Multiname and Multiscale Default Modeling

Transcription

1 Multiname and Multiscale Default Modeling Jean-Pierre Fouque University of California Santa Barbara Joint work with R. Sircar (Princeton) and K. Sølna (UC Irvine) Special Semester on Stochastics with Emphasis on Finance Kick-off-Workshop September 8-12, 2008 RICAM, Linz, Austria 1

2 Basics of Collaterized Debt Obligations (CDOs) e rt k f l (T k 1 )α l (T k T k 1 ) = k k e rt k (f l (T k 1 ) f l (T k )) (1 R) Predetermined payment times: T k, k {1, 2,,K} Constant risk-free short rate r Constant recovery R Tranches: {0-3, 3-7, 7-10, 10-15, 15-30}% (CDX tranches) α l is the yield associated with tranche l: the rate at which the insurance buyer pays for protection of tranche l f l (T k ) is the risk-neutral expected fraction of tranche l left at time T k (defaults between T k 1 and T k being accounted at T k ) 2

3 Doubly Stochastic Modeling We consider N obligors or underlying names. A particular obligor i defaults at the first arrival τ i of a Cox process with stochastic intensity or hazard rate X (i). Conditioned on the paths of the hazard rates, the default times τ i of the firms are independent, and the probability that obligor i has survived till time T, is given by exp( T 0 X(i) s ds). The unconditional survival probability is { IP {τ i > T } = IE e T } 0 X(i) s. Consider a subset {i 1,,i n } of obligors. The probability of the joint survival of this set till time T is then IP {τ i1 > T,,τ in > T } = IE { e n j=1 T } 0 X(i j ) s ds. 3

4 Vasicek Intensities and Survival Probabilities Negative intensities? Duffie-Singleton, Credit Risk 2003: the computational advantage with explicit solutions may be worth the approximation error associated with this Gaussian formulation. where the (W (i) t ( ) dx (i) t = κ i θ i X (i) t dt + σ i dw (i) t, ) are correlated Brownian motions with d W (i), W (j) = c ij dt. t We denote the survival probability for name i by { S i (T; x i ) = IP (τ i > T X (i) 0 = x i ) = IE e T 0 X(i) s ds X (i) 0 = x i }. We also denote the joint survival probability of all N names by { S(T;x, N) = IE e T N } 0 X(i) s ds X (1) 0 = x 1,,X (N) 0 = x N, 4

5 PDE for the Survival Probabilities From the Feynman-Kac formula it follows that the joint survival probability from time t till time T { u(t,x) = IE e T N } t X(i) s ds X t = x, solves the partial differential equation u t i,j=1 (σ i σ j c ij ) 2 u x i x j + κ i (θ i x i ) u x i ( N x i )u = 0, with terminal condition u(t,x) = 1. 5

6 Independent Case Assume first that the covariance matrix c is the identity matrix, corresponding to the components of the intensity process X being independent. Then, as is well known, or can be readily checked, the solution is given by the product-affine formula where B i (s) = s 0 A i (s) = e θ i u(t,x) = N A i (T t)e B i(t t)x i, e κiξ dξ = 1 e κ iτ, κ i s and θ d i = θ i σ2 i 2κ 2 i 0 κ ib i (ξ) dξ+ 1 2 σ2 i. s 0 B2 i (ξ) dξ = e ( θ d i (s B i(s))+ σ2 i 4κ i B 2 i (s) ), 6

7 Correlated Case In the general correlated case, we find with N u(t,x) = A c (T t) A i (T t)e B i(t t)x i, A c (s) = e 1 2 N N j (σ iσ j c ij ) s 0 B i(ξ)b j (ξ)dξ. The last integral is given explicitly by s 0 B i (ξ)b j (ξ) dξ = s κ i κ j B i (s) κ i (κ i + κ j ) B j (s) κ j (κ i + κ j ) B i(s)b j (s) (κ i + κ j ). 7

8 Symmetric Name Case The dynamics and the starting points of the intensities are the same for all the names. This is convenient to understand the effects of the correlation and the size of the portfolio. Specifically, we have ( = κ dx (i) t θ X (i) t ) dt + σ dw (i) t, X (i) 0 = x, with the parameters κ, θ σ and x assumed constant and positive. Moreover, we assume that the correlation matrix is defined by c ij = ρ X, for i j, with ρ X 0, and ones on the diagonal. Remark that such a correlation structure can be obtained by letting W (i) t = 1 ρ X W (i) t + ρ X W (0) t, where W (i), i = 0, 1,, N, are independent standard Brownian motions. 8

9 Explicit Survival Probabilities The joint survival probability for n given names, say the first n names, is S(T;x, n) = IE { e T with 0 (X(1) s } + +X(n) s ) ds X (1) 0 = x,,x (n) 0 = x = e n [θ (T B(T))+[1+(n 1)ρ X ]σ 2 B 2 (T)/(4κ)+xB(T)], B(T) = 1 e κt κ θ = θ [1 + (n 1)ρ X ] σ2 2κ 2. This expression shows explicitly how the joint survival probability depends on the correlation ρ X and the basket size n. Note in particular how the basket size enhances the correlation effect., 9

10 The Loss Distribution The loss distribution at time T of a basket of size N is given by its mass function p n = IP {(#names defaulted at time T) = n}, n = 0, 1,,N, and, in the symmetric correlated case, is explicitly p n = N n n S N+j n ( 1) j, n j j=0 using the short hand notation S n = S(T; (x,, x), n) for the joint survival probability of n names. This gives rise to an O(N 2 ) procedure for calculating the loss distribution. However, a direct implementation of this formula is not numerically stable due to catastrophic cancellation errors in finite precision arithmetics. 10

11 Alternative Implementation Note first that, from the explicit formula for the joint survival probabilities, we can write with S n = e n [θ (T B(T))+[1+(n 1)ρ X ]σ 2 B 2 (T)/(4κ)+xB(T)] = e d 1n+d 2 n 2, d 1 = d 1 (T, x) = θ(t B(T)) + xb(t) 1 2 σ2 (1 ρ X )B (2) (T), d 2 = d 2 (T) = 1 2 σ2 ρ X B (2) (T), B (2) (T) = T 0 B 2 (s) ds = (T B(T)) κ 2 B(T)2 2κ, where we assume that the model parameters are chosen so that d 1 > 0. 11

12 Alternative Implementation (continued) In the independent case ρ X = 0, we get the binomial distribution: p n = N n (1 e d 1 ) n e (N n)d 1 =: p n (d 1 ). In the general case, we can write { S n = IE e d 1n n } 2d 2 Z = IE { e n (d 1 + 2d 2 Z) }, for Z (or Z) a zero mean unit variance Gaussian random variable. Therefore, in the general case we find { p n = IE p n (d 1 + } 2d 2 Z). Thus, we get the loss distribution stably and fast by integrating (non-negative) binomial distributions with respect to the Gaussian density. 12

13 Alternative Implementation (final comments) We remark that p n = IE { p n (d 1 + } 2d 2 Z) essentially corresponds to conditioning with respect to the correlating Brownian motion W (0), in W (i) t = 1 ρ X W (i) t + ρ X W (0) t. The argument d 1 + 2d 2 Z will be negative for Z negative and with large magnitude. This reflects the fact that we are using a Vasicek model where the intensity may be negative. In the following numerical illustrations, we condition the Gaussian density to Z > d 1 / 2d 2, and choose parameters such that the complementary event has probability less than

14 Example with Constant Parameters and Strong Correlations 0.12 LOSS DISTRIBUTION INDEPENDENT STRONG CORRELATIONS TRANCHE PREMIA INDEPENDENT STRONG CORRELATIONS INDEPENDENT STRONG CORRELATIONS θ =.02, κ =.5, σ =.015, x =.02, T = 5, N = 125, r = 3%, R = 40%, ρ X = 0, or.75 14

15 Multiscale Stochastic Volatility Modeling Under the risk-neutral probability measure we assume the model dx (i) t = κ i (θ i X (i) t )dt + σ (i) t dw (i) t, for 1 i N where the W (j) s are correlated Brownian motions as before. The volatilities are stochastic and depend on a fast evolving factor Y and slowly evolving factor Z: σ (i) t = σ i (Y t, Z t ), where the functions σ i (y, z) are positive, bounded and bounded away from zero, and smooth in the second variable. The fast process is modeled for instance by dy t = 1 ε (m Y t)dt + ν 2 dw (y) ε t, ε << 1, d W (i), W (y) = ρ Y dt, for 1 i N. t 15

16 Multiscale Stochastic Volatility Modeling (continued) The slow factor evolves as dz t = δc(z)dt + δg(z) dw (z) t, 1/δ >> 1, d W (i), W (z) = ρ Z dt. t The functions and c and g are assumed to be smooth. We denote by ρ Y Z the correlation coefficient defined by d W (y), W (z) = ρ Y Z dt t We assume the coefficients ρ Y, ρ Z, ρ Y Z and the matrix (c ij ) are such that the joint covariance matrix of the Brownian motions W (i) (i = 1,,n), W (y) and W (z) is non-negative definite. 16

17 S(T;x, y, z, N) = IE { e N Joint Survival Probabilities T 0 X(i) s } ds X 0 = x, Y 0 = y, Z 0 = z. In this case the joint survival probability from time t { u ε,δ (t,x; y, z, N) = IE e T N } t X(i) s ds X t = x, Y t = y, Z t = z, solves the partial differential equation with the notation L ε,δ u ε,δ = 0, u ε,δ (T,x, y, z) = 1 L ε,δ = t + L (x,y,z) ( N ) x i where L (x,y,z) denotes the infinitesimal generator of the Markov process (X t, Y t, Z t ) under the risk-neutral measure., 17

18 Operator Notations L ε,δ = 1 ε L ε L 1 + L 2 + δm 1 + δm 2 + δ ε M 3, L 0 = ν 2 2 L 1 = 2νρ Y L 2 = t M 1 = ρ Z g(z) y 2 + (m y) y, i,j=1 2 σ i (y, z) x i y, 2 c ij σ i (y, z)σ j (y, z) + x i x j σ i (y,z) 2 x i z, M 2 = 1 2 g2 (z) 2 z 2 + c(z) z, M 3 = 2νg(z)ρ Y Z 2 y z. ( κ i (θ i x i ) N x i x i ), 18

19 Expansions Starting with u ε,δ = u ε 0 + δ u ε 1 + δ u ε 2 =, we get ( 1 ε L ) L 1 + L 2 u ε ε 0 = 0, u ε 0(T) = 1, ( 1 ε L ) L 1 + L 2 u ε ε 1 = (M 1 + ε 1 ) M 3 u ε 0, u ε 1(T) = 0. Then we expand u ε 0 = u 0 + ε u 1,0 + εu 2,0 + ε ε u 3,0 +. and get L 0 u 0 = 0, (choose u 0 independent of y), L 0 u 1,0 + L 1 u 0 =0 = 0, (choose u 1,0 independent of y), L 0 u 2,0 + L 1 u 1,0 =0 + L 2 u 0 = 0, (Poisson equation in u 2,0 ), Centering condition: L 2 u 0 = 0, where the triangular brackets denote an average with respect to the invariant distribution of the fast process Y. 19

20 Recall that L 2 = t +1 2 i,j=1 Leading Order Term 2 c ij σ i (y, z)σ j (y, z) + x i x j ( κ i (θ i x i ) N ) x i x i The leading order term u 0 of the joint survival probability is obtained by solving the constant volatility problem with the effective diffusion [d(z)] ij = d ij (z) := c ij σ i (y, z)σ j (y, z)φ(y) dy, with Φ being the invariant distribution for the Y process. Since the process Y evolves on the fast scale its leading order effect is obtained by averaging with respect to Φ. The process Z evolves on a relatively slow scale and at this level of approximation its effect corresponds to just evaluating this process at its current frozen level z. 20

21 Formula for the Leading Order Term The leading order term u 0 is the survival probability which solves where L e (d(z)) = t +1 2 L e (d(z))u 0 = 0, u 0 (T,x; z) = 1, i,j=1 2 d ij (z) + x i x j ( κ i (θ i x i ) N x i x i ) As previously in the constant volatility case, the solution is given by with u 0 (t,x; z) = A c (T t) N A i (T t)e B i(t t)x i, B i (s) = 1 e κ iτ, κ i A c (s) = e 1 2 N N j (d ij(z)) s 0 B i(ξ)b j (ξ) dξ. 21

22 Expansions (continued) Coming back to the expansion u ε 0 = u 0 + ε u 1,0 + εu 2,0 + ε ε u 3,0 +, we get u 2,0 = L 1 0 (L 2 L 2 ) u 0, L 0 u 3,0 + L 1 u 2,0 + L 2 u 1,0 = 0, (Poisson equation in u 3,0 ), so that the centering condition becomes L 1 u 2,0 + L 2 u 1,0 = 0, L 2 u 1,0 = L 1 u 2,0 = L 1 L 1 0 (L 2 L 2 ) u 0 22

23 Correction u 1,0 due to the fast volatility factor Y We introduce the operator A 1,0 = L 1 L 1 0 (L 2 L 2 ), which will be given explicitly below. The function u 1,0 (t,x, z) solves the inhomogeneous problem L e (d(z))u 1,0 = A 1,0 u 0, u 1,0 (T,x; z) = 0. Thus, u 1,0 solves a linear equation with the effective operator L e (d(z)), but now the problem involves a source term A 1,0 u 0, defined in terms of the leading order survival probability u 0, and with a zero terminal condition. 23

24 Expansions (continued) Coming back to the term δ u ε 1, we have ( 1 ε L ) L 1 + L 2 u ε ε 1 = (M 1 + ε 1 ) M 3 u ε 0, u ε 1(T) = 0. We expand u ε 1 = u 0,1 + ε u 1,1 + εu 2,1 +, and keep only the terms of order one, to obtain as previously: L 2 u 0,1 = M 1 u 0 where we have used that M 3 takes derivatives with respect to y. 24

25 Correction u 0,1 due to the slow volatility factor Z In this case we introduce the operator and obtain: A 0,1 = M 1, The function u 0,1 (t,x, z) is the solution of the problem L e (d(z))u 0,1 = A 0,1 u 0, u 0,1 (T,x; z) = 0, which is again a source problem with respect to the operator L e (d(z)) and with a zero terminal condition. 25

26 Explicit Formulas and Group Parameters We introduce the symmetric matrix Ψ(y, z) satisfying the Poisson equations L 0 Ψ i1,i 2 = c i1,i 2 σ i1 (y, z)σ i2 (y, z) d i1,i 2 (z), and the coefficients V ε 3 (z, i 1, i 2, i 3 ) = ε ρ Y ν 2 σ i3 Ψ i1 i 2 y One then obtains that the scaled operator εa 1,0 can be written εa1,0 = ε L 1 L 1 0 (L 2 L 2 ) = i 1,i 2,i 3 =1. V3 ε 3 (z, i 1, i 2, i 3 ). x i1 x i2 x i3 26

27 Explicit Formula for the Correction ε u 1,0 Setting ε u 1,0 (t,x; z) = D(T t)u 0 (t,x; z), leads to the following ODE for D: D = i 1,i 2,i 3 =1 V ε 3 (z, i 1, i 2, i 3 )B i1 B i2 B i3, D(0) = 0, which gives εu1,0 = i 1,i 2,i 3 =1 T V3 ε (z, i 1, i 2, i 3 ) 0 B i1 (s)b i2 (s)b i3 (s) ds u 0. It depends on the underlying model structure in a complicated way, but only the effective market group parameters V ε 3 ( ) are needed to compute the fast time scale correction u 1,0. 27

28 Explicit Formula for the Correction δ u 0,1 Using M 1 = ρ Z g(z) N σ i(y, z) 2 x i z, we find that δa0,1 = δ M 1 = where we introduced the coefficients V δ V δ 1 (z, i) = δ g(z)ρ Z σ i. 2 1 (z, i), z x i As for the fast correction, it follows that δ u0,1 = ( 1 2 i 1 =1 V δ 1 (z, i 1 ) i 2 =1 i 3 =1 d T dz (d i 1 i 2 ) 0 B i1 (v) v 0 B i1 (s)b i2 (s) ds dv ) u 0. 28

29 Accuracy Result For any fixed t < T, x IR N and y, z IR, ( u ε,δ (t,x, y, z) u 0 (t,x, z) + ε u 1,0 (t,x, z) + δ u 0,1 (t,x, z)) = O(ε+δ), where u ε,δ is the solution of the original problem L ε,δ u ε,δ = 0, u ε,δ (T,x, y, z) = 1, and u 0, ε u 1,0 and δ u 0,1 are given by the formulas above. One of the difficulties is that the potential ( x i i) in ( N L ε,δ = t + L (x,y,z) x i ) is unbounded from above since the X (i) s are unbounded from below., 29

30 The transformation Accuracy Result (sketch of proof) u ε,δ (t,x, y, z) = M ε,δ (t, y, z) N e B i(t t)x i reduces to a Feynman Kac equation for M with a bounded potential in (y, z), bounded time-dependent coefficients, and smooth terminal condition M ε,δ (T, y, z) = 1. The proof given in the one-dimensional equity case by Fouque Papanicolaou Sircar Sølna is then easily extended. 30

31 Stochastic Volatility Effects in the Symmetric Case In the symmetric case with σ (i) t σ(y t, Z t ), we obtain S(T;x, y, z, n) = u ε,δ (0; (x,,x), y, z, n) ũ ε,δ (0; (x,,x), z, n) = [ ] (1 + D ε (T; z, n) + D δ (T; z, n))e n θ (z)(t B(T))+[1+(n 1)ρ X ] σ 2 (z) B2 (T) 4κ +xb(t) where σ 2 (z) = σ(, z) 2, D ε (T; z, n) = v 3 (z)n 2 (1 + (n 1)ρ X )B (3) (T), B (3) (T) = D δ (T; z,n) = v 1 (z)n 2 (1 + (n 1)ρ X ) B (3) (T), B(3) (T) = v 1 (z) = δ 2 g(z)ρ Z σ(, z) z L 0 Ψ = σ 2 (y, z) σ 2 (, z), T 0 T σ 2 (, z), v 3 (z) = ε 2 ρ Y ν 0 θ (z) = θ [1 + (n 1)ρ X ] σ2 (z) 2κ 2. B 3 (s) ds, B(s)B (2) (s) ds, Ψ(, z) σ(,z) y 31

32 Stochastic Volatility Effects in the Symmetric Case (continued) Using the constant volatility notation, we have with S(T; x, z, n) ( )) (1 + n 3 ρ X v 3 (z)b (3) (T) + v 1 (z) B (3) (T) e nd 1(T,x,z)+n 2 d 2 (T,z), d 1 (T,x,z) = θ(t B(T)) + xb(t) 1 2 (1 ρ X) σ(z) 2 B (2) (T), d 2 (T, z) = 1 2 ρ X σ(z) 2 B (2) (T), d 2 (T,z) = d 2 (T, z) + (1 ρ X ) ( ) v 3 (z)b (3) (T) + v 1 (z) B (3) (T), 32

33 Stochastic Volatility Effects on the Loss Distribution Recall the case with constant volatility and ρ X = 0: p n (x ) = N n n e x (N+j n) ( 1) j n j j=0 = N n (1 e x ) n e (N n)x. In the general case with ρ X 0, we introduce as before Z N(0, 1), we replace d 2 by d 2, to obtain the loss distribution: { p n = IE p n (d d ( ) 2 Z) + ρ X (v 3 B (3) + v B(3) 1 ) p n (d 1 + } 2 d 2 Z), Note that n=0 p n (x) = d3 dx 3 n=0 p n (x) = 0 p n = 1 (but p n may be < 0). n=0 33

34 Effect of Combined Name Correlation and Stochastic Volatility LOSS DISTRIBUTION CONSTANT PARAMETERS STOCHASTIC VOLATILITY TRANCHE PREMIA CONSTANT PARAMETERS STOCHASTIC VOLATILITY CONSTANT PARAMETERS STOCHASTIC VOLATILITY θ =.03, κ =.5, σ =.02, x =.03, T = 5, N = 125, r = 3%, R = 40%, ρ X =.01, and v 3 = , v 1 =

35 Summary Models with Vasicek intensities are computationally tractable. Stochastic volatility can be implemented via perturbation techniques (regular and singular). Strong effect on loss distribution and on senior tranches. Generalizations: Fluctuations in the hazard rate level (θ stochastic). Short rate term structure effect (r stochastic). Name heterogeneity via grouping. 35