Portfolio Distribution Modelling and Computation. Harry Zheng Department of Mathematics Imperial College

Transcription

1 Portfolio Distribution Modelling and Computation Harry Zheng Department of Mathematics Imperial College Workshop on Fast Financial Algorithms Tanaka Business School Imperial College July 4,

2 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 X n where X 1,..., X n are Bernoulli random variables. 2

3 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

4 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

5 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 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

6 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

7 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

8 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 ( 1) k V k 1 U 1 + ( 1) k+1 V k 8

9 Recursive Method II (Andersen-Sidenius-Basu, 2003) For j = 1,..., n, define random variables L j = X 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

10 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

11 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

12 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

13 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 = Glasserman Example 0.1 MC CLT Probability Total Loss 13

14 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) = , and p 2 (2) = Probability MC CLT AE Loss per Position 14

15 Comparison of Basket CDS Pricing λ ρ k MC EE MC EE MC EE MC EE 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

16 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

17 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

18 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

19 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

20 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

21 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

22 Basket CDS Pricing Data: n = 10, T = 3, Δ = 0.5, r = 0.05, and R = 0.5. b a k MC AP MC AP MC AP MC AP

23 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

24 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