On the Estimation and Application of Max-Stable Processes

On the Estimation and Application of Max-Stable Processes Zhengjun Zhang Department of Statistics University of Wisconsin Madison, WI 53706, USA Co-author: Richard Smith EVA 2009, Fort Collins, CO Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 1 / 41

Outline 1 Motivations 2 Multivariate extremes and max-stable processes 3 Tail dependence 4 Estimation 5 Financial applications 6 Did Gaussian copula cause Wall Street crash? Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 2 / 41

Motivations 1 there are spatial ( multivariate ) and temporal extremal dependence, 2 observations are fat tailed and clustered when extreme events occur, 3 want to calculate VaR or optimize assets allocation under market recession or market expansion, etc. 4 Some well-known extreme values theorists believe the way forward is in terms of multivariate EVT, but workable and advanced parametric models do not exist. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 4 / 41

Extremal co-movements in nature 450 400 750 700 ELD EUR 350 650 600 Height (cm) 300 250 Height (cm) 550 500 200 450 400 150 100 Time (hour) ELD EUR 350 300 Time (hour) 650 600 ELD EUR 390 380 ELD EUR 550 370 500 360 Height (cm) 450 Height (cm) 350 340 400 330 350 320 300 310 250 Time (hour) 300 Time (hour) Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 5 / 41

Extremal co-movements in financial market 10 Negative Daily Return Divided by Estimated Standard Deviation, 1977 2004 5 Neg Log Return 0 5 10 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 SPOT EXCHANGE RATE, JPY/USD 10 5 Neg Log Return 0 5 10 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 SPOT EXCHANGE RATE, CAD/USD 10 5 Neg Log Return 0 5 10 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 SPOT EXCHANGE RATE, GBP/USD Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 6 / 41

Basic definitions Suppose {X i = (X i1,...,x id ),i = 1,2,...} is a D-dimensional i.i.d. random vectors with distribution F. Let M nd = max{x id,1 i n}. If there exist normalizing constants a n > 0,b n such that P{M nd a nd x d + b nd,d = 1,...,D} H(x), then the distribution H is called a D-dimensional multivariate extreme value distribution and F is said to belong to the domain of attraction of H, which we write F D(H). Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 8 / 41

Multivariate extremes Representations: Pickands, de Haan and Resnick, Deheuvels (1970s) gave general representation formulae for MEVDs (see Resnick (1987) book for full description). However these formulae are too general to be useful for statistics. Statistics: Much work on parametric subfamilies (Tawn, Coles,...) and on nonparametric estimation methods but these work well only for small D. Problem 1: What to do about large D? (e.g. D 100 for a typical portfolio) Problem 2: How to extend these methods to take into account also time-series dependence within each series? Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 9 / 41

Max-Stable Processes Suppose {Y id,i = 0,±1,±2,d = 1,...,D} is a D-dimensional time series with discrete time index i. W.l.o.g. we may assume P{Y id y} = e 1/y for 0 < y < (unit Fréchet assumption). The process is max-stable if for any i = i 1,i 1 + 1,...,i 2 and any positive set of values {y id, i = i 1,...,i 2, d = 1,...,D}, we have P{Y id y id, i = i 1,...,i 2, d = 1,...,D} = P n {Y id ny id, i = i 1,...,i 2, d = 1,...,D}. De Haan (1982) for definition. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 10 / 41

Representations For univariate processes, such a characterization was provided by Deheuvels (1983). This was generalized by Smith and Weissman (1996) to the following: any max-stable process with unit Fréchet margins may be approximated by a multivariate maxima of moving maxima process, or M4 for short, with the representation Y id = max l=1,2,... max a l,k,dz l,i k, < i <, d = 1,...,D, <k< where {Z l,i, l = 1,2,..., < i < } are independent unit Fréchet random variables and a l,k,d are non-negative coefficients satisfying l=1 k= a l,k,d = 1 for each d. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 11 / 41

Finite representations Y id = max max a l,k,d Z l,i k, < i <, (1) 1 l L d K 1ld k K 2ld where L d, K 1ld, K 2ld are finite and the coefficients satisfy L d l=1 K 1ld k= K 1ld a l,k,d = 1 for each d. Approximation theory Zhang and Smith (2004), Zhang (2009) Illustration of M4 dependence Z 1, 100 Z 1, 99 Z 2, 100 Z 2, 99... Z L, 100 Z L, 99 a 1K b 1K Z 10 a 2K b 2K Z 20. a LK b LK Z L0 a 1,K 1 b 1,K 1 Z 11 a 2,K 1 b 2,K 1 Z 21. a L,K 1 b L,K 1 Z L1 a 10 b 10 Z 1K a 20 b 20 Z 2K. a L0 b L0 Z LK Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 12 / 41

100 20 18 16 14 12 10 90 80 70 60 50 40 30 20 10 8 6 4 2 100 90 80 70 60 50 40 30 20 10 100 90 80 70 60 50 40 30 20 10 35 30 25 20 15 10 5 Illustration of M4 processes (a) 0 0 100 200 300 400 (d) (b) 0 41 42 43 44 45 46 0 102 103 104 105 106 107 (c) (e) 0 256 257 258 259 260 261 0 312 313 314 315 316 317 Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 13 / 41

Distributional properties P(Y id y) = e 1/y, P(Y id y id,y i+1,d y i+1,d ) [ L d 2+K 1ld = exp max { a l,1 m,d y l=1m=1 K 2ld id P(Y 1d y 1d,Y 1d y 1d ) [ max(l d,l d ) = exp l=1 1+max(K 1ld,K 1ld ) m=1 max(k 2ld,K 2ld ), a l,2 m,d y i+1,d } ], max { a l,1 m,d, a l,1 m,d } ], y 1d y 1d Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 14 / 41

The bivariate tail dependence index Definition The bivariate tail dependence index λ = lim u xf P(X > u Y > u). (2) λ > 0 tail dependent, otherwise tail independent Sibuya (1960), de Haan and Resnick (1977), Embrechts, McNeil, and Straumann (2002), Zhang (2004). Definition A sequence of variables {X 1, X 2,..., X n } is called lag-k tail dependent if λ k = lim u xf P(X 1 > u X k+1 > u) > 0, lim P(X 1 > u X k+j > u) = 0, j > 1, u x F Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 16 / 41

Illustration of bivariate tail (in)dependence 4 4 u 3 3 2 2 u 1 1 N(0,1) 0 N(0,1) 0 1 1 2 2 3 3 4 4 2 0 2 4 N(0,1) 4 4 2 0 2 4 N(0,1) 4 4 3 3 2 2 1 1 N(0,1) 0 N(0,1) 0 1 1 2 2 3 3 4 4 2 0 2 4 N(0,1) 4 4 2 0 2 4 N(0,1) Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 17 / 41

Tail dependence in M4 Cross sections: Lag-k in time: max(l d,l d ) λ dd = 2 l=1 λ d(k) = 2 1+max(K 1ld,K 1ld ) m=1 max(k 2ld,K 2ld ) L d 1+k+K 1ld l=1 max{a l,1 m,d,a l,1 m,d }, m=1 K 2ld max{a l,1 m,d,a l,1+k m,d }. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 18 / 41

Illustration of bivariate tail (in)dependence 3 Bivariate Normal (ρ=0.8) 3 Example (L=10) 2.5 2.5 2 2 1.5 1.5 1 1 1.5 2 2.5 3 1 1 1.5 2 2.5 3 3 Bivariate t (ρ=0.8, N=4) 3 Example (L=50) 2.5 2.5 2 2 1.5 1.5 1 1 1.5 2 2.5 3 1 1 1.5 2 2.5 3 3 Gumbel Copula (α=0.5) 3 Example (L=300) 2.5 2.5 2 1.5 2 1.5 1 1 1.5 2 2.5 3 1 1 1.5 2 2.5 3 Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 19 / 41

Connecting to real data: marginal transformation Suppose the transformed variables are Y id, then one may have Y id = Y id + ε id where Y id is a unit Fréchet random variable and ε id is an error term which is normally distributed. ε id 0 a.s. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 21 / 41

Estimation difficulties The existence of signature patterns essentially means that standard methods such as maximum likelihood are not applicable. Earlier methods were based on simpler forms of model,but ran into essentially the same difficulty. For example, the one-dimensional model Y i = max k a i k Z k is called the moving maximum process. The max ARMA processes of Davis and Resnick (1989, 1993) are special cases of this. Hall, Peng and Yao (2002) estimated moving maxima processes through the empirical distribution function, thus avoiding the issue of degeneracy. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 22 / 41

Bivariate joint distribution approach b d (x) = log [ P(Y 1d 1, Y 2d x) ] b d (x) = L d [ a l,k2ld,d + max(a l,k2ld 1,d, a l,k 2ld,d x l=1 + +max(a l, K1ld,d, a l, K 1ld +1,d x d = 1,...,D, ) (3) )+ a l, K 1ld,d x ], b dd (x) = log [ P(Y 1d 1,Y 1d x) ], q d (x) = xb d (x) Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 23 / 41

[ b d (x) = L d l=1 a l,k2ld,d + max(a l,k2ld 1,d, a l,k 2ld,d x ) b 1d (x) = + +max(a l, K1ld,d, a l, K 1ld +1,d x )+ a l, K 1ld,d x j = 1,...,m,d = 1,...,D max(l d,l d ) l=1 1+max(K 1ld,K 1ld ) m=1 max(k 2ld,K 2ld ) j = 1,...,m,d = 2,...,D ], max(a l,1 m,d, a l,1 m,d x ), (4) This can be written in vector and matrix form for suitable choices of xs. b = Ca, (5) or (C T C) 1 C T b = a. (6) Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 24 / 41

Replace b functions by their empirical counterparts, we can estimate the parameters, i.e. [ b d (x) = L d l=1 â l,k2ld,d + max(â l,k2ld 1,d, âl,k 2ld,d x ) b 1d (x) = + +max(â l, K1ld,d, âl, K 1ld +1,d x )+ âl, K 1ld,d x j = 1,...,m,d = 1,...,D max(l d,l d ) l=1 1+max(K 1ld,K 1ld ) m=1 max(k 2ld,K 2ld ) j = 1,...,m,d = 2,...,D ], max(â l,1 m,d, âl,1 m,d x ), (7) b = Ĉâ, (8) or (ĈT Ĉ) 1 Ĉ T b = â. (9) Convergence questions Does b a.s. b imply Ĉ a.s. C and â a.s. a??? Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 25 / 41

Definitions of a.s. convergence up to a row permutation That b = Ca up to a column permutation matrix P means b = CPP T a. That a sequence of random matrices C n converges to a matrix C a.s. almost surely up to a column permutation means C n CP, where P is a column permutation matrix. That a sequence of column random vectors a n converges to a vector a almost surely up to a row a.s. permutation means a n P T a, where P T is a row permutation matrix. That a q dimensional vector a with all elements being positive is equivalent to a three-dimensional (L (K + 1) D) array ã with all elements being non-negative means all elements of ã can be obtained in the following way: (1) t = 1; (2) For d=1 to D; For l=1 to L; For k=1 to K + 1; if ã l,k,d > 0; ã l,k,d = a(t); t=t+1; Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 26 / 41

Lemma of products convergence Suppose S is a set with finite number of distinct values, b is a p dimensional vector, and suppose three is a unique p q, (p q), matrix C and a unique vector a such that b = C a up to a column permutation, where the elements of C belong to S, C T C being invertible; all elements a (k) of a are positive and a is equivalent to a three-dimensional array ã whose elements satisfy the three conditions of Proposition 2.1 and l,k ã l,k,d = 1, d = 1,...,D. Suppose {b n,n = 1,2,...} is a sequence of random vectors, {a n,n = 1,2,...} is a sequence of random vectors and its elements a n(k) are positive and a n is equivalent to a three-dimensional array ã n whose elements satisfy the three conditions of Proposition 2.1 and l,k ã n,l,k,d = 1, d = 1,...,D, and {Cn,n = 1,2,...} is a sequence of random matrices satisfying Cn T C n being invertible and each element of C n belonging to S. Suppose b n = Cna n, n = 1,2,... b a.s. n b, as n, then Cn a.s. C P, a a.s. n P T a, as n. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 27 / 41

Main theorem If all ratios a l,j,d a for all l and j j are distinct for each d = 1,...,D, and l,j,d nonzero existing ratios a l,k,1 a for all l, l and k are distinct for each l,k,d d = 2,...,D, of the multivariate processes {Y id }, then there exist and {x 1d, x 2d,..., x md, d = 1,..., D}, {x 1d, x 2d,..., x m d, d = 2,..., D}, such that the estimator â of a satisfies n(â a) L max l,d K 1ld +max l,d K 2ld +1 N(0,BΘ[Σ+ where B = (C T C) 1 C T. k=1 {W k + W T k }]ΘT B T ) Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 28 / 41

Social, Political and Economic Concerns Mr. A. Greenspan: work that characterizes the distribution of extreme events would be useful as well." Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 30 / 41

0.06 Negative Daily Return 1977 2004 0.04 Neg Log Return 0.02 0 0.02 0.04 0.06 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 SPOT EXCHANGE RATE, JPY/USD 0.02 0.015 0.01 Neg Log Return 0.005 0 0.005 0.01 0.015 0.02 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 SPOT EXCHANGE RATE, CAD/USD 0.06 0.04 Neg Log Return 0.02 0 0.02 0.04 0.06 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 SPOT EXCHANGE RATE, GBP/USD Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 31 / 41

The GARCH model Suppose time series regression has the form where y t = φ 0 + φ 1 y t 1 + +φ p y t p + u t, u t = h t v t, h t = κ + δ 1 h t 1 + +δ r h t r + θ 1 u 2 t 1 + + θ su 2 t s, v t N(0,1). These three formula together are called GARCH(r, s) model. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 32 / 41

0.03 Estimated Conditional Standard Deviation Using GARCH(1,1) 0.02 0.01 0 0 1000 2000 3000 4000 5000 6000 7000 8000 SPOT EXCHANGE RATE, JPY/USD 0.01 0.005 0 0 1000 2000 3000 4000 5000 6000 7000 8000 SPOT EXCHANGE RATE, CAD/USD 0.02 0.015 0.01 0.005 0 0 1000 2000 3000 4000 5000 6000 7000 8000 SPOT EXCHANGE RATE, GBP/USD Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 33 / 41

10 Negative Daily Return Divided by Estimated Standard Deviation, 1977 2004 5 Neg Log Return 0 5 10 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 SPOT EXCHANGE RATE, JPY/USD 10 5 Neg Log Return 0 5 10 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 SPOT EXCHANGE RATE, CAD/USD 10 5 Neg Log Return 0 5 10 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 SPOT EXCHANGE RATE, GBP/USD Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 34 / 41

Data transformation procedure Set up a threshold value u = 1.2 Throw away" all values below the threshold Fit the generalized extreme value distribution (GEV) to the data. { H(x) = exp (1+ξ x µ } ψ ) 1/ξ +, (10) Transform the pseudo-observations into unit Fréchet scales Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 35 / 41

Series N u µ log(ψ) ξ (SE) (SE) (SE) JPY/USD 564 4.163003-0.058268 0.075472 (0.187203) (0.126526) (0.044431) CAD/USD 500 3.336733-0.361246 0.071919 (0.137698) (0.127546) (0.046693) GBP/USD 535 3.781497-0.308770 0.002758 (0.145603) (0.119488) (0.043877) Table: Estimations of parameters in GEV using standardized negative return series. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 36 / 41

Model selections a 1, 1,d Z 1,i 1, a 1,0,d Z 1,i,.. Y id = max a 6, 1,d Z 6,i 1, a 6,0,d Z 6,i, a 7,0,d Z 7,i,. a 11,0,d Z 11,i, d = 1,2,3, (11) where a l,k,d = 0 for {l = 1,2,9, d = 2,3}; {l = 3,4,10, d = 1,3}; {l = 5,6,11, d = 1,2}; {l = 7, d = 3}; {l = 8, d = 2}. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 37 / 41

Signature JPY/USD CAD/USD GBP/USD l a l, 1,1 a l,0,1 a l, 1,2 a l,0,2 a l, 1,3 a l,0 1 0.0201 0.0061 (0.0808) (0.0567) 2 0.0130 0.0262 (0.0809) (0.2210) 3 0.0634 0.0099 (0.1382) (0.0496) 4 0.0270 0.0536 (0.0779) (0.2171) 5 0.0313 0.02 (0.0705) (0.05 6 0.0118 0.01 (0.0767) (0.12 Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 38 / 41

Conclusions 1 The use of M4 processes to the modeling of financial time series data is new. The main goal here is to propose approaches which can efficiently model the extreme observations of multivariate time series which are both inter-serially and temporally tail dependent. 2 Ultimately the test of such methods will be whether they can be used for more reliable risk calculations than established methods such as RiskMetrics. 3 There are also many variations on the basic method which deserve to be explored. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 39 / 41

Zhang and Huang (2006), Zhang and Shinki (2006), Zhang, Zhao, and Zhou (2009) Expected return in dollars (a) Variance Covariance Approach x 104 5 Dji 4.5 4 3.5 3 NASDAQ SP500 Expected return in dollars (b) Recession, MCM4 Approach x 104 5 Dji 4.5 4 3.5 3 NASDAQ SP500 Expected return in dollars (c) Expansion, MCM4 Approach x 104 5 Dji 4.5 4 3.5 3 NASDAQ SP500 2.5 1.1 1.2 1.3 1.4 1.5 1.6 1.7 VaR x 10 4 2.5 1.6 1.8 2 2.2 2.4 2.6 2.8 VaR x 10 4 2.5 1.6 1.8 2 2.2 2.4 VaR x 10 4 Figure: VaR comparison of portfolios of different combinations. Z. Zhang (UW-Madison) Max-stable processes June 23, 2009 41 / 41