Portfolio Distribution Modelling and Computation. Harry Zheng Department of Mathematics Imperial College h.zheng@imperial.ac.uk

Portfolio Distribution Modelling and Computation Harry Zheng Department of Mathematics Imperial College h.zheng@imperial.ac.uk Workshop on Fast Financial Algorithms Tanaka Business School Imperial College July 4, 2007 1

Sum of Stochastic Processes Let X 1,..., X n be some stochastic processes defined on some filtered probability space (Ω, F, F {0 t T }, P ). Define L(t) = X 1 (t) + + X n (t). Objective: find the probability distribution function of L(t) for 0 t T. In this talk we assume X i (t) default indicator process, i.e., X i (t) = 1 {τi t} where τ i is default time of name i. If t = T is fixed then L = X 1 + + X n where X 1,..., X n are Bernoulli random variables. 2

Applications Basket CDS (credit default swaps): L(t) is number of defaults by time t. Let {τ k } be ordered default times. Then the distribution of kth default time τ k is given by P (τ k t) = P (L(t) k). Expectation of k default contingent leg equals E[e rτ k 1 {τ k T } ]. CDO (collateralized debt obligations): L(t) is total portfolio loss. Expected tranche loss L k (t) at t is defined by L k (t) = E [min{max{l(t) A k, 0}, B k A k }] VaR (minimum loss) and CVaR (average loss): (α confidence level) VaR α (t) = inf{x : P (L(t) x) α}. CVaR α (t) = E [ L(t) L(t) VaR α (t) ]. 3

Correlation Modeling Factor model (Duffie-Singleton (1991), Lando (1998)). Defaults of individual names depend on some common state variables, given realization of state variables, defaults are independent to each other. Infectious model (Davis-Lo (2001), Jarrow-Yu (2001)). Default of one name may increase probabilities of other names to default. Copula model (Embrechts et al (1999, 2001)). Joint distribution is characterized by a copula function and marginal distributions, i.e., F (x 1,..., x n ) = C(F 1 (x 1 ),..., F n (x n )). Common copulas are Gaussian, t, exponential, Gumbel, Clayton, Frank copulas. Gaussian copula is used in CreditMetrics. Other models: particle system (Giesecke-Weber (2003)), queuing network (Davis-Rodriguez (2004)), etc. 4

Default indicator and latent variable: Homogeneous CreditMetrics X i = 1 Y i c and Y i = ρz + 1 ρ 2 ɛ i where c := Φ 1 (p) default threshold, ρ correlation factor, p unconditional default probability, and Z, ɛ 1,..., ɛ n independent N(0, 1) random variables. Unconditional default probability: P (X i = 1) = (Y i c) = Φ(c) = p. Conditional default probability: p(z) := P (Y i c Z = z) = Φ Φ 1 (p) ρz 1 ρ 2 Portfolio loss distribution: given Z, L = X 1 + + X n binomial variable, P (L = k) = E Z [P (L = k Z)] = E Z n k p(z) k (1 p(z)) n k. 5

Default indicator and latent variable: Heterogeneous Factor Model X i = 1 Y i c i and Y i = a i Z + b i ɛ i where a 1, b i, c i constant, Z, ɛ i independent with distributions F Z, F i. Monte Carlo method: easy. Conditional default probability: given Z = z, p i (z) := P (Y i c i Z = z) = F i Conditional loss distribution: given Z, P (L = k Z) = S =k p i(z) i S i S c i a i z. b i (1 p i (Z)) where S subset of {1,..., n} containing exactly k elements. Summation is taken over ( ) n k different ways in which numbers can be chosen. 6

MGF Method (Gregory-Laurent, 2003) Moment Generating Functions: Given Z, E[u L ] = n k=0 P (L = k)u k E[u L Z] = E[u X 1 Z] E[u X n Z] = n (1 p i(z) + p i (Z)u) i=1 Therefore, n P (L = k)u k = E[ n (1 p i(z) + p i (Z)u)] k=0 i=1 RHS is polynomial of degree n. Expand formally and compare coefficients to get probabilities P (L = k). 7

Reformulation: Recursive Method I (Hull-White, 2004) U k := P (L = k Z) = π(z) S =k c i(z) i S where π(z) = n (1 p i(z)), c i (Z) = p i(z) i=1 1 p i (Z) Recurrence relationship: define then U 1 = V 1 2U 2 = V 1 U 1 V 2. V k := n i=1 c i(z) k, ku k = V 1 U k 1 V 2 U k 2 + + ( 1) k V k 1 U 1 + ( 1) k+1 V k 8

Recursive Method II (Andersen-Sidenius-Basu, 2003) For j = 1,..., n, define random variables L j = X 1 + + X j Recursive relation: for i = 0, 1,..., n 1, build portfolio distribution incrementally from the relation L j+1 = L j + X j+1 as follows: P (L j+1 = k Z) = P (L j = k Z)P (X j+1 = 0 Z) +P (L j = k 1 Z)P (X j+1 = 1 Z) with starting point P (L 0 = k) = 1 {k=0}. We can find conditional probability P (L n = k Z) in n iterations and the whole conditional distribution in n 2 iterations. 9

Large Portfolio Approximations Homogeneous portfolio: When n Vasicek (1987) applied strong law of large numbers to get limit distribution P ( ˉL ˉx) = Φ Φ 1 (p) + 1 ρ 2 Φ 1 (ˉx) ρ where ˉL percentage of default names in portfolio and 0 ˉx 1. Limiting distribution is highly skewed with heavy tails. Heterogeneous portfolio. Glasserman (2003) used Lagendre-Fenchel transformation to get the approximation Φ 1 (p) + 1 ρ P (L x) Φ 2 Φ 1 (x/n) ρ where 0 x n. The approximation works well if underlying names are uniformly distributed, but suffers large errors for highly structured portfolios. 10

0th order (normal) approximation Edgeworth Expansion P (L x) E Z Φ x μ(z) σ(z) where μ(z) = n i=1 p i (Z) and σ 2 (Z) = n i=1 p i (Z)(1 p i (Z)). 1st order approximation where P (L x E Z Φ x μ(z) σ(z) + H x μ(z) σ(z) H(x) = 6 1 σ(z) 3 γ(z)(1 x 2 )φ(x) γ(z) = n i=1 p i(z)(1 p i (z))(1 2p i (z)). kth order expansion: expressed in Chebyshev-Hermite polynomials and moments of X i s up to order k + 2, and approximation error is o(n k/2 ). 11

Saddle Point Approximation Daniels (1987), Martin-Thompson-Browne (2001) Conditional moment generating function: MGF of L, given Z, is given by M L (s; Z) = E(e sl Z) = n i=1 M X i (s; Z). Conditional saddle point: Define ˆt solution to K (ˆt; Z) = t for all t, where K(s; Z) = log M L (s; Z) and ˆt depend on t and Z. Conditional density function: Saddle point approximation f(t; Z) = (2πK (ˆt; Z)) 1/2 exp(k(ˆt; Z) tˆt). P (L x) E Z [ x f(t; Z)dt]. Excellent approximation for tail distributions, but not for whole distributions, good in computing VaR, CVaR. 12

Glasserman (2003) Example Portfolio consists of two homogeneous subportfolios, each characterized by one-factor CreditMetrics, and homogeneous approximation is poor. Data: U 1 = 1, U 2 = 1, N 1 = 150, N 2 = 850, p 1 = 0.05, p 2 = 0.001, ρ 1 = 0.8, ρ 2 = 0.7. 1 Glasserman Example 0.1 MC CLT Probability 0.01 0.001 0.0001 0.00001 1 20 39 58 77 96 115 134 153 172 191 210 229 248 Total Loss 13

Dembo, Deuschel, Duffie (2004) Example 2 homogeneous subportfolios, c i exponential variables, Dembo et al used large deviations method. Data: K = 2, N 1 = 5000, N 2 = 5000, β 1 = 0.01, β 2 = 0.1, J = 2, p 1 = 0.7, p 2 = 0.3, p 1 (1) = 0.001, p 2 (1) = 0.004, p 1 (2) = 0.0015, and p 2 (2) = 0.1. 1 0.1 Probability 0.01 0.001 MC CLT 0.0001 AE 0.00001 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 Loss per Position 14

Comparison of Basket CDS Pricing λ ρ k 1 2 3 4 5 10 20 30 40 50 0.01 0.1 MC 3802 1944 1245 867 626 144 8 0 0 0 EE 4049 1996 1271 879 631 142 8 0 0 0 0.01 0.3 MC 1927 1116 788 603 480 203 55 18 6 2 EE 2075 1135 795 605 481 202 56 18 5 2 0.01 0.6 MC 817 551 434 364 315 188 95 55 33 20 EE 854 553 434 363 313 188 95 56 34 21 0.03 0.3 MC 4789 2942 2205 1784 1497 805 337 156 73 32 EE 4986 2958 2203 1776 1492 804 335 157 73 32 The comparison of the hybrid method and the Monte Carlo method for the kth-to-default swap rates (in basis point) of a homogeneous basket CDS, where maturity T = 5, names n = 100, payment interval Δ i = 0.25, recovery rate δ = 0.4, interest rate r = 0.05, default time distribution F (t) = 1 e λt, Z and ɛ i are independent standard normal variables. 15

Infectious Defaults (Davis-Lo, 2001) A firm may default directly or may default due to the default of some other firm and the infection. Let Z i = 1 if firm i defaults and 0 otherwise. Then Z i = X i + (1 X i ) where X i, Y ji are Bernoulli random variables. 1 (1 X j Y ji ) j i 16

Interacting Default Intensities (Jarrow-Yu, 2001) Upward (downward) jumps in default intensities of non-defaulted firms at the default time of one of default-correlated firms. Default times τ i associated with names i, i = 1,..., n, defined by τ i = inf{t > 0 : t 0 λ i(s)ds E i } where E i are independent standard exponential random variables. The default intensity processes λ i (t) are given by (Jarrow and Yu (2001)) λ i (t) = a i0 + j i a ij 1 {τj t} where a ij are constants. The joint distribution of the default times is defined recursively through each other, a looping default phenomenon. Yu (2004) uses total hazard construction to get joint distribution of default times of 3 names and suggests simulation method for pricing basket CDSs. 17

Markovian Approach I (Frey-Backhaus, 2004) Denote X(t) = (X 1 (t),..., X n (t)) default indicator process, taking values in state space E = {0, 1} n, noting E = 2 n. Transition probabilities p(t, s, x, y) for 0 t s and x, y E satisfy backward Kolmogorov equations (ODEs) p(t, s, x, y) + Gp(t, s, x, y) = 0, p(s, s, x, y) = 1 {x=y} t and forward Kolmogorov equations (ODEs) p(t, s, x, y) = G p(t, s, x, y) = 0, p(t, t, x, y) = 1 {y=x} s where G infinitesimal generator and G adjoint operator. Main challenge is to solve Kolmogorov equations with dimension 2 n, although for homogeneous portfolio the dimension can be reduced to n. 18

Markovian Approach II (Herbertsson-Rootzen, 2006) Arrange elements of E in increasing order wrt number of defaults k to form an upper triangular generating matrix Q (dimension E E and very sparse). E.g. n = 2 and E arranged as (0, 1), (1, 0), (0, 1), (1, 1) then Q = (a 10 + a 20 ) a 10 a 20 (a 20 + a 21 ) a 20 + a 21 (a 10 + a 12 ) a 10 + a 12 0 Compute matrix exponential exp(qt) to get transition probabilities of states. Basket CDS and CDOs can be expressed in matrix-analytic forms.. Main challenge is to compute matrix exponential with dimension 2 n. (2006) used series expansion method and uniformization method. HR 19

Hazard Construction Method (Zheng, 2007) Using the total hazard method, we find The density function of τ as for 0 < t 1 < t 2 < < t n, where c = n ( i 1 i=1 f(t 1, t 2,..., t n ) = ce (w 1t 1 + +w n t n ) a im) and w k = ( k 1 a km) ( m=0 m=0 n m=k+1 a mk ) for k 1. R n + can be divided into n! regions according to order of (t 1, t 2,..., t n ). The pdf f in other regions can be derived with permutation. If n = 2 then f(t 1, t 2 ) = a 10 (a 20 + a 21 )e (a 10 a 21 )t 1 (a 20 +a 21 )t 2 if t 1 < t 2 a 20 (a 10 + a 12 )e (a 20 a 12 )t 2 (a 10 +a 12 )t 1 if t 2 < t 1 20

Homogeneous Interacting Intensities Assume that λ i (t) = a + n b1 {τj t}. j=1,j i Denote τ k the kth default time. The density function of τ k is given by f τ k(t) = k 1 j=0 α k,je β jt where β j = (n j)(a + jb) and α k,j constant depending on a and b. Example f τ 1(t) = nae nat f τ 2(t) = α 20 e nat + α 21 e (n 1)(a+b)t. Contagion has no effect on first default, but affects all subsequent defaults. 21

Basket CDS Pricing Data: n = 10, T = 3, Δ = 0.5, r = 0.05, and R = 0.5. b 0 0.003 0.3 30 a k MC AP MC AP MC AP MC AP 0.01 1 0.0506 0.0506 0.0506 0.0506 0.0507 0.0506 0.0505 0.0506 2 0.0056 0.0056 0.0071 0.0071 0.0436 0.0435 0.0504 0.0505 5 0.0000 0.0000 0.0000 0.0000 0.0344 0.0343 0.0503 0.0504 8 0.0000 0.0000 0.0000 0.0000 0.0271 0.0270 0.0502 0.0504 10 0.0000 0.0000 0.0000 0.0000 0.0181 0.0181 0.0501 0.0502 0.1 1 0.5057 0.5058 0.5052 0.5058 0.5063 0.5058 0.5061 0.5058 2 0.2133 0.2131 0.2172 0.2172 0.3865 0.3861 0.5041 0.5037 5 0.0148 0.0147 0.0179 0.0178 0.2722 0.2721 0.5013 0.5010 8 0.0001 0.0001 0.0002 0.0002 0.2068 0.2068 0.4990 0.4986 10 0.0000 0.0000 0.0000 0.0000 0.1406 0.1403 0.4959 0.4955 22

Self-Exciting Point Processes Consider homogeneous self-exciting point processes λ i (t) = a + j i The joint density function is given by f(t 1,..., t n ) = n m=1 m 1 (a + be (t τ j) 1 {τj t}. j=1 be (t m t j ) )e ( n l=1 at l + n l 1 l=2 j=1 b(1 e (t l t j ) ) for t 1 <... < t n. The kth default time distribution is given by P (τ k > t) = e nat + k 1 i=1 for some functions g(t; t 1,..., t i ). t 0 t t 1... t t i 1 g(t; t 1,..., t i )dt 1... dt i 23

Summary All computation difficulties for general heterogeneous portfolio distributions are essentially due to dimensionality. Exact method: how to effectively use convolution method or Fourier transform method of high dimensions? Approximation method: how to improve the accuracy without comprising efficiency for both the whole and the tail distributions? Markovian method: how to solve Kolmogorov equations or compute matrix exponentials of high dimensions? Total hazard method: how to compute multivariate integrals of high dimensions? 24