stein's method and zero bias transformation: application...

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

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


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


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


Stein method and Zero bias transformation


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 )].


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.


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

)).


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

)).


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 )].


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.


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.


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))

.


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


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).


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 .


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.


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


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.


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.


Applications to CDOs

In collaboration with David KURTZ (Calyon, Londres).


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.


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.


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


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


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.


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


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


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


Conclusion remark and perspective

Tests for VaR and sensitivity remain satisfactory

Perspective: remove the deterministic recovery ratecondition in the Poisson case.


stein's method and zero bias transformation: application...

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