Steins method and zero bias transformation: Application to CDO - - PowerPoint PPT Presentation

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 ≈ 100 underlying names susceptible to default risk. Key term to calculate: the cumulative loss LT = n

i=1 Ni(1 − Ri)1

1{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 in the market-adopted framework

Ying Jiao Stein’s method and zero bias transformation for CDOs

Factor models

Standard model in the market: factor model with conditionally independent defaults. Normal factor model: Gaussian copula approach {τi ≤ t} = {ρiY +

• 1 − ρ2

i Yi ≤ N −1(αi(t))} where

Yi, Y ∼ N(0, 1) and independent, and αi(t) is the average probability of default before t. Given Y (not necessarily normal), LT is written as sum of independent random variables LT follows binomial law for homogeneous portfolios Central limit theorems: Gaussian and Poisson approximations

Ying Jiao Stein’s method and zero bias transformation for CDOs

Some approximation methods

Gaussian approximation in finance:

Total loss of a homogeneous portfolio (Vasicek (1991)), normal approximation of binomial distribution Option 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 for non-homogeneous portfolio Our solution: combine Gaussian and Poisson approximations and propose a corrector term in each case. Mathematical tools: Stein’s method and zero bias transformation

Ying Jiao Stein’s method and zero bias transformation for CDOs

Stein method and Zero bias transformation

Ying Jiao Stein’s method and zero bias transformation for CDOs

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[X1

1{X>x}]/σ2. Observation of Stein (1972): if Z ∼ N(0, σ2), then E[Zf(Z)] = σ2E[f ′(Z)].

Ying Jiao Stein’s method and zero bias transformation for CDOs

Stein method and normal approximation

Stein’s equation (1972) For the normal approximation of E[h(X)], Stein’s idea is to use fh 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)

• ≤ σ2f ′′

h E

• |X ∗ − X|
• .

To estimate the approximation error, it’s important to measure the “distance” between X and X ∗ and the sup-norm of fh and its derivatives.

Ying Jiao Stein’s method and zero bias transformation for CDOs

Example and estimations

Example (Asymmetric Bernoulli distribution) : P(X = q) = p and P(X = −p) = q with q = 1 − p. So E(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)
• =

1 2σ2 E

• X sG(X s)
• where G(x) =

x

0 g(t)dt, X s = X −

X and X is an independent copy of X. In particular, E[|X ∗ − X|] =

1 4σ2 E

• |X s|3
• ∼ O

1

√n

• .

E[|X ∗ − X|k] =

1 2(k+1)σ2 E

• |X s|k+2
• ∼ O
• 1

√nk

• .

Ying Jiao Stein’s method and zero bias transformation for CDOs

Sum of independent random variables

Goldstein-Reinert, 97 Let W = X1 + · · · + Xn where Xi are independent r.v. of mean zero and Var(W) = σ2

W < ∞. Then

W ∗ = W (I) + X ∗

I ,

where P(I = i) = σ2

i /σ2

• W. 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)σ2

W

n

i=1 E

• |X s

i |k+2

• ∼ O
• 1

√nk

• .

Ying Jiao Stein’s method and zero bias transformation for CDOs

Conditional expectation technic

To obtain an addition order in the estimations, for example, E[XI|X1, · · · , Xn] = n

i=1 σ2

i

σ2

W Xi, then,

E[E[XI|X1, · · · , Xn]2] = n

i=1 σ6

i

σ4

W is of order O( 1

n2 ).

However, E[X 2

I ] is of order O( 1 n).

This technic is crucial for estimating E[g(XI, X ∗

I )] and

E[f(W)g(XI, X ∗

I )].

Ying Jiao Stein’s method and zero bias transformation for CDOs

First order Gaussian correction

Theorem Normal approximation ΦσW (h) of E[h(W)] has corrector : Ch = 1 2σ4

W n

• i=1

E[X 3

i ]ΦσW

x2 3σ2

W

− 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 up to fourth order.

Ying Jiao Stein’s method and zero bias transformation for CDOs

Remarks

For i.i.d. asymmetric Bernoulli: corrector of order O( 1

√n);

corrected error of order O( 1

n).

In the symmetric case, E[X ∗

I ] = 0, so Ch = 0 for any h. Ch

can be viewed as an asymmetric corrector. Skew play an important role.

Ying Jiao Stein’s method and zero bias transformation for CDOs

Ingredients of proof

development around the total loss W E[h(W)] − ΦσW (h) = σ2

WE[f ′′ h (W)(X ∗ I − XI)]

+ σ2

WE

• 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 technic E[f ′′

h (W)XI] = cov

• f ′′

h (W), E[XI|X1, · · · , Xn]

• .

write ΦσW (f ′′

h ) in term of h and obtain

Ch = σ2

WE[X ∗ I ]ΦσW

• f ′′

h

• =

1 σ2

W E[X ∗

I ]ΦσW

• x2

3σ2

W − 1

• xh(x)
• .

Ying Jiao Stein’s method and zero bias transformation for CDOs

Function “call”

Call function Ck(x) = (x − k)+: absolutely continuous with C′

k(x) = 1

1{x>k}, but no second order derivative. Hypothesis not satisfied. Same corrector

• E[(W − k)+] − ΦσW ((x − k)+) − 1

3E[X ∗

I ]kφσW (k)

• ∼ O(1

n). Error bounds depend on |f ′

Ck|, |xf ′′ Ck|.

Ingredients of proof:

write f ′′

Ck as more regular function by Stein’s equation

Concentration inequality (Chen-Shao, 2001)

Corrector Ch = 0 when k = 0, extremal values when k = ±σ2

Ying Jiao Stein’s method and zero bias transformation for CDOs

Normal or Poisson?

The methodology works for other distributions. Stein’s method in Poisson case (Chen 1975): a r.v. Λ taking positive integer values follows Poisson distribution P(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

Poisson correction

Poisson corrector of PλW (h) for E[h(W)] : CP

h = λW

2 PλW

• ∆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) = 1 1{x=k−1}, then CP

h =

σ2

W − λW

2(⌊k⌋ − 1)!e−λW λ⌊k⌋−1

W

.

Ying Jiao Stein’s method and zero bias transformation for CDOs

Numerical tests: E[(Wn − k)+]

E[(Wn − k)+] in function of n σW = 1 and k = 1 (maximal Gaussian correction) for homogeneous portfolio. Two graphes : p = 0.1 and p = 0.01 respectively. Oscillation of the binomial curve and comparison of different approximations.

Ying Jiao Stein’s method and zero bias transformation for CDOs

Numerical tests: E[(Wn − k)+]

Legend : binomial (black), normal (magenta), corrected normal (green), Poisson (blue) and corrected Poisson (red). p = 0.1, n = 100.

50 100 150 200 250 300 350 400 450 500 0.08 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

Numerical tests : E[(Wn − k)+]

Legend : binomial (black), normal (magenta), corrected normal (green), Poisson (blue) et corrected Poisson (red). p = 0.01, n = 100.

50 100 150 200 250 300 350 400 450 500 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 n binomial normal normal with correction poisson poisson with correction

Ying Jiao Stein’s method and zero bias transformation for CDOs

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

18σ8

W

− 5x4 6σ6

W

+ 5x2 2σ4

W

• (h(x) − ΦσW (h))
• E[X ∗

I ]2

+ 1 2ΦσW x4 4σ6

W

− 3x2 2σ4

W

• h(x)
• E[(X ∗

I )2] − E[X 2 I ]

• .

(5) Since the call function is not regular enough, it’s difficult to control errors.

Ying Jiao Stein’s method and zero bias transformation for CDOs

Applications to CDOs

In collaboration with David KURTZ (Calyon, Londres).

Ying Jiao Stein’s method and zero bias transformation for CDOs

Normalization for loss

Percentage loss lT = 1

n

n

i=1(1 − Ri)1

1{τi≤T} K = 3%, 6%, 9%, 12%. Normalization for ξi = 1 1{τi≤T} : let Xi =

ξi−pi

√

npi(1−pi) where

pi = P(ξi = 1). Then E[Xi] = 0 and Var[Xi] = 1

n.

Suppose Ri = 0 and pi = p for simplicity. Let p(Y) = pi(Y) = P(τi ≤ T|Y). Then E

• (lT−K)+|Y
• =
• p(Y)(1 − p(Y))

n E

• (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 and proportional.

Ying Jiao Stein’s method and zero bias transformation for CDOs

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

Numerical tests : Domain of validity

Legend : corrected Gaussian error (blue), corrected Poisson 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 ≈ 15 Maximal error : when k near the average loss; overlapping area of the two approximations

Ying Jiao Stein’s method and zero bias transformation for CDOs

Comparison with saddle-point method

Legend : Gaussian error (red), saddle-point first order error (blue dot), saddle-point second order error (blue). np=5 and 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

Numerical tests : stochastic Ri

Conditional error of E[(lT − K)+], in function of K/p for Gaussian approximation error with stochastic Ri. Ri : Beta distribution with expectation 50% and variance 26%, 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

Numerical tests : stochastic Ri

Gaussian approximation better when p is large as in the standard 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

Real-life CDOs pricing

Parametric correlation model of ρ(Y) to match the smile. Method adapted to all conditional independent models, not

• nly in the normal factor model case

Numerical 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

Real-life CDOs pricing

Error of prices in bp (10−4) for corrected Gauss-Poisson approximation 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

Conclusion remark and perspective

Tests for VaR and sensitivity remain satisfactory Perspective: remove the deterministic recovery rate condition in the Poisson case.

Ying Jiao Stein’s method and zero bias transformation for CDOs