1 Exact and Efficient Simulation of Correlated Defaults Kay Giesecke Management Science & Engineering Stanford University giesecke@stanford.edu www.stanford.edu/ ∼ giesecke Joint work with H. Takada, H. Kakavand, and M. Mousavi Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 2 Corporate defaults cluster Joint work with F. Longstaff, S. Schaefer and I. Strebulaev 18 16 14 Value−Weighted Default Rate (Percent) 12 10 8 6 4 2 0 1880 1900 1920 1940 1960 1980 2000 Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 3 Correlated default risk Important applications • Risk management of credit portfolios – Prediction of correlated defaults and losses – Portfolio risk measures: VaR etc. • Optimization of credit portfolios • Risk analysis, valuation, and hedging of portfolio credit derivatives – Collateralized debt obligations (CDOs) Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 4 Default timing • Consider a portfolio of n defaultable assets – Default stopping times τ i relative to (Ω , F , P ) and F t = I ( τ i ≤ t ) – Default indicators N i – Vector of default indicators N = ( N 1 , . . . , N n ) • The portfolio default process 1 n · N counts defaults – At the center of many applications Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 5 Bottom-up model of default timing • Name i defaults at intensity λ i � · – A martingale is given by N i − 0 (1 − N i s ) λ i s ds – λ i represents the conditional default rate: for small ∆ > 0 λ i t ∆ ≈ P ( i defaults during ( t, t + ∆] | F t ) • The vector process λ = ( λ 1 , . . . , λ n ) is the modeling primitive – Component processes are correlated: diffusion, common or correlated or feedback jumps – Large literature Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 6 Model computation • We require E ( f ( N T )) for T > 0 and real-valued f on { 0 , 1 } n – P (1 n · N T = k ) – P ( τ i > t ) for constituents i • Semi-analytical transform techniques – Limited to (one-) factor doubly-stochastic intensity models • Monte Carlo simulation – Larger class of intensity models – Treatment of more complex instruments such as cash CDOs Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 7 Simulation by time-scaling • Widely used � t – τ i has the same distribution as inf { t : 0 λ i s ds = Exp (1) } – In practice: approximate λ i on discrete-time grid, integrate, and record the hitting time of the integrated process • Potential problems – Discretization may introduce bias ∗ Magnitude? ∗ Computational effort ∗ Allocation of resources – Can be computationally burdensome (often n ≥ 100 ) Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 8 Time-scaling vs. exact methods Distribution of 1 100 · N 2 0.14 Time−Scaling Exact 0.12 0.1 0.08 Probability 0.06 0.04 0.02 0 0 5 10 15 20 Number of Defaults Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 9 Exact and efficient simulation • Our approach has two parts 1. Construct a time-inhomogeneous, continuous-time Markov chain M ∈ { 0 , 1 } n with the property that M t = N t in law 2. Estimate E ( f ( N T )) = E ( f ( M T )) by simulating M – Exact : avoids intensity discretization – Efficient : adaptive variance reduction scheme • Powerful simulation engine applicable to many intensity models in the literature Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 10 Multivariate Markovian projection Proposition • Let M be a Markov chain that takes values in { 0 , 1 } n , starts at 0 n , has no joint transitions in any of its components and whose i th component has transition rate h i ( · , M ) where t I ( τ i > t ) | N t = B ) , h i ( t, B ) = E ( λ i B ∈ { 0 , 1 } n Then M t = N t in distribution: B ∈ { 0 , 1 } n P ( M t = B ) = P ( N t = B ) , • Related univariate results in Br´ emaud (1980), Arnsdorf & Halperin (2007), Cont & Minca (2008), Lopatin & Misirpashaev (2007) Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 11 Markov counting process • M is a Markov point process in its own filtration G • The Markov counting process 1 n · M has G -intensity n − 1 � 1 n · h ( t, M t ) = H ( t, k ) I ( T k ≤ t < T k +1 ) k =0 where h = ( h 1 , . . . , h n ) , and ( T k ) is the strictly increasing sequence of event times of 1 n · M , and H ( t, k ) = 1 n · h ( t, M T k ) , t ≥ T k • Compare: original portfolio default process 1 n · N has F -intensity n t I ( τ i > t ) � λ i i =1 Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 12 Markov counting process • The G inter-arrival intensities H ( t, k ) of the Markov counting process 1 n · M are deterministic • Exact simulation of arrival times of 1 n · M – Time-scaling method based on H ( t, k ) – Equivalently, inverse method based on � � T k + s � P ( T k +1 − T k > s | G T k ) = exp H ( t, k ) dt T k – Sequential acceptance/rejection based on H ( t, k ) • Exact simulation of the component I k ∈ { 1 , 2 , . . . , n } of M in which the k th transition took place: P ( I k = i | G T k − ) = h i ( T k , M T k − 1 ) H ( T k , k − 1) Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 13 Variance reduction • Interested in P (1 n · N T = k ) for large k – Need to force mimicking chain M into rare-event regime • Selection/mutation scheme – Evolve R copies ( V r p ) of M over grid p = 0 , 1 , . . . , m under P – At each p , select R particles by sampling with replacement 1 δ 1 n · ( V r p − V r � � P ( particle r selected ) = exp p − 1 ) Rη p � R where η p = 1 r =1 exp[ δ 1 n · ( V r p − V r p − 1 )] and δ > 0 R – Final estimator of P (1 n · N T = k ) corrects for selections R η 0 · · · η m − 1 � I (1 n · V r m = k ) exp ( − δ 1 n · V r m ) R r =1 Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 14 Selection/mutation scheme • The selection mechanism adaptively forces the mimicking Markov chain M into the rare-event regime – Del Moral & Garnier (2005, AAP) – Carmona & Crepey (2009, IJTAF) – Carmona, Fouque & Vestal (2009, FS) – Twisting of Feynman-Kac path measures – Well-suited to deal with different model specifications • Mutations are generated under the reference measure P via the exact A/R scheme • Estimators are unbiased • Choice of R , m and δ Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 15 Calculating the projection t I ( τ i > t ) | N t = B ) for given ( λ 1 , . . . , λ n ) • Need h i ( t, B ) = E ( λ i • We show how to calculate h i ( t, B ) for a range of t + α i · Y t – Multi-factor doubly-stochastic models λ i t = X i t + E ( α i · Y t | F t ) – Multi-factor frailty models λ i t = X i – Self-exciting models λ i t = X i t + c i ( t, N t ) in terms of the transform � t � � �� Z ∈ { X i , Y } φ ( t, u, z, Z ) = E exp − u Z s ds − zZ t , 0 • This extends the reach of our exact method to most models in the literature, and beyond Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 16 Numerical results Self-exciting intensity model for n = 100 • Suppose the intensities λ i = X i + � n j � = i β ij N j – Extends Jarrow & Yu (2001), Kusuoka (1999), Yu (2007) – Feedback specification can be varied – Analytical solutions not known • Suppose the idiosyncratic factor follows the Feller diffusion � dX i t = κ i ( θ i − X i t dW i X i t ) dt + σ i t where ( W 1 , . . . , W n ) is a standard Brownian motion • Parameters selected randomly (relatively high credit quality) Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 17 Numerical results Projection for self-exciting intensity model • The projected intensity is given by t I ( τ i > t ) | N t = B ) h i ( t, B ) = E ( λ i n − ∂ z φ ( t, 1 , z, X i ) | z =0 � � � = (1 − B i ) β ij B j + φ ( t, 1 , 0 , X i ) j � = i • The transform φ ( t, u, z, X i ) is in closed form, and so is h i ( t, B ) – Can add compound Poisson jumps without reducing tractability – General affine jump diffusion dynamics Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 18 Numerical results Simulation results for E (( C 1 − 3) + ) where C 1 = N 1 · 1 n Method Trials Steps Bias SE RMSE Time Exact 5,000 N/A 0 0.0239 0.0239 0.10 min 7,500 N/A 0 0.0193 0.0193 0.15 10,000 N/A 0 0.0165 0.0165 0.20 50,000 N/A 0 0.0073 0.0073 1.69 100,000 N/A 0 0.0052 0.0052 5.51 1,000,000 N/A 0 0.0016 0.0016 463.78 Time 5,000 71 0.0735 0.0246 0.0775 1842.15 Scaling 7,500 87 0.0697 0.0199 0.0725 2628.33 10,000 100 0.0174 0.0171 0.0244 3255.12 Kay Giesecke
Exact and Efficient Simulation of Correlated Defaults 19 Numerical results Convergence of RMS errors 0.08 Exact Time−scaling 0.07 0.06 0.05 RMSE 0.04 0.03 0.02 0.01 0 −1 0 1 2 3 4 5 10 10 10 10 10 10 10 Total simulation time (minutes) Kay Giesecke
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