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Pricing of CDO’s Based on the Multivariate Wang Transform*

ASTIN 2009 Colloquium @ Helsinki (02 June, 2009)

Masaaki KijimaTokyo Metropolitan University/ Kyoto University

Email: kijima@tmu.ac.jphttp://www.comp.tmu.ac.jp/kijimam

* Joint Work with Shin-ichi Motomiya and Yoichi Suzuki, Credit Pricing Corporation (CPC), Tokyo, Japan

Pricing Principles for Insurance Risk

Esscher transform, Wang Transform

Extension to the Multivariate Setting

The Pricing of CDO’s

Image of CDO, Synthetic CDO, Standard Model

Proposed Models for the Pricing of CDO’s

Numerical Examples

Plan of my Talk

In the actuarial literature, a popular method for the pricing offinancial and insurance risks, among others, is

the Esscher transform:where stands for risk adjustment.

]e[/]e[)( XX EXEX δδπ −−=0>δ

0 ],))(([)( 1 >+ΦΦ= − δδxFxF Q

Recently, Wang (2002, ASTIN) proposed a pricing method based on the following transformation:

QFF →The distortion describes risk adjustment.

Premium Principles for Insurance Risk

Note: Insurance markets are incomplete and exhibit fat-tailed distributions, as for CDO markets.

0 ,]e[]e[)( >= −

−

δπ δ

δ

X

X

EXEX

]e[ XE δ−

Also, Wang (2002, ASTIN) showed that the transform is the only transform among the family of distortions that can recover CAPM and the Black-Scholes formula for options.

]))(([)( );(d)( 1 δπ +ΦΦ== −∫ xFxFxFxX QQ

Main drawback of the Esscher transform for the practical use (in the discrete-time setting) is that the MGF must exist (counterpart of Novikov’s condition).

Recall that

Esscher:

Wang:

The Wang transform has a merit in this aspect.

]e[e ],[)( Z

Z

EYEY λ

λ

ηηπ −

−

==

The Buhlmann's equilibrium pricing model (1980, ASTIN):- pure risk-exchange economy- exponential utility with distinct risk aversion index- agent j faces a risk of potential loss He derived the following equilibrium pricing formula for risk Y

jX

∑∑ =−

=− ==

n

j jn

j jXZ1

11

1 , λλwhere

Z is the aggregate risk and is the risk aversion index of the representative agent in the market.

0>λ

The Esscher transform can be reduced from the Buhlmann'sformula by assuming , whence it has a soundeconomic interpretation.

ZY <<

It can be shown that the Wang transform is the same as the Esscher transform for normally distributed risks.

Moreover, Wang (2003, ASTIN) showed that his transform can be derived from Buhlmann's formula even for general risk under some assumptions on Z. Hence, the Wang transform also has a sound economic interpretation.

However, actuarial pricing formulas (including the Esscher and Wang) are not linear, yielding arbitrage opportunities.

)()()( YbXabYaX πππ +≠+

Kijima (2006, ASTIN) developed a multivariate Wang transform based on the Buhlmann's equilibrium pricing formula.

The multivariate Wang transform

The normal case:

jZjkj

n

k kjkjjj

nnnQ

w

xFy

yyxxF

λσλρ

ρλ

ρ ==Σ

+Φ=

Φ=

∑ =−

),(

)]([

),,(),,(

11

11 KK ),0( ρΣnN:CDF of

),( , ,1

ZXCw jj

kjn

k kjjZ

jj σ

λρλλσλσσ

=== ∑ =

In this case, we have

and )/)(()( jjjjj xxF σμ−Φ=

jjjjjnnnQ ZXCxyyyxxF σλμ /)],([ ),,,(),,( 11 +−=Φ=∴ KK

),( ZXC jjj λμμ −→

Risk aversion index of the representative agent

Wang changesrisk premium

⎥⎦

⎤⎢⎣

⎡+= ∫ )(

d)(0 TB

MttB

cMEV TT tQ

Pricing of TranchesCDO

)(tL

D

A

Remaining Principal at time t for the tranche

: Money Market Account

Note: We assume that the recovery is zero.

})({})({ 1))((1)( DtLAAtLt tLDADM <≤< −+−=

∫=t

s srtB 0

de)(

The present value of cash flows arising from the tranche.

}{11)( ,)1( ,)()( tiiii

n

i ii itNFMtNMtL ≤==−==∑ τδ

Assumed to be constantIn general,

Standard Model: One-Factor Gaussian Copula Model

),,,( 21 nXXX K

),,( 21 nτττ KWe need to model in the bottom-up setting.

)(log)( ,)( tVtXxtXt iiii =≤⇔≤τ

Consider the Merton’s structural model:

where V stands for the firm value.

So, instead of modeling the correlated ),,( 21 nτττ K

it should be easier to model the correlated

),,,( 21 nXXX K

)(}{}{)( xKxXQtQtF iii =≤=≤= τ

iiiii UUUUX ⊥−+= ,1 2ρρ

iXwhere has the CDF H(x), U has G(x), and has K(x) iU

Example 1 (The industry standard model): U’s follow a normal. Then K(x) is also normal, and

By definition,

Example 2 (Hull and White, 2004): U’s follow a t distribution. Then K(x) needs to be evaluated numerically

Supposing for all i, (to be calibrated)ρρ =i

Take to be latent variables such that:

under Q

))((1iii XF Φ= −τ

∑ =+−=≡=⇒

n

j jiiiijijii ZXCCXXCNX1

21),( ,),( ),1,0(~ ρρρρρ

∑ ==

n

j jXZ1

),,(),,(

)( ),,,()(

111

11:

nnd

n

jinnnQ

CXCXXX

CxCxxK

λλ

ρρλλ ρρ

−−=⇔

=Σ++Φ=∗∗

Σ

KK

K

iiiii UUUUX ⊥−+= ,1 2ρρ

As for CreditMetrics, we start with assuming that under P

: follow N(0,1)

: Aggregate risk

Change of Measures from P to Q

)(}{}{)( iiiQ

i CxxXQtQtF λτ +Φ=≤=≤= ∗

Apply the multivariate Wang.

risk-adjusted log-firm values

By definition,

)))((,)),((()( 111

1: n

Qn

Qn

Q tFtFtF −−Σ ΦΦΦ=∴ Kρ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−ΦΦ=≤=≤=

−∗

2

1

1))((}|{}|{)|(

i

iQ

iii

Qi

UtFUxXQUtQUtqρ

ρτ

Conditional on U in the one-factor model, we have

)))((,)),((()( 111

1: n

Qn

Qn

Q tFtFtF −−Σ ΦΦΦ= Kρ

risk aversion index)()( i

Qi CxtF λ+Φ=

: joint CDF of under Qs'τ: marginal CDF of under Qiτ

The risk premium is embedded in the marginal default CDF that is calibrated from market quotes for CDS’s

( ) uuutqtF n

iQi

Qi d)()|()(

1φ∫ ∏

∞

∞− == : the standard model if

By independence, it follows that

ρρ =i

Derived from Merton’s structural model and Buhlmann’s principle, whence it has a sound economic interpretation.

Risk-Adjusted Gaussian Copula Model

In the standard model, the correlation is the only free parameter that can be calibrated from market quotes.

ρρ =i

The correlation should be thought of as the risk premium.

However, then, what is the default correlation?

Note: In the Gaussian model, the change of measures does not change the variance-covariance structure.

The correlation parameters should be estimated under P and the risk premium is calibrated from market quotes under Q

Fact: The CDO market is segmented into tranches according to investors’ preferences against risks.

)(tFQi

When evaluating the tranche, we use

)))((,,))((()( 1111

1: nDn

QnD

Qn

Q CtFCtFtF λλρ

+Φ+ΦΦ= −−Σ K

risk premium for CDS of name i)()( i

Qi CxtF λ+Φ= : marginal CDF of under Qiτ

Parameter estimation and calibration

jiijij ρρρρρ ==Σ );(

),,(),,( 1111 nDnnDd

n CCXCCXXX λλλλ −−−−=∗∗ KK

The risk premium for tranche with detachment D

where iiiiii UUUX ερρρ +=−+= 21

Use the simple regression to estimate

: calibrated from market quotes for CDS’s

Dλ : calibrated from market quotes for the CDO tranche

Assume:

iρ

0

50

100

150

200

250

300

350

400

450

500

2006

年9月

2006

年10

月20

06年

11月

2006

年12

月20

07年

1月20

07年

2月20

07年

3月20

07年

4月20

07年

5月20

07年

6月20

07年

7月20

07年

8月20

07年

9月20

07年

10月

2007

年11

月20

07年

12月

2008

年1月

2008

年2月

2008

年3月

3-6

6-9

9-12

12-22

Prices of iTraxx Tranches (since 2006/9)

Averaged Correlation under P (since 2006/9)

相関（平均）推移

0.46

0.48

0.5

0.52

0.54

0.56

0.58

2006

年9月

2006

年10

月20

06年

11月

2006

年12

月20

07年

1月20

07年

2月20

07年

3月20

07年

4月20

07年

5月20

07年

6月20

07年

7月20

07年

8月20

07年

9月20

07年

10月

2007

年11

月20

07年

12月

2008

年1月

2008

年2月

2008

年3月

ρ 系列1

Calibrated Risk Premium Curves since 2007/6/30Transition of λCurve（2007_06-2008_03）

-0.015

-0.013

-0.011

-0.009

-0.007

-0.005

-0.003

-0.001

0.001

0.003

0.005

3% 6% 9% 12% 22%

Detachment Point

λ

2007_06

2007_07

2007_08

2007_09

2007_10

2007_11

2007_12

2008_01

2008_02

2008_03

Risk-Adjusted t Copula Model

The risk-adjusted Gaussian model can fit perfectly market quotes for all tranches of standard CDO’s by calibrating the risk adjustment parameters Dλ

Also, financial markets often exhibit fat-tailed distributions.

Note: We have interpreted as the risk aversion index for tranche D of the representative agent in the market.

It may be more plausible to assume that is increasing in D

Dλλ +

Dλ

DbaD log+=λWe consider, e.g., the case

Introduce the t copula framework

Easy to calculate

0 )],)(([)( >+Φ= θθα YxExF YQ

))(()( 1 xFGx −=αwhere Y > 0 and for some G(x).

Extension by Kijima and Muromachi (2008, IME)

In particular, Y = 1 implies the Wang transform.

: risk premium

))](([)]))((([)( 1:

1 xFtPYxFGExF YQ −

−− =+Φ= νθνθ

νχν /2=YAlso, when , we have

where, denotes the CDF of a non-central t distribution.)(: xP δν

0 )],))((([)))((( 11 ≥+Φ>+ΦΦ −− θθθ YxFGExF Y

However, against our expectation, we can prove

Therefore, following the Wang’s idea, we propose

jZjkjn

k kjkjjj

nnQ

wxFy

yytxxF

λσλρρλ ρ

ν

==Σ+Φ=

=

∑ =− ),( ,)]([

),,(),,(

11

11 KK

jDjQjjn

Q CtFttF λββρν +Φ== −

Σ ))(( ),()( 1,:

),,,( 21 nτττ KThe Proposed Model: For

However, this formula involves a double integral.

Adopt the following approximation

Pricing of CDO’s based on the 2-parameter Wang Transform

under Q

iiiiiiiDi

ii UUXYC

YXX 22

: 1 ,/)( ,)()(

ρρνχνλλν ν −+==+−=∗

Assume

Define21

)( ,)(

)()(i

ii

i

iii

uuY

uUuρ

ρδνδξ

−=

+=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

+Φ=≤=≤=

−∗

2

1

)(:1

))((}|{}|{)|(i

iDQ

iUii

Qi

CtFPUxXQUtQUtqi ρ

λτ δν

Calibration Results

A-1.DJ iTraxx 5-year Index Tranches（EUR） 2004/8/23Tranches 0-3% 3-6% 6-9% 9-12% 12-22% RMSE

Market mid Price 25.5% 146.0 60.3 36.3 7.7

Bid/ask spread 1.3% 10.0 5.5 5.5 3.5Jump-diffusion intensities 25.0% 145.0 58.6 38.1 17.7 0.34Pure diffusion intensities 30.0% 187.1 27.4 3.5 0.1 5.11Gaussian copula 27.4% 222.3 52.5 13.8 1.6 4.58RFL Gaussian copula 25.3% 148.9 52.4 43.4 17.9 0.90

Double-t copula 24.0% 153.4 56.5 32.4 17.4 0.84

Risk-Adjusted t Copula 25.8% 145.2 49.1 28.4 17.1 1.15

Fitted parameters :μ＝３

λ＝0.0045×ln（Detachment Point）-0.0372

D2004/8/23

2005/12/5

A-2.DJ iTraxx 5-year Index Tranches（EUR） 2005/12/5Tranches 0-3% 3-6% 6-9% 9-12% 12-22% RMSE

Market mid Price 26.3% 80.6 23.1 10.3 5.8

Bid/ask spread 0.6% 3.3 2.6 2.0 1.3Jump-diffusion intensities 28.7% 86.3 18.7 14.4 10.4 2.88Pure diffusion intensities 32.5% 104.3 8.9 0.8 0.0 6.99Gaussian copula 34.6% 99.9 2.9 0.1 0.0 8.44RFL Gaussian copula 27.0% 83.2 9.4 7.4 7.3 2.54

Double-t copula 29.8% 101.1 24.4 13.2 6.6 3.99

Risk-Adjusted t Copula 26.5% 77.2 18.5 12.6 8.3 1.37

Fitted parameters :μ＝1

λ＝0.024×ln（Detachment Point）-0.3675

22

Concluding Remarks

1. We showed that, contrary to the criticism, the one-factor Gaussian copula model is consistent with Buhlmann'seconomic premium principle, whence it has a sound economic interpretation.

2. Based on this finding, we developed an alternative within the Buhlmann's framework. Namely,

we introduce the risk aversion index for each tranche to be calibrated, while keeping the correlation structure as given under the actual probability measure;we also apply the Student t copula.

3. Numerical experiments reveal that our model provide a better fit than the existing models in the literature.

Thank You for Your Patience