Synthetic CDO Pricing Using the Student t Factor Model with … · 2014-05-15 · FACULTY OF...

FACULTY OF COMMERCE & ECONOMICS - ACTUARIAL STUDIES

Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 1/28

Synthetic CDO Pricing Using the Student tFactor Model with Random Recovery

UNSW Actuarial Studies Research Symposium 2006

University of New South Wales

Tom Hoedemakers — Yuri Goegebeur — Jurgen Tistaert

http://www.actuarial.unsw.edu.au



Overview

1. Introduction — Motivation

2. Setting and Notations

3. Modelling the Joint Default Time Distribution with Copulas

4. Random Recovery and Portfolio Distribution

5. Large Portfolio Approximation

6. Numerical Illustration

7. Conclusion and Possible Extensions




Single-Name CDS Cash Flows




What is a CDO?

A CDO (collateralized debt obligation) is a portfolio of debt securities that is splitinto tranches of debt subordination. Each tranche protects the tranches senior to itfrom losses on the underlying portfolio.

Typical tranches aren equity tranche (no subordination)

n mezzanine tranche

n senior tranche

n super senior tranche

Individual tranches can be rated by a rating agency.




CDO: Protection Buyers and Sellers

test




CDO: Credit Market Size

05

1015

1997 1999 2001 2003 2005

Global Credit Derivatives Outstanding

(In trillions of U.S. dollars)

test

050

100

150

200

250

300

1995 1997 1999 2001 2003 2005

Global Issuance of Collateralized Debt Obligations:

Cash Versus Synthetic (In billions of U.S. dollars)

CashSynthetic

050

100

150

200

250

1995 1997 1999 2001 2003 2005

Global Issuance of Collateralized Debt Obligations

by Underlying Assets (in billions of U.S. dollars)

Asset-backet securities and other structured productsNoninvestment-grade bonds and leveraged loansInvestment-grade bondsBank balance sheets assetsOther assets

020

4060

Tranche Notional Value Versus Economic Risk Transfer

(as a percent of the reference pool)

NotionalRisk Transfer

3%-7% 7%-10%

Mezzanine tranches

10%-15%15%-30%

Senior tranches

30%-100%

Super-senior tranche

Corporate bonds and loans

Credit derivatives

0% - 3%

Equity tranche




Literature

n Benchmark model: one factor Gaussian modelu Li (2000)u Used by all major investments banks to communicate quotes

n The oponentsu Within the “copula family”

n Student t (O’Kane & Schloegl, Lindskog & McNeil),Clayton (Schönbucher),double t (Hull & White), multifactor Gaussian, Marshall-Olkin (Giesecke,Lindskog & McNeil), random factor loadings (Andersen & Sidenius)

u Intensity modelsn Affine Jump Diffusion (Hutt), Gamma (Joshi), Stochastic Networks (Davis,

Backhaus & Frey, Giesecke),u Structural models




Setting and Notations

n Notationu i = 1, . . . , n creditsu T1, . . . , Tn default timesu Ni nominal of credit i

u Ri recovery rateu I[0,T ](Ti) default indicatoru Li = Ni(1 − Ri) loss given default

n Semi-explicit pricing for CDO tranchesu Default payments are based on the accumulated losses on the pool of credits:

L(T ) =

n∑

i=1

LiI[0,T ](Ti)

u Tranche premiums only involves call options on the accumulated losses:

E[(L(T ) − K)+]




Setting and Notations

n Default occurs whenever a stochastic variable Xi (or a stochastic process indynamic models) lies below a critical threshold Ci at the end of time period [0, T ].

n In the Merton model default occurs when the value of the assets of a firm fallsbelow the value of the firms’ liabilities.

n In order to apply these models at portfolio level we require a multivariate versionof a firm-value model.

n In the factor copula model the critical variable Xi is interpreted as the latentdefault time of company i.

n In a CDO the cash flows are functions of the whole random vectorX = (X1, . . . , Xn). To evaluate a CDO, all we need is today’s (risk neutral) jointdistribution of the Xi’s:

P (X1 < C1, . . . , Xn < Cn)




Copulas

n A copula captures the nonparametric, scale-invariant, distribution-free natureof the association between random variables.

n Theorem 1 Sklar (1959) Let X′ = (X1, . . . , Xn) be a random vector with joint

distribution function FX and marginal distribution functions FXi, i = 1, . . . , n.

Then there exists a copula C such that for all x ∈ Rn

FX(x) = C(FX1(x1), . . . , FXn

(xn)). (1)

• If FX1, . . . , FXn

are all continuous then C is unique.• Given a copula C and marginal distribution functions FX1

, . . . , FXn, the function

FX as defined by (1) is a joint distribution function with margins FX1, . . . , FXn

.

n The critical variables Xi adopt a Gaussian/student t copula:

CΦR(u1, u2, . . . , un; R) = ΦR

(Φ−1(u1), Φ

−1(u2), . . . , Φ−1(un)

)

Ct(ν,R)(u1, u2, . . . , un; R) = tν,R

(t−1ν (u1), t

−1ν (u2), . . . , t

−1ν (un)

)




Gaussian vs. Student t Copula

1000 samples of Gaussian and student t copula with Kendall’s τ = 0.5




Default Time Distribution

n To reduce the high dimensionality of the modelling problem a set of commonfactors is chosen (assumed to drive the default dependency between firms)

n Assume that the variable Yi depends on a single common factor Z:

Yi = aiZ +√

1 − ‖ai‖2εi,

where Z, ε1, . . . , εn, are independent N(0,1) random variables.

n The critical variable

XiD=

Yi for Gaussian factor model√

νW Yi for student t factor model

with W ∼ χ2ν independent of Y1, . . . , Yn

n If we didn’t choose a factor structure for the Xi, we would have to estimate all the12n(n − 1) elements of the covariance matrix of the critical variables.




Default Time Distribution

n The latent default times are related to the original default times in the followingway

Ti ≤ t ⇔ FTi(Ti) ≤ FTi

(t) ⇔ Ui ≤ FTi(t)

⇔ Xi ≤

Φ−1(FTi(t)) for Gaussian factor model

t−1ν (FTi

(t)) for student t factor model

n The model is calibrated to observable market prices of credit default swaps, i.e.the default thresholds are chosen so that they produce risk neutral defaultprobabilities implied by quoted credit default swap spreads:

Ci =

Φ−1(FTi(t)) for Gaussian factor model

t−1ν (FTi

(t)) for student t factor model




Factor Models

n The ith issuer defaults if Xi ≤ Ci or

εi ≤

Ci−aiZ√1−‖ai‖2

for Gaussian factor model√

Wν

Ci−aiZ√1−‖ai‖2

for student t factor model

n The probability that the ith issuer defaults

P (Xi ≤ Ci|Z = z) = Φ

(

Ci − aiz√

1 − ‖ai‖2

)

for Gaussian factor model

P (Xi ≤ Ci|Z = z, W = w) = Φ

(√wν Ci − aiz

√

1 − ‖ai‖2

)

for student t factor model




Random Recovery

n The joint default time distribution describes the joint default behavior of thedebtors underlying the CDO structure and hence completely determines the CDOcash flows.

n At default only a fraction of the nominal can be recovered. The recovery rates areassumed to be random and follow the cumulative Gaussian recovery modelproposed by Andersen and Sidenius (2004):

Ri = Φ(µi + biZ + ξi),

where ξi ∼ N(0, σ2ξi

), i = 1, . . . , n. The error terms ξ1, . . . , ξn are assumed to beindependent from each other and also independent from Z, W and ε1, . . . , εn.

n The interdependence between default of the ith obligor and recovery on the jth

obligor is controlled by ai · bj .




Recovery Rate Density Functions with σ2

ξ = 0.25

recovery rate

dens

ity

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

b=0.5b=sqrt(.5)b=sqrt(.75)b=sqrt(.9)

(a)

recovery rate

dens

ity

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4


(b)

recovery rate

dens

ity

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5


(c)




Portfolio Loss Distribution L(T )

• The conditional portfolio loss distribution:

P (L(T ) ≤ `|Z = z, W = w) =(

FL̃1|z,w ∗ · · · ∗ FL̃n|z,w

)

(`), 0 ≤ ` ≤ N,

where N =∑n

i=1 Ni,

FL̃i|z,w(`) = P (L̃i ≤ `|Z = z, W = w)

= 1 − P (Ti ≤ T |Z = z, W = w)(1 − P (Li ≤ `|Z = z, W = w)),

and

P (Ti ≤ T |Z = z, W = w) = Φ

(√wν t−1

ν (FTi(T )) − aiz

√

1 − ‖ai‖2

)

P (Li ≤ `|Z = z, W = w) = Φ

µi + biZ − Φ−1

(

1 − `Ni

)

σξi




Portfolio Loss Distribution L(T ): recursion

• Let Ki denote the loss given default of obligor i expressed in loss units u

For ` = 1, . . . , `maxi = bNi/uc:

P (Ki = 0|Z = z, W = w) = P (Li ≤ 0|Z = z, W = w) = 0

P (Ki = `|Z = z, W = w) = P (Li ≤ `u|Z = z, W = w)

−P (Li ≤ (` − 1)u|Z = z, W = w)

• Let K(T ) denote the portfolio loss over [0, T ] expressed in loss units u and let P (i)

denote the distribution of K(T ) for the first i obligors. Then

P (i)(K(T ) = `|Z = z, W = w)

=

min(`maxi ,`)∑

k=0

P (i−1)(K(T ) = ` − k|Z = z, W = w)P (K̃i = k|Z = z, W = w)

• The recursion starts from the boundary case of an empty portfolio for whichP (0)(K(T ) = `|Z = z, W = w) = δ`,0




Portfolio Loss Distribution L(T ): FFT

• Characteristic function of K(T ):

E(

eitK(T ))

= E

(n∏

i=1

eitKiI[0,T ](Ti)

)

= E

n∏

i=1

E(

eitKiI[0,T ](Ti)|Z, W)

︸︷︷︸

∗

with

∗ =

`maxi∑

l=0

(eit`P (Ti ≤ T |Z = z, W = w) + P (Ti > T |Z = z, W = w)

)

× P (Ki = `|Z = z, W = w)

= 1 − P (Ti ≤ T |Z = z, W = w)

1 −`max

i∑

l=0

eit`P (Ki = `|Z = z, W = w)

• Apply an inverse Fourier transform to the sequence E(ei2πkK(T )/(`max+1))(k = 0, . . . , `max =

∑ni=1 `max

i )




LHP Approximation

n Approximate the real reference credit portfolio with a portfolio consisting of a largenumber of equally weighted identical instruments (having the same term structureof default probabilities, recovery rates, and correlations to the common factor)

n Homogeneous portfolio: ai = a and Ci = C for all i and the notional amounts andrecovery rates R are the same for all issuers.

n Basel II agreement: “infinitely granular” portfolios

n LHP using a one factor Gaussian copula has become a standard model inpractice.↪→ Closed-form analytic synthetic CDO pricing formula

n This model fails to fit the prices of different CDO tranches simultaneously: lack oftail dependence of the Gaussian copula↪→ Use student t copula




Stochastic Order Bounds

n Theorem: (Kaas et al., 2000)

For any (X1, · · · , Xn) and any Z, we have that

Sl :=n∑

i=1

E [Xi | Z] ≤cx

n∑

i=1

Xi ≤cx

n∑

i=1

F−1Xi

(U) =: Sc

n Assume that all E [Xi | Z] are ↗ functions of Z

⇒ Sl is a comonotonic sum.

n E[(Sl − d)+] ≤ E[(S − d)+] ≤ E[(Sc − d)+]

n The stop-loss premiums of a sum of comonotonic random variables can easily beobtained from the stop-loss premiums of the terms:

E[(Sc − d)+] =

n∑

i=1

E[(

Xi − F−1Xi

(FSc (d)))

+

]

,(F−1

Sc (0) < d < F−1Sc (1)

)




Stop-loss Premium

E [(X − d)+] =

∫ ∞

d

FX(x)dx, −∞ < d < ∞

↪→ =surface above the distribution function, from d on.




Stop-loss Premium

E [(X − d)+] =

∫ ∞

d

FX(x)dx, −∞ < d < ∞

↪→ =surface above the distribution function, from d on.0.

00.

20.

40.

60.

81.

0

F (x )X

E[(X−d ) ]+

d




LHP Approximation

n Xi := Ni(1 − Ri)I[0,T ](Ti): loss at time T associated with name i

n Assume Xi’s are independent conditionally upon Z and identically distributed:

1

n

n∑

i=1

Xi −→ E[Xi|Z]︸︷︷︸

LPA

n L(T ) = X1 + · · · + Xn: aggregated loss at time T

n Convex order bounds:n∑

i=1

E[Xi|Z] ≤cx L(T ) ≤cx

n∑

i=1

F−1Xi

(U),

with E[Xi|Z] = Ni(1 − Ri)P (Ti ≤ T |Z).




LHP Approximation

n Loss on the portfolio as a percentage of the total portfolio notional:

L = (1 − R)1

n

n∑

i=1

I{Xi≤C}

n The probability that the ith issuer defaults

P (Xi ≤ Ci|η) = Φ

(η√

1 − a2

)

,

with

η =

C − aZ for Gaussian factor model

C√

Wν − aZ for student t factor model




LHP Approximation

n Law of large numbers:

1

n

n∑

i=1

I{Xi≤C} −→ Φ

(η√

1 − a2

)

n LPH: Assume that exactly this fraction of issuers defaults for eachrealization of η, so that L ≈ h(η), with h : R → [0, 1] and

h(x) = (1 − R)Φ

(x√

1 − a2

)

n Distribution of L:P [L ≤ θ] = Fη(h−1(θ)), θ ∈ [0, 1]

n CDO premiums can be easily computed!




LPH Approximation: student t factor model

Distribution of the mixing variable η

n P [W ≥ x] = Γ(

ν2 , x

2

)

n P [η ≤ t|Z] = I{Z≥− ta} + I{Z≤− t

a}Γ(

ν2 , ν(t+aZ)2

2C2

)

n Cumulative distribution function Fη(t):

P [η ≤ t] = Φ

(t

a

)

+1√2π

∫ −t

a

−∞

Γ

(ν

2,ν(t + au)2

2C2

)

e−u2

2 du

n Density fη(t):

fη(t) =1

a√

π2ν+12 Γ( ν

2 )

∫ +∞

0

e−1

2a2 (t−C√

wν

)2wν2−1e−

w2 dw




Conclusion and Possible Extensions

n Synthetic CDO Pricingu Dependence between default times is modelled through Student t copulas.u Factor approach leading to semi-analytic pricing expressions that ease model

risk assessment.u The loss amounts — or equivalently, the recovery rates — associated with

defaults are random.u Closed-form solutions for the loss distribution can be derived under the LHP

approximation.

n Possible extensions:u other copulas: skewed t copula, grouped t copula, . . .u stable distributionsu moment matching techniquesu . . .




References

[1] Andersen, L. and Sidenius, J., 2004. Extensions to the Gaussian copula: random recovery andrandom factor loading. Journal of Credit Risk, 1, 29-70.

[2] Andersen, L., Sidenius, J. and Basu, S., 2003. All your hedges in one basket. Risk, Nov. 67-72.

[3] Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J., 2004. Statistics of extremes - theory andapplications. Wiley Series in Probability and Statistics.

[4] Demarta, S. and McNeil, A.J., 2005. The t copula and related copulas. To appear InternationalStatistical Review.

[5] Gibson, M.S., 2004. Understanding the risk of synthetic CDOs. Technical report Federal ReserveBoard.

[6] Joe, H., 1997. Multivariate models and dependence concepts. Chapman & Hall.

[7] Kotz, S., Balakrishnan, N. and Johnson, N., 2000. Continuous multivariate distributions. Wiley.

[8] Li, D., 2000. On Default Correlation: a Copula Approach. Journal of Fixed Income. 9, 43-54.

[9] Nelsen, R.B., 1999. An introduction to copulas. Springer.

[10] Sklar, A., 1959. Fonctions de repartition a n dimensions et leurs marges. Publications de l’Institut deStatistique de l’Université de Paris, 8, 229-231.


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