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1 ( ) r-taka@maritime.kobe-u.ac.jp IBIS2009

2 (m/s)

3 Daily returns of the S&P 500 index Gilli & Këllezi (2006). Computational Econometrics 27(1).

4 Daily returns of the S&P 500 index

6 (GEV) (GP) GEV GP

8 block maxima) (threshold exceedances or Peaks Over Threshold)

10 F X F X 1, X 2,..., X n Z n = max { X 1, X 2,..., X n } = max 1 i n X i n Z n F x F Z n x F = sup{x : F (x) < 1}, n.

11 Z n F a n > 0, b n R (n = 1, 2,...) G(x) Z P ( Zn b n Z n b n a n L Z, n. a n ) x P (Z x) = G(x) G (extreme value distribution) F G (domain of attraction) F D(G) (a n, b n )

12 (1) G(x) (2) F D(G) W n = min 1 i n X i W n = min { } { } X 1, X 2,..., X n = max X1, X 2,..., X n W G W (x) = 1 G( x)

13 ( ) ( ) Zn b n P x = P (Z n a n x + b n ) = P max X i a n x + b n 1 i n a n = P (X i a n x + b n, i = 1, 2,..., n) = n i=1 P (X i a n x + b n ) = = F n (a n x + b n ). n i=1 F (a n x + b n )

14 F f(x) Exp(1) F (x) = 1 e x, f(x) = e x, x 0. F n (x + log n) = { { } n 1 e (x+log n)} n = 1 + e x n a n = 1 b n = log n e e x = exp( exp( x)), n. ( < x < ).

15 Pareto ( α > 0) F (x) = 1 1 x α, f(x) = αx α 1, x 1. F n (n 1/α x) = { 1 1 (n 1/α x) α } n = { 1 + x α n } n a n = n 1/α b n = 0 e x α = exp( x α ), n. (x > 0).

16 ( α > 0) F (x) = 1 (1 x) α, f(x) = α(1 x) α 1, 0 x 1. F n (n 1/α x + 1) = { { } n 1 ( n 1/α x) α} n = 1 + ( x)α n e ( x)α = exp( ( x) α ), n. (x 0). a n = n 1/α b n = 1 = x F

17 [Fréchet (1927) Fisher and Tippett (1928) Gnedenko (1943); Trinity Theorem ] G(x) Gumbel Λ(x) = exp( exp( x)), x R, Fréchet Φ α (x) = ( )Weibull Ψ α (x) = { 0, x 0, exp( x α ), x > 0, { exp( ( x) α ), x 0, 1, x > 0, α > 0, α > 0.

18 [Gnedenko (1943) de Haan (1970)] F D(Φ α ) x F = & lim x 1 F (tx) 1 F (x) = t α, t > 0. F D(Ψ α ) x F < & lim x xf 1 F (x F (x F x)t) 1 F (x) = t α, t > 0. 1 F (x + ts(x)) F D(Λ) s( ) > 0 s.t. ( ) lim x xf 1 F (x) ( ) = xf x = s(x) = (1 F (y))dy <, x < x F xf x = e t. (1 F (y))dy/(1 F (x)) satisfies ( ).

19 von Mises-Jenkinson ( ) G ξ (x) = exp { (1 + ξx) 1/ξ}, 1 + ξx > 0. (generalized extreme value, GEV) ξ R ξ = 0 G 0 (x) = lim ξ 0 G ξ (x) = exp{ exp( x)} = Λ(x) Φ α (x) = G 1/α (α(x 1)), Ψ α (x) = G 1/α (α(x + 1)).

20 (max stable) n A n > 0 B n R G n ξ (A nx + B n ) = G ξ (x) 3

21 F n (a n x + b n ) G ξ (x). n a n x + b n = z P (Z n z) = F n (z) G ξ ( z bn a n ). G ξ n

22 F n (a n x + b n ) = [ 1 { 1 F (a n x + b n ) }] n = [ 1 n { 1 F (a n x + b n ) }/ n ] n exp [ lim n n { 1 F (a n x + b n ) }] lim n n{1 F (a nx + b n )} = log G ξ (x) = (1 + ξx) 1/ξ. x = 0 lim n n{1 F (b n)} = 1. 2 lim n n{1 F (a n x + b n )} n{1 F (b n )} = (1 + ξx) 1/ξ.

23 P (X > a n x + b n X > b n ) = 1 F (a nx + b n ) 1 F (b n ) (1 + ξx) 1/ξ. P (X b n a n x X > b n ) 1 (1 + ξx) 1/ξ. H ξ (x) = 1 + log G ξ (x) = 1 (1 + ξx) 1/ξ (Generalized Pareto, GP) ( y b n P (X b n y X > b n ) H ξ a n ). H ξ b n

24 F F u F u (y) = P (X u y X > u), 0 y x F u [Pickands (1975)] F D(G ξ ) lim u x F F u (a(u)y) = H ξ (y), y 0, F u (a(u)y) < 1. a( ) a n n

25 G ξ H ξ ξ ξ < 0 ξ = 0 ξ > 0 Pareto t Cauchy D(G ξ ), / D(G ξ ). F (x) = 1 1/ log x, x e = F / D(G ξ ).

26 GEV (generalized extreme value) GEV(µ, σ, ξ) < µ < σ > 0 < ξ < { [ ( )] } z µ 1/ξ ( ) z µ G(z) = exp 1 + ξ = G ξ, 1 + ξ(z µ)/σ > 0. σ σ G ξ GEV(0, 1, ξ) G ξ (z) = exp [ (1 + ξz) 1/ξ], 1 + ξz > 0, µ σ ξ

27 G(z) ξ < 0 Weibull z < µ σ/ξ ξ = 0 Gumbel < z < G 0 ((z µ)/σ) = lim ξ 0 G ξ ((z µ)/σ) = exp{ exp[ (z µ)/σ]} ξ > 0 Fréchet z > µ σ/ξ G ξ (z) g ξ (z) = { } (1 + ξ z) 1/ξ 1 exp (1 + ξ z) 1/ξ, 1 + ξz > 0, ξ 0, exp { z exp( z) }, < z <, ξ = 0,

28 GEV( 2.5, 1, 0.4) 0 GEV(0, 1, 0) GEV(2.5, 1, 0.4) 0

29 GEV) {z 1, z 2,..., z n } GEV(µ, σ, ξ) { [ ( )] } z µ 1/ξ ( ) z µ G(z) = exp 1 + ξ = G ξ, 1 + ξ(z µ)/σ > 0, σ σ

30 GEV(µ, σ, ξ) ξ 0 n [ ( )] zi µ l(µ, σ, ξ) = n log σ (1 + 1/ξ) log 1 + ξ σ i=1 n [ ( )] 1/ξ zi µ 1 + ξ, σ i=1 1 + ξ(z i µ)/σ > 0, i = 1,..., n. ξ = 0 l(µ, σ) = n log σ n i=1 ( zi µ σ ) n i=1 exp { ( zi µ σ )}. ( µ, σ, ξ) ( µ, σ)

31 GEV(µ, σ, ξ) (Prescott and Walden, 1980) n σ 2 ξ 2 ξ 2 p ξ { 2 Γ(2 + ξ) p } σξ(p/ξ q) 1 2 Γ(2 + ξ) + p σ σ 2 [ Γ(2 + ξ) 1 π2 6 + ξ ( 1 γ + 1 ξ + q p ξ 1 + γ ] ) 2 2q ξ + p ξ 2 (µ, σ, ξ) Γ( ) Ψ(r) = d log Γ(r)/dr p = (1 + ξ) 2 Γ(1 + 2ξ), γ = Euler q = Γ(2 + ξ){ψ(1 + ξ) + (1 + ξ)/ξ},

32 {GEV(µ, σ, ξ), < µ <, σ > 0, < ξ < } ξ > 0.5 Smith, 1985 ξ < ξ < 0.5 (µ, σ, ξ)

33 ( ) GEV(µ, σ, ξ) 1 1/T R T ( ) RT µ G(R T ) = G ξ = 1 1 σ T µ + σ [{ log(1 1/T ) } ξ ]/ 1 ξ, ξ 0, R T = µ + σ [ log { log(1 1/T ) }], ξ = 0. R T (return period) T (return level) T = 200 R n T

34 R T R T = µ + σ { y ξ T 1}/ ξ, ξ 0, µ + σ { log y T }, ξ = 0, ( µ, σ, ξ) y T = log(1 1/T ) V ( R T ) RT T V R T V ( µ, σ, ξ) [ RT T RT = µ, R T σ, R ] T ξ [ = 1, (y ξ T 1)/ξ, σy ξ T ( log y T )/ξ σ(y ξ ] T 1)/ξ2 ( µ, σ, ξ)

35 ξ < 0 R = µ σ/ξ R = µ σ/ ξ R T = [ 1, 1/ξ, σ/ξ 2 ]

36 ξ R T ξ ξ = ξ 0 l(µ, σ, ξ 0 ) µ σ ξ 95% { ξ : 2 { l( µ, σ, ξ) } max µ, σ l(µ, σ, ξ)} χ 2 1 (0.05) = { = ξ : max µ, σ l(µ, σ, ξ) l( µ, σ, ξ) } 1.921

37 R T µ = R T σ [ y ξ T 1] /ξ (µ, σ, ξ) (R T, σ, ξ) n [ )] n [ )] 1/ξ l(r T, σ, ξ) = n log σ (1+1/ξ) i=1 log y ξ T + ξ ( zi R T σ i=1 y ξ T + ξ ( zi R T σ R T 95% { R T : 2 { l( R T, σ, ξ) max l(r T, σ, ξ) } } χ 2 1 (0.05) σ, ξ { = R T : max l(r T, σ, ξ) l( R T, σ, ξ) } χ 2 1 (0.05)/2 σ, ξ

38 (m/s) (1982) (2002) GEV (2004)

39 (m/s) (m/s)

40 GEV(µ, σ, ξ) ( µ, σ, ξ) = (15.349, 2.550, 0.111) V = ( µ, σ, ξ) PP plot QQ plot

41 Probability Plot Quantile Plot Empirical Empirical Model Model

42 Return Level Plot Density Plot Return Level f(z) Return Period z

43 ξ = > 0 Fréchet ξ 95% ξ ± = [ 0.037, 0.259]. 95% [ 0.021, 0.274]

44 Profile Log-likelihood Shape Parameter ξ ξ = > 0 95% [ 0.021, 0.274]

45 200 R % R 200 R 200 ± 1.96 R T 200 V T 200 = [25.75, 41.73]. 95% [28.29, 46.90]

46 Profile Log-likelihood Return Level 200 R 200 = % [28.29, 46.90]

47 GP (generalized Pareto) GP(σ, ξ) σ > 0 < ξ < H(y) = 1 ( 1 + ξ y ) 1/ξ ( ) y = Hξ, 1 + ξy/σ > 0. σ σ H ξ GP(1, ξ) H ξ (y) = 1 (1 + ξy) 1/ξ, 1 + ξy > 0, σ ξ

48 H(y) ξ < 0 0 < y < σ/ξ ξ = 0 0 < y < H 0 (y/σ) = lim ξ 0 H ξ (y/σ) = 1 e y/σ ξ > 0 Pareto 0 < y < H ξ (y) (1 + ξ y) 1/ξ 1, 1 + ξy > 0, ξ 0, h ξ (y) = exp( y), 0 < y <, ξ = 0,

49 GP(1, ξ) ξ = 0.4, 0, 0.4

50 (GP) {y 1, y 2,..., y n } GP(σ, ξ) H(y) = 1 ( 1 + ξ y ) 1/ξ ( ) y = Hξ, 1 + ξy/σ > 0, σ σ

51 GP(σ, ξ) ξ 0 n l(σ, ξ) = n log σ (1 + 1/ξ) log(1 + ξ y i /σ), i=1 1 + ξ y i /σ > 0, i = 1, 2,..., n. ξ = 0 l(σ) = n log σ 1 σ n i=1 y i. ( σ, ξ) σ

52 GP(σ, ξ) Fisher n [ (1 + ξ)/σ 2 1/σ ] (1 + ξ)(1 + 2ξ) 1/σ 2 ξ > 1/2 V n = 1 [ 2σ 2 (1 + ξ) σ(1 + ξ) n σ(1 + ξ) (1 + ξ) 2 Smith, 1985 ]

53 Y GP(σ, ξ) ξ < 1 E(Y ) = y+ y+ (1 H(y))dy = 0 0 y + = sup{y : H(y) < 1} Y v Y > v GP(σ + ξv, ξ) (v > 0) P (Y v > y Y > v) = = 1 H(y + v) ( 1 H(v) y 1 + ξ σ + ξv ( 1 + ξ y ) 1/ξ σ dy = σ 1 ξ. = ( 1 + ξ(y + v)/σ ) 1/ξ ) 1/ξ. ( 1 + ξv/σ ) 1/ξ

54 e(v) = E(Y v Y > v) Y (mean excess) ẽ(v) Y (median excess) P (Y v ẽ(v) Y > v) = 1/2 Y v Y > v GP(σ + ξv, ξ) e(v) = σ + ξv 1 ξ ẽ(v) = = σ 1 ξ + ξ 1 ξ v, σ(2 ξ 1)/ξ + (2 ξ 1)v, ξ 0, σ log 2, ξ = 0.

55 u u

56 (mean excess plot) u x [1], x [2],..., x [Nu ] x max u < x max u, 1 N u N u i=1 (x [i] u) : u < x max u u Y GP(σ, ξ) = E(Y v Y > v) = σ 1 ξ + ξ 1 ξ v.

57 ξ σ u < x max {x [i] u} N u i=1 GP(σ u, ξ) σ u ξ u (u, σ ) (u, ξ) σ = σ u ξu u σ ξ u Y v Y > v GP(σ + ξv, ξ) (v > 0), σ v = σ + ξv.

58 F F (VaR p ) = 1 p (Value at Risk) F (x) F (x) = (1 F (u))f u (y) + F (u), y = x u. u F u GP H ξ ( ) x u F (x) (1 F (u))h ξ + F (u), x u. σ

59 = ζ u = 1 F (u) VaR p = u + σ ( ζu ξ p ) ξ 1, u ( σ, ξ) ζ u N u /n n N u u VaR p = u + σ ( ) ξ ζ u 1 ξ p

60 VaR p ( ζ u, σ, ξ) [ ζ V = u (1 ζ u )/n 0 T ] 0 V Nu 0 T = (0, 0) VaR p V ( VaR p ) VaR T p V VaR p, VaR T p = [ VaRp ζ u, VaR p σ, VaR ] p ξ ( ζ u, σ, ξ)

61 GEV ξ 95% {ξ : max σ l(σ, ξ) l( σ, ξ) 1.921} VaR p σ = (VaR p u)ξ (ζ u /p) ξ 1 l(var p, ξ) VaR p max ξ l(var p, ξ) ζ u

62 M. Gilli & E. Këllezi An Application of Extreme Value Theory for Measuring Financial Risk. S&P 500

63 Daily returns of the S&P 500 index

64 Mean Excess u

65 Mean Excess u

66 Modified Scale Threshold Shape Threshold σ ξ

67 u = σ = (0.0359), ξ = (0.0474)

68 Probability Plot Quantile Plot Empirical Empirical Model Model S&P 500

69 Profile Log-likelihood Shape Parameter ξ 95% [0.052, 0.238]. ξ = [0.044, 0.230]. )

70 Profile Log-likelihood Return Level VaR % [2.411, 2.610].

71 Stephenson, A. and Gilleland, E. (2006). Software for the analysis of extreme events: The current state and future directions. Extremes 8, Extremes 1998 (2004) Vol. 52, No. 1 ( )

72 Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes, Theory and Applications. Wiley. Coles, S. G. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer. Embrechts, P., Klüppelberg, C. and Mikosch, T. (2001). Modelling Extremal Events for Insurance and Finance, 3rd ed. Springer. Falk, M., Hüsler, J. and Reiss, R.-D. (2004). Laws of Small Numbers: Extremes and Rare Events, 2nd ed. Birkhäuser.

73 de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer. Kotz, S. and Nadarajah, S. (2000). Extreme Value Distributions: Theory and Applications. Imperial College Press. Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York. McNeil, J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management. Princeton University Press.

74 Reiss, R.-D. and Thomas, M. (2007). Statistical Analysis of Extreme Values, with Applications to Insurance, Finance, Hydrology and Other Fields, 3rd ed. Birkhäuser. Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer-Verlag, Berlin. Resnick, S. I. (2007). Hervy-Tail Phenomena. Springer.

75 (2004) (2008) II. (2008). 21 II (2008) (5)

76 (2004). r Fisher, R. A. and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Cambridge Philos. Soc. 24, Fréchet, M. (1927). Sur la loi de probabilité de l écart maximum. Ann. Soc. Math. Polon. 6, Gnedenko, B. (1943). Sur la distibution limite du terme maximum d une serie aleatoire. Ann, Math. 44, Translated and

77 reprinted in: Breakthroughs in Statistics, Vol.I, 1992, eds. S. Kotz and N. L. Johnson, Springer-verlag, pp Gumbel, E. J. (1958). Statistics of Extremes. Columbia University Press. de Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Math. Centre Tracts, Vol.32, Amsterdam. de Haan, L. (1976). Sample extremes: an elementary introduction. Statist. Neerlandica 30,

78 Jenkinson, A. F. (1955). The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quart. J. R. Met. Soc. 81, Mises, R. von (1936). La distribution de la plus grande de n valeurs. Rev. Math. Union Interbalcanique 1, Reproduced in Selected Papers of Richard von Mises, Amer. Math. Soc. 2 (1964), pp Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3,

79 Prescott, P. and Walden, A. T. (1980). Maximum likelihood estimation of the parameters of the generalized extreme value distribution. Biometrika 67, Smith, R. L. (1985). Maximum likelihood estimation in a class of nonregular cases. Biometrika 72, Smith, R. L. (1986). Extreme value theory based on the r largest annual events. Journal of Hydrology 86, Smith, R. L. (1987). Estimating tails of probability distributions. Ann. Statist. 15,

80 Smith, R. L. (1989). Extreme value analysis of environmental time series: an example based on ozone data (with discussion). Statistical Science 4, Smith, R. L. (1990). Extreme value theory. In Handbook of Applicable Mathematics VII, Ed. W. Ledermann, Chichester, Wiley. pp Weissman, I. (1978). Estimation of parameters and large quantiles based on the k largest observations. J. Amer. Statist. Assoc. 73,

1 Tokyo Daily Rainfall (mm) Days (mm)

1 Tokyo Daily Rainfall (mm) Days (mm) ( ) r-taka@maritime.kobe-u.ac.jp 1 Tokyo Daily Rainfall (mm) 0 100 200 300 0 10000 20000 30000 40000 50000 Days (mm) 1876 1 1 2013 12 31 Tokyo, 1876 Daily Rainfall (mm) 0 50 100 150 0 100 200 300 Tokyo,