25 / 68
Dependence Copula Function
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.
Paola Mosconi Lecture 4 26 / 68
Dependence Copula Function
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.
Paola Mosconi Lecture 4 27 / 68
CDOs: Stylized Facts
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
Paola Mosconi Lecture 4 28 / 68
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)
Paola Mosconi Lecture 4 29 / 68
CDOs: Stylized Facts Stylized Facts
(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.
Paola Mosconi Lecture 4 30 / 68
CDOs: Stylized Facts Stylized Facts
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
Paola Mosconi Lecture 4 31 / 68
CDOs: Stylized Facts Stylized Facts
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)
Paola Mosconi Lecture 4 32 / 68
CDOs: Stylized Facts Stylized Facts
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)
Paola Mosconi Lecture 4 33 / 68
CDOs: Stylized Facts Stylized Facts
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.
Paola Mosconi Lecture 4 34 / 68
CDOs: Stylized Facts Stylized Facts
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)]
Paola Mosconi Lecture 4 35 / 68
CDOs: Stylized Facts Stylized Facts
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)
Paola Mosconi Lecture 4 36 / 68
CDO Pricing
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
Paola Mosconi Lecture 4 37 / 68
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.
Paola Mosconi Lecture 4 38 / 68
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 .
Paola Mosconi Lecture 4 39 / 68
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
Paola Mosconi Lecture 4 40 / 68
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.
Paola Mosconi Lecture 4 41 / 68
CDO Pricing One Factor Gaussian Copula
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!
Paola Mosconi Lecture 4 42 / 68
CDO Pricing One Factor Gaussian Copula
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!
Paola Mosconi Lecture 4 43 / 68
CDO Pricing One Factor Gaussian Copula
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)
Paola Mosconi Lecture 4 44 / 68
CDO Pricing One Factor Gaussian Copula
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
Paola Mosconi Lecture 4 45 / 68
CDO Pricing One Factor Gaussian Copula
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.
Paola Mosconi Lecture 4 46 / 68
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)
Paola Mosconi Lecture 4 47 / 68
CDO Pricing Large Homogeneous Portfolio (LHP)
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)
Paola Mosconi Lecture 4 48 / 68
CDO Pricing Large Homogeneous Portfolio (LHP)
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)
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CDO Pricing Large Homogeneous Portfolio (LHP)
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
)
.
Paola Mosconi Lecture 4 50 / 68
CDO Pricing Large Homogeneous Portfolio (LHP)
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)
Paola Mosconi Lecture 4 51 / 68
CDO Pricing Large Homogeneous Portfolio (LHP)
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)
Paola Mosconi Lecture 4 52 / 68
CDO Pricing Large Homogeneous Portfolio (LHP)
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)
Paola Mosconi Lecture 4 53 / 68
Market Quotation Standard
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
Paola Mosconi Lecture 4 54 / 68
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,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
Paola Mosconi Lecture 4 55 / 68
Market Quotation Standard
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
Paola Mosconi Lecture 4 56 / 68
Market Quotation Standard
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)
Paola Mosconi Lecture 4 57 / 68
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.
Paola Mosconi Lecture 4 58 / 68
Market Quotation Standard Compound Correlation
Compound Correlation II
Figure: Example of compound correlation for the DJ-iTraxx. Source: Brigo and Mercurio(2006)
Paola Mosconi Lecture 4 59 / 68
Market Quotation Standard Compound Correlation
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)
Paola Mosconi Lecture 4 60 / 68
Market Quotation Standard Compound 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)
Paola Mosconi Lecture 4 61 / 68
Market Quotation Standard Compound Correlation
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).
Paola Mosconi Lecture 4 62 / 68
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.
Paola Mosconi Lecture 4 63 / 68
Market Quotation Standard Base Correlation
Base Correlation II
Figure: Example of base correlation for the DJ-iTraxx. Source: Brigo and Mercurio (2006)
Paola Mosconi Lecture 4 64 / 68
Market Quotation Standard Base Correlation
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).
Paola Mosconi Lecture 4 65 / 68
Selected References
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
Paola Mosconi Lecture 4 66 / 68
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)
Paola Mosconi Lecture 4 67 / 68
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
Paola Mosconi Lecture 4 68 / 68
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