Stein s method and zero bias transformation: Application to CDO pricing ESILV and Ecole Polytechnique Joint work with N. El Karoui
Introduction CDO a portfolio credit derivative containing 100 underlying names susceptible to default risk. Key term to calculate: the cumulative loss L T = n i=1 N i(1 R i )11 {τi T } where N i is the nominal of each name and R i is the recovery rate. Pricing of one CDO tranche: call function E[(L T K ) + ] with attachment or detachment point K. Motivation: fast and precise numerical method for CDOs in the market-adopted framework
Factor models Standard model in the market: factor model with conditionally independent defaults. Normal factor model: Gaussian copula approach {τ i t} = {ρ i Y + 1 ρ 2 i Y i N 1 (α i (t))} where Y i, Y N(0, 1) and independent, and α i (t) is the average probability of default before t. Given Y (not necessarily normal), L T is written as sum of independent random variables L T follows binomial law for homogeneous portfolios Central limit theorems: Gaussian and Poisson approximations
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
Stein method and Zero bias transformation
Zero bias transformation Goldstein-Reinert, 1997 -X a r.v. of mean zero and of variance σ 2 -X has the zero-bias distribution of X if for all regular enough f, E[Xf (X)] = σ 2 E[f (X )] (1) -distribution of X unique with density p X (x) = E[X11 {X>x} ]/σ 2. Observation of Stein (1972): if Z N(0, σ 2 ), then E[Zf (Z )] = σ 2 E[f (Z )].
Stein method and normal approximation Stein s equation (1972) For the normal approximation of E[h(X)], Stein s idea is to use f h defined as the solution of h(x) Φ σ (h) = xf (x) σ 2 f (x) (2) where Φ σ (h) = E[h(Z )] with Z N(0, σ 2 ). Proposition (Stein): If h is absolutely continuous, then E[h(X)] Φ σ (h) = σ 2 E [ f h (X ) f h (X)] σ 2 f 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 f h and its derivatives.
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 2σ 2 E[ X s G(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 E [ X s 3] ( O ( )) n 1 4σ. 2 E[ X X k 1 ] = E [ X s k+2] ( O ( )) 1 2(k+1)σ 2 k. n
Sum of independent random variables Goldstein-Reinert, 97 Let W = X 1 + + X n where X i are independent r.v. of mean zero and Var(W ) = σw 2 <. Then W = W (I) + X I, where P(I = i) = σi 2 /σw 2. W (i) = W X i. Furthermore, W (i), X i, Xi and I are mutually independent. Here W and W are not independent. E [ W W k] 1 = n 2(k+1)σW 2 i=1 E[ Xi s k+2] ( O ( )) 1 k. n
Conditional expectation technic To obtain an addition order in the estimations, for example, E[X I X 1,, X n ] = n σi 2 i=1 X i, then, σ 2 W E[E[X I X 1,, X n ] 2 ] = n i=1 σ 6 i σ 4 W is of order O( 1 n 2 ). However, E[XI 2] is of order O( 1 n ). This technic is crucial for estimating E[g(X I, XI )] and E[f (W )g(x I, XI )].
First order Gaussian correction Theorem Normal approximation Φ σw (h) of E[h(W )] has corrector : ( ) C h = 1 n ( x 2σW 4 E[Xi 3 2 ) ]Φ σw 3σW 2 1 xh(x) i=1 (3) where h has bounded second order derivative. The corrected approximation error is bounded by E[h(W )] Φ σw (h) C h α(h, X1,, X n ) where α(h, X 1,, X n ) depends on h and moments of X i up to fourth order.
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 C h = 0 for any h. C h can be viewed as an asymmetric corrector. Skew play an important role.
Ingredients of proof development around the total loss W E[h(W )] Φ σw (h) = σw 2 [ + σw 2 E E[f h (W )(XI X I )] ( ξw + (1 ξ)w ) ξ(w W ) 2]. f (3) h by independence, E[f h (W )X I ] = E[X I ]E[f h (W )] = E[XI ]Φ σ W (f h ) + E[X I ]( E[f h (W )] Φ σ W (f h )). use the conditional expectation technic E[f h (W )X I] = cov ( f h (W ), E[X I X 1,, X n ] ). write Φ σw (f h ) in term of ) h and obtain C h = σw 2 E[X I ]Φ σ W ( f h (( = 1 E[X σw 2 I ]Φ x 2 σ W 3σW 2 ) ) 1 xh(x).
Function call Call function C k (x) = (x k) + : absolutely continuous with C k (x) = 11 {x>k}, but no second order derivative. Hypothesis not satisfied. Same corrector E[(W k) + ] Φ σw ((x k) + ) 1 3 E[X I ]kφ σ W (k) O( 1 n ). Error bounds depend on f C k, xf C k. Ingredients of proof: write f C k as more regular function by Stein s equation Concentration inequality (Chen-Shao, 2001) Corrector C h = 0 when k = 0, extremal values when k = ±σ 2
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)] := A P g(λ). Poisson zero bias transformation X for X with E[X] = λ: E[Xg(X)] = E[A P g(x )] Stein s equation : h(x) P λ (h) = xg(x) A P g(x) (4) where P λ (h) = E[h(Λ)] with Λ P(λ).
Poisson correction Poisson corrector of P λw (h) for E[h(W )] : C P h = λ W 2 P λ W ( 2 h ) E [ X I X I ] where h(x) = h(x + 1) h(x) and P(I = i) = λ i /λ W. Proof: combinatory technics for integer valued r.v. For h = (x k) +, 2 h(x) = 11 {x=k 1}, then C P h = σ2 W λ W 2( k 1)! e λ W λ k 1 W.
Numerical tests: E[(W n k) + ] E[(W n k) + ] in function of n σ W = 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.
Numerical tests: E[(W n k) + ] Legend : binomial (black), normal (magenta), corrected normal (green), Poisson (blue) and corrected Poisson (red). p = 0.1, n = 100. 0.15 0.14 binomial normal normal with correction poisson poisson with correction 0.13 0.12 0.11 0.1 0.09 0.08 0 50 100 150 200 250 300 350 400 450 500 n
Numerical tests : E[(W n k) + ] Legend : binomial (black), normal (magenta), corrected normal (green), Poisson (blue) et corrected Poisson (red). p = 0.01, n = 100. 0.22 0.2 binomial normal normal with correction poisson poisson with correction 0.18 0.16 0.14 0.12 0.1 0.08 0 50 100 150 200 250 300 350 400 450 500 n
Higher order corrections The regularity of h plays an important role. Second order approximation is given by C(2, h) = Φ σw (h) + C h + Φ σw ( ( x 6 18σ 8 W + 1 2 Φ σ W ( ( x 4 4σ 6 W 5x 4 6σW 6 + 5x 2 2σW 4 3x 2 ) 2σW 4 h(x) ) ) (h(x) Φ σw (h)) E[XI ]2 ) (E[(X I )2 ] E[X 2 I ]). Since the call function is not regular enough, it s difficult to control errors. (5)
Applications to CDOs In collaboration with David KURTZ (Calyon, Londres).
Normalization for loss Percentage loss l T = 1 n n i=1 (1 R i)11 {τi T } K = 3%, 6%, 9%, 12%. Normalization for ξ i = 11 {τi T } : let X i = ξ i p i where npi (1 p i ) p i = P(ξ i = 1). Then E[X i ] = 0 and Var[X i ] = 1 n. Suppose R i = 0 and p i = p for simplicity. Let p(y ) = p i (Y ) = P(τ i T Y ). Then E [ (l T K ) + Y ] p(y )(1 p(y )) = E [ (W k(n, p(y ))) + Y ]. n where W = n i=1 X i and k(n, p) = (K p) n. p(1 p) Constraint in Poisson case: R i deterministic and proportional.
Numerical tests: domain of validity Conditional error of E[(l T K ) + ], in function of K /p, n = 100. Inhomogeneous portfolio with log-normal p i of mean p.
Numerical tests : Domain of validity Legend : corrected Gaussian error (blue), corrected Poisson error (red). np=5 and np=20. np=5 0.020% 0.015% 0.010% 0.005% 0.000% -0.005% 0% 100% 200% 300% -0.010% Error Gauss Error Poisson np=20 0.012% 0.010% 0.008% 0.006% 0.004% 0.002% 0.000% -0.002% 0% 100% 200% 300% -0.004% -0.006% -0.008% Error Gauss Error Poisson Domain of validity : np 15 Maximal error : when k near the average loss; overlapping area 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=5 and np=20. np=5 np=20 0.040% 0.030% 0.020% 0.010% 0.000% 0% 100% 200% 300% -0.010% -0.020% -0.030% 0.025% 0.020% 0.015% 0.010% 0.005% 0.000% -0.005% 0% 100% 200% 300% -0.010% -0.015% -0.020% -0.040% Gauss Saddle 1 Saddle 2-0.025% Gauss Saddle 1 Saddle 2
Numerical tests : stochastic R i Conditional error of E[(l T K ) + ], in function of K /p for Gaussian approximation error with stochastic R i. R i : Beta distribution with expectation 50% and variance 26%, independent (Moody s). Corrector depends on the third order moment of R i. Confidence interval 95% by 10 6 Monte Carlo simulation.
Numerical tests : stochastic R i Gaussian approximation better when p is large as in the standard case. np=5 and np=20. np=5 0.004% 0.003% 0.002% 0.001% 0.000% -0.001% 0% 100% 200% 300% -0.002% -0.003% -0.004% -0.005% Lower MC Bound Gauss Error Upper MC Bound np=20 0.006% 0.004% 0.002% 0.000% -0.002% 0% 100% 200% -0.004% -0.006% 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, not only in the normal factor model case Numerical 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-Poisson approximation with realistic local correlation. Expected loss 4% 5% Break Even Error 1.40 1.20 1.00 0.80 0.60 0.40 0.20 - -0.20 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 rate condition in the Poisson case.