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 L T = � n i = 1 N i ( 1 − R i ) 1 1 { τ i ≤ 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 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 � 1 − ρ 2 i Y i ≤ N − 1 ( α i ( t )) } where { τ i ≤ t } = { ρ i Y + 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 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 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 [ X 1 1 { X > x } ] /σ 2 . Observation of Stein (1972): if Z ∼ N ( 0 , σ 2 ) , then E [ Zf ( Z )] = σ 2 E [ 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 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 ) � � | X ∗ − X | ≤ σ 2 � f ′′ h � E . 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. 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 , � � � � 1 g ( X ∗ − X ) X s G ( X s ) = E 2 σ 2 E � x 0 g ( t ) dt , X s = X − � X and � where G ( x ) = X is an independent copy of X . � | X s | 3 � � � 1 �� In particular, E [ | X ∗ − X | ] = 1 ∼ O 4 σ 2 E . √ n � | X s | k + 2 � � � �� E [ | X ∗ − X | k ] = 1 1 ∼ O 2 ( k + 1 ) σ 2 E . √ n k Ying Jiao Stein’s method and zero bias transformation for CDOs
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 ) = σ 2 W < ∞ . Then W ∗ = W ( I ) + X ∗ I , W . W ( i ) = W − X i . Furthermore, W ( i ) , where P ( I = i ) = σ 2 i /σ 2 X i , X ∗ i and I are mutually independent. Here W and W ∗ are not independent. � | W ∗ − W | k � � n � i | k + 2 � � � �� 1 | X s 1 = ∼ O . E i = 1 E √ n k 2 ( k + 1 ) σ 2 W 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 [ X I | X 1 , · · · , X n ] = � n σ 2 W X i , then, i i = 1 σ 2 E [ E [ X I | X 1 , · · · , X n ] 2 ] = � n σ 6 W is of order O ( 1 n 2 ) . i i = 1 σ 4 I ] is of order O ( 1 However, E [ X 2 n ) . This technic is crucial for estimating E [ g ( X I , X ∗ I )] and E [ f ( W ) g ( X I , X ∗ I )] . Ying Jiao Stein’s method and zero bias transformation for CDOs
First order Gaussian correction Theorem Normal approximation Φ σ W ( h ) of E [ h ( W )] has corrector : �� x 2 � n � � 1 E [ X 3 C h = i ]Φ σ W − 1 xh ( x ) (3) 2 σ 4 3 σ 2 W W i = 1 where h has bounded second order derivative. The corrected approximation error is bounded by � � � � � E [ h ( W )] − Φ σ W ( h ) − C h � ≤ α ( h , X 1 , · · · , X n ) where α ( h , X 1 , · · · , X n ) depends on � h ′′ � and moments of X i 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 C h = 0 for any h . C h 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 )] − Φ σ W ( h ) = σ 2 W E [ f ′′ h ( W )( X ∗ I − X I )] � ξ ( W ∗ − W ) 2 � � ξ W + ( 1 − ξ ) W ∗ � f ( 3 ) + σ 2 W E . h by independence, E [ f ′′ h ( W ) X ∗ I ] = E [ X ∗ I ] E [ f ′′ h ( W )] = � � E [ X ∗ I ]Φ σ W ( f ′′ h ) + E [ X ∗ E [ f ′′ h ( W )] − Φ σ W ( f ′′ I ] h ) . use the conditional expectation technic � � E [ f ′′ f ′′ h ( W ) X I ] = cov h ( W ) , E [ X I | X 1 , · · · , X n ] . write Φ σ W ( f ′′ h ) in term of h and obtain � � �� � � x 2 C h = σ 2 W E [ X ∗ f ′′ 1 W E [ X ∗ I ]Φ σ W = I ]Φ σ W W − 1 xh ( x ) . h σ 2 3 σ 2 Ying Jiao Stein’s method and zero bias transformation for CDOs
Function “call” Call function C k ( 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 ) + ] − Φ σ W (( x − k ) + ) − 1 � ∼ O ( 1 3 E [ X ∗ I ] k φ σ W ( k ) 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 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 [Λ g (Λ)] = λ E [ 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 ) . Ying Jiao Stein’s method and zero bias transformation for CDOs
Poisson correction Poisson corrector of P λ W ( h ) for E [ h ( W )] : � � � � h = λ W C P ∆ 2 h X ∗ 2 P λ W I − X I E 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 ) = 1 1 { x = k − 1 } , then σ 2 W − λ W 2 ( ⌊ k ⌋ − 1 )! e − λ W λ ⌊ k ⌋− 1 C P h = . W Ying Jiao Stein’s method and zero bias transformation for CDOs
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. Ying Jiao Stein’s method and zero bias transformation for CDOs
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 binomial normal normal with correction poisson 0.14 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 Ying Jiao Stein’s method and zero bias transformation for CDOs
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 binomial normal normal with correction poisson 0.2 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 Ying Jiao Stein’s method and zero bias transformation for CDOs
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