Estimating Joint Default Probabilities by Efficient Importance - PowerPoint PPT Presentation

Estimating Joint Default Probabilities by Efficient Importance Sampling Chuan-Hsiang Han Dept. of Quantitative Finance Natl. Tsing-Hua University, Taiwan WSAF, Hong Kong July 3, 2009

Outline • Credit Risk Modeling: Classical Models • Monte Carlo Simulations, Importance Sam- pling, and Large Deviations • Homogenization by Singular Perturbation and Effect of (Stochastic) Correlation 1

Motivation I: Loss Density Function A loss random variable is defined by N � L ( T ) = c i I ( τ i ≤ T ) , i =1 so that, say the VaR, can be calculated from solving P ( L ( T ) > VaR α ) = α. 2

Motivation II: Evaluation of Credit Swaps I E { (1 − R ) × B (0 , τ ) × I ( τ < T ) } premium = �� N � j =1 △ j − 1 , j × B (0 , t j ) × I ( τ > t j ) I E Notations: τ : default time, R : recovery rate, B (0 , t ): discount factor, △ j − 1 , j : time incre- ment. CDS: τ is the time to default of an asset. CDO: τ is often an order statistics of τ 1 , τ 2 , · · · , τ n . 3

Modeling Default Times • Intensity-Based (Reduced Form) View firm’s default as extraneous and model the hazard rate of the firm. � t � � P ( τ ≤ t ) = F ( t ) = 1 − exp 0 h ( s ) ds I − . • Asset Value-Based (Structural Form) Firm assets follow correlated geometric Brow- nian motions: dS i t = µ i S i t dt + σ i S i t dW i t , d < W i , W j > = ρ ij dt. Joint default event: Π n i =1 I ( S i T ≤ B i ) (Mer- ton’s Model). 4

Then we ask a basic question � � Π n JDP = E i =1 I ( τ i ≤ T ) ? And hope this leads to the estimation of (1) P ( L ( T ) = i ) = p i , (2) P ( τ ≤ T ) , where τ is the k-th oder statis- tics of { τ i } n i =1 . JDP = joint default probability 5

Correlation under Reduced Form Model: Copula Method ∗ � n � τ i = F − 1 1. Default Time: ( U i ) i =1 , U ’s i are [0,1]-uniform random variables. 2. Copula is a distribution function on [0 , 1] n with uniform marginal distributions. 3. Through a copula function , one can build up correlations between default times. ∗ Cherubini, Luciano, Vecchiato (2004), Nel- son(2006). 6

Characterization of Default Events Gaussian Copula Factor Model (Laurent and Gregory (2003)) { τ i = F − 1 (Φ( W i )) ≤ T } i � � � 1 − ρ 2 i Z i ≤ Φ − 1 ( F i ( T )) = W i = ρ i Z 0 +   � Φ − 1 ( F i ( T )) − 1 − ρ 2 i z i     =  when Z i = z i  Z 0 ≤ ρ i      Z i ≤ Φ − 1 ( F i ( T )) − ρ i z 0     =  when Z 0 = z 0 . � 1 − ρ 2   i 7

(Conditional) Importance Sampling �� n � Estimate the JDP I E i =1 I ( τ i ≤ T ) by (1) Condition on common factor � � �� i =1 I ( Z i ≤ c i − ρ i Z 0 E ( u 1 , ··· ,u n ) Π n )Π n ˜ i =1 L ( Z i ; u i ) | Z 0 I E I � 1 − ρ 2 i (2) Condition on marginal factors � � � � 1 − ρ 2 � �� Z 0 ≤ c i − i Z i Π n ˜ I E I E L ( Z 0 ; u ) | Z 1 , · · · , Z n i =1 I ρ i (3) Direct Change of Measure �� n � i =1 I ( W i ≤ c i )Π n ˜ i =1 L ( W i ; w i ) I E , where c i = Φ − 1 ( F i ( T )) and L ( · , · ) the Likeli- hood ratio. 8

Comparison of Variance Reduction 9

Asymptotic Optimality of Direct Change of Measure Let W be a centered multivariate normal JDP = E { I ( W < c ) } = E µ { I ( W < c )Π n i =1 L ( W i ; µ i ) } , where L ( w ; µ ) = e µ 2 / 2 e − µw is the likelihood function. Theorem The variance of I { W<c } Π n i =1 L ( W i ; µ i ) is optimally minimized at µ = c when each component in the vector − c is sufficiently large. 10

More Applications • Computation of tail probability for multi- variate normals. (versus a Matlab program mvncdf.m , based on Genz and Bretz (’99)) • Connection to Merton’s structural-form model model in high dimension to estimate E { Π n I i =1 I ( S iT ≤ B i ) } • Extend to Student-t copula (Conditional IS) (see also mvtcdf.m ) 11

Tail Probability Estimation: N=5, ρ =0.5 Multivariate Normal IS versus QMC D -log (P_MC) -log (P_IS) -log(P_MATLAB) 30.0 0 1.05 1.05 1.04 -0.5 1.64 1.67 1.68 -1 2.55 2.51 2.52 22.5 -1.5 3.70 3.59 3.57 -2 4.87 4.85 -log(P,10) 15.0 -2.5 6.34 6.37 -3 8.11 8.13 -3.5 10.14 10.14 7.5 -4 12.43 12.40 -4.5 14.88 14.92 -5 17.65 17.68 0 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -5.5 -6 -5.5 20.49 20.71 D: Default Level -6 23.99 23.98 -log (P_MC) -log (P_IS) -log(P_MATLAB) 12

Tail Probability Estimation: N=5, ρ =0.5 d.f.=10 (Multivariate Student-T) IS versus QMC D -log (P_MC) -log (P_IS) -log (P_MATLAB) 10.0 0.0 1.046 1.041 1.04 -0.5 1.622 1.644 1.636 7.5 -1.0 2.330 2.335 2.317 -log(P,10) -1.5 3.018 3.082 3.021 5.0 -2.0 3.553 3.676 3.717 -2.5 4.398 5.102 4.358 2.5 -3.0 4.398 4.866 4.914 文字 -3.5 5.41 5.43 0 -4.0 5.712 5.836 0.0 -0.5 -1.5 -2.5 -3.5 -4.5 -5.5 D: Default Level -4.5 6.693 6.236 -5.0 7.650 6.907 -5.5 7.076 8.038 -6.0 8.166 7.077 -log (P_MC) -log (P_IS) -log (P_MATLAB) 13

Credit Risk Modeling: Structural Form Approach Multi-Names Dynamics: for 1 ≤ i ≤ n dS it = µ i S it dt + σ i S it dW it , � � d W it , W jt = ρ ij dt. Each default time τ i for the i th name is de- fined as τ i = inf { t ≥ 0 : S it ≤ B i } , where B i denotes the i th debt level. The i th default event is defined as { τ i ≤ T } . 14

Joint Default Probability: First Passage Time Problem in High Dim. � � Π n Q: How to compute JDP = I E i =1 I ( τ i ≤ T ) under structural-form models? Explicit Formulas exist for 1-name case (Black and Cox ’76) and 2-name case (Zhou ’01). Note that Carmona, Fouque, and Vestal (’08) dealed with a similar problem by means of Interacting Particle Systems. 15

Multi-Dimensional Girsanov Theorem Given the Radon-Nikodym derivative �� T � T � W s − 1 0 || h ( s,S s ) || 2 ds dI P 0 h ( s,S s ) · d ˜ P = Q h 2 T = e , d ˜ I � t ˜ W t = W t + 0 h ( s, S s ) ds is a vector of Brown- ian motions under ˜ I P . Thus � Π n i =1 I ( τ i ≤ T ) Q h � DP = ˜ I E . T 16

Monte Carlo Simulations: Importance Sampling An importance sampling method is devel- oped to satisfy ˜ E { S iT |F 0 } = B i , i = 1 , · · · , n. I The new measure is characterized by solving j =1 ρ ij h j = µ i σ i − ln B i /S i 0 the linear system Σ i σ i T so that by Girsanov Theorem E { Π n JDP = ˜ I i =1 I ( τ i ≤ T ) Q T } . 17

Trajectories under different measures Single Name Case 18

Single Name Default Probability B BMC Exact Sol Importance Sampling 50 0.0886 (0.0028) 0.0945 0.0890 (0.0016) 7 . 7 ∗ 10 − 5 7 . 2 ∗ 10 − 5 (2 . 3 ∗ 10 − 6 ) 20 0 (0) 1 . 3 ∗ 10 − 30 1 . 8 ∗ 10 − 30 (3 . 4 ∗ 10 − 31 ) 1 0 (0) The number of simulations is 10 4 and the Euler discretization takes time step size T/ 400, where T is one year. Other parameters are S 0 = 100 , µ = 0 . 05 and σ = 0 . 4 . Standard errors are shown in parenthesis. 19

Asymptotic Optimality Efficient Importance Sampling Assume ln ( B i /S i 0 ) = − 1 for each 1 ≤ i ≤ N . ε Let the JDP, P ε , and the second moment, M 2 ε under a new measure, are defined by � � �� Π N P ε = I E 0 ≤ t ≤ T S it ≤ B i inf i =1 I � � � � Π N ˜ M 2 ε = I E 0 ≤ t ≤ T S it ≤ B i inf Q T i =1 I Theorem: By M 2 ε ≈ ( P ε ) 2 for small ε (spa- tial scale) we observe the optimality of cho- sen measure. 20

Sketch of Proof: Single-Name Default Prob. Approximation   µ − σ 2 � � t + σW t ≤ B  inf 2 0 ≤ t ≤ T S t = S 0 e I P  ( scaling in space by ln ( B/S 0 ) = − 1 ε ) µ − σ 2 � � � � �� = I E 0 ≤ t ≤ T ε inf t + εσW t ≤ − 1 I 2 := P ε − 1 � � ≈ exp . ( by Freidlin-WentzellThm ) ε 2 2 σ 2 T 21

Importance Sampling: 2 nd Moment Approximation � � � � W T − h 2 T e 2 h ˜ ˜ inf I E I 0 ≤ t ≤ T S t ≤ B µ − σ 2 � � t + σ ˜ W t , h = µ σ − ln B/S 0 2 − σh S t = S 0 e σ T     µ − σ 2 � � t + σ ˆ 2 + σh W t ≤ B  e h 2 T  inf ˆ = I E 0 ≤ t ≤ T S 0 e  I  22

2 nd Moment Approximation (Cont.) 2 µ − σ 2 � � � � � � �� + 1 ˆ t + εσ ˆ = I E inf ε W t ≤ − 1 I 2 T 0 ≤ t ≤ T � 2 � σ + 1 r ( scaling by ln ( B/S 0 ) = − 1 T εσT ε ) × e := M 2 ε − 1 � � ≈ exp . ( by F-W Thm ) ε 2 σ 2 T Observe that M 2 ε ≈ ( P ε ) 2 when ε is small. 23

Three-Names Joint Default Probability BMC Importance Sampling ρ 0 . 0049(6 . 98 ∗ 10 − 4 ) 0 . 0057(1 . 95 ∗ 10 − 4 ) 0.3 3 . 00 ∗ 10 − 4 (1 . 73 ∗ 10 − 4 ) 6 . 40 ∗ 10 − 4 (6 . 99 ∗ 10 − 5 ) 0 2 . 25 ∗ 10 − 5 (1 . 13 ∗ 10 − 5 ) -0.3 0(0) Parameters are S 10 = S 20 = S 30 = 100 , µ 1 = µ 2 = µ 3 = 0 . 05, σ 1 = σ 2 = 0 . 4 , σ 3 = 0 . 3 and B 1 = B 2 = 50 , B 3 = 60. Effect of Correlations! Debt to Asset-Value Ratios ( B i /S i 0 ) are not small. 24

Loss Density Function: 25 Names Diffusion Model Note: Consider both survival and default prob- abilities. Applications: Pricing CDOs, Risk Manage- ment of credit portfolios, etc. 25