pricing of cdo’s based on the multivariate wang transform · - agent j faces a risk of potential...
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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: [email protected]://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