stein's method and zero bias transformation: application...
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Stein’s method and zero bias transformation:Application to CDO pricing
Ying Jiao
ESILV and Ecole Polytechnique
Joint work with N. El Karoui
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Introduction
CDO — a portfolio credit derivative containing ≈ 100underlying names susceptible to default risk.
Key term to calculate: the cumulative lossLT =
∑ni=1 Ni(1− Ri)11{τi≤T} where Ni is the nominal of
each name and Ri is the recovery rate.
Pricing of one CDO tranche: call function E[(LT − K )+]with attachment or detachment point K .
Motivation: fast and precise numerical method for CDOs inthe market-adopted framework
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Factor models
Standard model in the market: factor model withconditionally independent defaults.
Normal factor model: Gaussian copula approach
{τi ≤ t} = {ρiY +√
1− ρ2i Yi ≤ N−1(αi(t))} where
Yi ,Y ∼ N(0,1) and independent, and αi(t) is the averageprobability of default before t .
Given Y (not necessarily normal), LT is written as sum ofindependent random variablesLT follows binomial law for homogeneous portfoliosCentral limit theorems: Gaussian and Poissonapproximations
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Some approximation methods
Gaussian approximation in finance:Total loss of a homogeneous portfolio (Vasicek (1991)),normal approximation of binomial distributionOption pricing: symmetric case (Diener-Diener (2004))Difficulty: n ≈ 100 and small default probability.
Saddle point method applied to CDOs(Martin-Thompson-Browne,Antonov-Mechkov-Misirpashaev): calculation time fornon-homogeneous portfolioOur solution: combine Gaussian and Poissonapproximations and propose a corrector term in each case.Mathematical tools: Stein’s method and zero biastransformation
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Stein method and Zero bias transformation
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Zero bias transformation
Goldstein-Reinert, 1997
-X a r.v. of mean zero and of variance σ2
-X ∗ has the zero-bias distribution of X if for all regular enough f ,
E[Xf (X )] = σ2E[f ′(X ∗)] (1)
-distribution of X ∗ unique with density pX∗(x) = E[X11{X>x}]/σ2.
Observation of Stein (1972): if Z ∼ N(0, σ2), thenE[Zf (Z )] = σ2E[f ′(Z )].
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Stein method and normal approximation
Stein’s equation (1972)
For the normal approximation of E[h(X )], Stein’s idea is to usefh defined as the solution of
h(x)− Φσ(h) = xf (x)− σ2f ′(x) (2)
where Φσ(h) = E[h(Z )] with Z ∼ N(0, σ2).
Proposition (Stein): If h is absolutely continuous, then∣∣E[h(X )]− Φσ(h)∣∣ =
∣∣σ2E[f ′h(X ∗)− f ′h(X )
]∣∣≤ σ2‖f ′′h ‖ E
[|X ∗ − X |
].
To estimate the approximation error, it’s important tomeasure the “distance” between X and X ∗ and thesup-norm of fh and its derivatives.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Example and estimations
Example (Asymmetric Bernoulli distribution) :P(X = q) = p and P(X = −p) = q with q = 1− p. SoE(X ) = 0 and Var(X ) = pq. Furthermore, X ∗ ∼ U[−p,q].
If X and X ∗ are independent, then, for any even function g,
E[g(X ∗ − X )
]=
12σ2 E
[X sG(X s)
]where G(x) =
∫ x0 g(t)dt , X s = X − X̃ and X̃ is an
independent copy of X .
In particular, E[|X ∗ − X |] = 14σ2 E
[|X s|3
] (∼ O
( 1√n
)).
E[|X ∗ − X |k ] = 12(k+1)σ2 E
[|X s|k+2] (
∼ O( 1√
nk
)).
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Sum of independent random variables
Goldstein-Reinert, 97Let W = X1 + · · ·+ Xn where Xi are independent r.v. of meanzero and Var(W ) = σ2
W <∞. Then
W ∗ = W (I) + X ∗I ,
where P(I = i) = σ2i /σ
2W . W (i) = W − Xi . Furthermore, W (i),
Xi , X ∗i and I are mutually independent.
Here W and W ∗ are not independent.E[|W ∗ −W |k
]= 1
2(k+1)σ2W
∑ni=1 E
[|X s
i |k+2] (
∼ O( 1√
nk
)).
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Conditional expectation technic
To obtain an addition order in the estimations, for example,
E[XI |X1, · · · ,Xn] =∑n
i=1σ2
iσ2
WXi , then,
E[E[XI |X1, · · · ,Xn]2] =∑n
i=1σ6
iσ4
Wis of order O( 1
n2 ).
However, E[X 2I ] is of order O( 1
n ).This technic is crucial for estimating E[g(XI ,X ∗I )] andE[f (W )g(XI ,X ∗I )].
Ying Jiao Stein’s method and zero bias transformation for CDOs
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First order Gaussian correction
TheoremNormal approximation ΦσW (h) of E[h(W )] has corrector :
Ch =1
2σ4W
n∑i=1
E[X 3i ]ΦσW
(( x2
3σ2W− 1)
xh(x)
)(3)
where h has bounded second order derivative.The corrected approximation error is bounded by∣∣∣E[h(W )]− ΦσW (h)− Ch
∣∣∣ ≤ α(h,X1, · · · ,Xn)
where α(h,X1, · · · ,Xn) depends on ‖h′′‖ and moments of Xi upto fourth order.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Remarks
For i.i.d. asymmetric Bernoulli: corrector of order O( 1√n );
corrected error of order O( 1n ).
In the symmetric case, E[X ∗I ] = 0, so Ch = 0 for any h. Chcan be viewed as an asymmetric corrector.Skew play an important role.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Ingredients of proof
development around the total loss W
E[h(W )]− ΦσW (h) = σ2W E[f ′′h (W )(X ∗I − XI)]
+ σ2W E[f (3)h
(ξW + (1− ξ)W ∗)ξ(W ∗ −W )2
].
by independence, E[f ′′h (W )X ∗I ] = E[X ∗I ]E[f ′′h (W )] =E[X ∗I ]ΦσW (f ′′h ) + E[X ∗I ]
(E[f ′′h (W )]− ΦσW (f ′′h )
).
use the conditional expectation technicE[f ′′h (W )XI ] = cov
(f ′′h (W ),E[XI |X1, · · · ,Xn]
).
write ΦσW (f ′′h ) in term of h and obtain
Ch = σ2W E[X ∗I ]ΦσW
(f ′′h)
= 1σ2
WE[X ∗I ]ΦσW
((x2
3σ2W− 1)
xh(x))
.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Function “call”
Call function Ck (x) = (x − k)+: absolutely continuous withC′k (x) = 11{x>k}, but no second order derivative.Hypothesis not satisfied.
Same corrector
∣∣E[(W − k)+]− ΦσW ((x − k)+)− 13
E[X ∗I ]kφσW (k)∣∣ ∼ O(
1n
).
Error bounds depend on |f ′Ck|, |xf ′′Ck
|.
Ingredients of proof:write f ′′Ck
as more regular function by Stein’s equationConcentration inequality (Chen-Shao, 2001)
Corrector Ch = 0 when k = 0, extremal values whenk = ±σ2
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Normal or Poisson?
The methodology works for other distributions.Stein’s method in Poisson case (Chen 1975): a r.v. Λtaking positive integer values follows Poisson distributionP(λ) iif E[Λg(Λ)] = λE[g(Λ + 1)] := APg(Λ).Poisson zero bias transformation X ∗ for X with E[X ] = λ:E[Xg(X )] = E[APg(X ∗)]
Stein’s equation :
h(x)− Pλ(h) = xg(x)−APg(x) (4)
where Pλ(h) = E[h(Λ)] with Λ ∼ P(λ).
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Poisson correction
Poisson corrector of PλW (h) for E[h(W )] :
CPh =λW
2PλW
(∆2h
)E[X ∗I − XI
]where ∆h(x) = h(x + 1)− h(x) and P(I = i) = λi/λW .
Proof: combinatory technics for integer valued r.v.For h = (x − k)+, ∆2h(x) = 11{x=k−1}, then
CPh =σ2
W − λW
2(bkc − 1)!e−λWλ
bkc−1W .
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Numerical tests: E[(Wn − k)+]
E[(Wn − k)+] in function of nσW = 1 and k = 1 (maximal Gaussian correction) forhomogeneous portfolio.Two graphes : p = 0.1 and p = 0.01 respectively.Oscillation of the binomial curve and comparison ofdifferent approximations.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Numerical tests: E[(Wn − k)+]
Legend : binomial (black), normal (magenta), correctednormal (green), Poisson (blue) and corrected Poisson(red). p = 0.1,n = 100.
0 50 100 150 200 250 300 350 400 450 5000.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
n
binomial normal normal with correction poisson poisson with correction
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Numerical tests : E[(Wn − k)+]
Legend : binomial (black), normal (magenta), correctednormal (green), Poisson (blue) et corrected Poisson (red).p = 0.01,n = 100.
0 50 100 150 200 250 300 350 400 450 5000.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
n
binomial normal normal with correction poisson poisson with correction
Corrected Poisson approximation remains precise even forp = 0.1.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Higher order corrections
The regularity of h plays an important role.Second order approximation is given by
C(2,h) = ΦσW (h) + Ch
+ ΦσW
(( x6
18σ8W− 5x4
6σ6W
+5x2
2σ4W
)(h(x)− ΦσW (h))
)E[X ∗I ]2
+12
ΦσW
(( x4
4σ6W− 3x2
2σ4W
)h(x)
)(E[(X ∗I )2]− E[X 2
I ]).
(5)
Since the call function is not regular enough, it’s difficult tocontrol errors.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Applications to CDOs
In collaboration with David KURTZ (Calyon, Londres).
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Normalization for loss
Percentage loss lT = 1n∑n
i=1(1− Ri)11{τi≤T}K = 3%,6%,9%,12%.Normalization for ξi = 11{τi≤T} : let Xi = ξi−pi√
npi (1−pi )where
pi = P(ξi = 1). Then E[Xi ] = 0 and Var[Xi ] = 1n .
Suppose Ri = 0 and pi = p for simplicity. Letp(Y ) = pi(Y ) = P(τi ≤ T |Y ). Then
E[(lT−K )+|Y
]=
√p(Y )(1− p(Y ))
nE[(W−k(n,p(Y )))+|Y
].
where W =∑n
i=1 Xi and k(n,p) = (K−p)√
n√p(1−p)
.
Constraint in Poisson case: Ri deterministic andproportional.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Numerical tests: domain of validity
Conditional error of E[(lT − K )+], in function of K/p,n = 100.
Inhomogeneous portfolio with log-normal pi of mean p.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Numerical tests : Domain of validity
Legend : corrected Gaussian error (blue), correctedPoisson error (red). np=5 and np=20.
np=5
-0.010%
-0.005%
0.000%
0.005%
0.010%
0.015%
0.020%
0% 100% 200% 300%
Error Gauss Error Poisson
np=20
-0.008%-0.006%-0.004%-0.002%0.000%0.002%0.004%0.006%0.008%0.010%0.012%
0% 100% 200% 300%
Error Gauss Error Poisson
Domain of validity : np ≈ 15Maximal error : when k near the average loss; overlappingarea of the two approximations
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Comparison with saddle-point method
Legend : Gaussian error (red), saddle-point first order error(blue dot), saddle-point second order error (blue). np=5and np=20.
np=5
-0.040%
-0.030%
-0.020%
-0.010%
0.000%
0.010%
0.020%
0.030%
0.040%
0% 100% 200% 300%
Gauss Saddle 1 Saddle 2
np=20
-0.025%
-0.020%
-0.015%
-0.010%
-0.005%
0.000%
0.005%
0.010%
0.015%
0.020%
0.025%
0% 100% 200% 300%
Gauss Saddle 1 Saddle 2
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Numerical tests : stochastic Ri
Conditional error of E[(lT − K )+], in function of K/p forGaussian approximation error with stochastic Ri .
Ri : Beta distribution with expectation 50% and variance26%, independent (Moody’s).
Corrector depends on the third order moment of Ri .Confidence interval 95% by 106 Monte Carlo simulation.
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Numerical tests : stochastic Ri
Gaussian approximation better when p is large as in thestandard case. np=5 and np=20.
np=5
-0.005%-0.004%-0.003%-0.002%-0.001%0.000%0.001%0.002%0.003%0.004%
0% 100% 200% 300%
Lower MC Bound Gauss Error Upper MC Bound
np=20
-0.006%
-0.004%
-0.002%
0.000%
0.002%
0.004%
0.006%
0% 100% 200%
Lower MC Bound Gauss Error Upper MC Bound
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Real-life CDOs pricing
Parametric correlation model of ρ(Y ) to match the smile.Method adapted to all conditional independent models, notonly in the normal factor model caseNumerical results compared to the recursive method(Hull-White)Calculation time largely reduced: 1 : 200 for one price
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Real-life CDOs pricing
Error of prices in bp (10−4) for corrected Gauss-Poissonapproximation with realistic local correlation.Expected loss 4%− 5%
Break Even Error
-0.20
-
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0-3 3-6 6-9 9-12 12-15 15-22 0-100
Break Even Error
Ying Jiao Stein’s method and zero bias transformation for CDOs
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Conclusion remark and perspective
Tests for VaR and sensitivity remain satisfactory
Perspective: remove the deterministic recovery ratecondition in the Poisson case.
Ying Jiao Stein’s method and zero bias transformation for CDOs