Slides lausanne-2013-v2

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Slides lausanne-2013-v2

  1. 1. Arthur CHARPENTIER - Unifying copula families and tail dependence Unifying standard multivariate copulas families (with tail dependence properties) Arthur Charpentier charpentier.arthur@uqam.ca http ://freakonometrics.hypotheses.org/ inspired by some joint work (and discussion) with A.-L. Fougères, C. Genest, J. Nešlehová, J. Segers January 2013, H.E.C. Lausanne 1
  2. 2. Arthur CHARPENTIER - Unifying copula families and tail dependence Agenda• Standard copula families◦ Elliptical distributions (and copulas)◦ Archimedean copulas◦ Extreme value distributions (and copulas)• Tail dependence◦ Tail indexes◦ Limiting distributions◦ Other properties of tail behavior 2
  3. 3. Arthur CHARPENTIER - Unifying copula families and tail dependence CopulasDefinition 1A copula in dimension d is a c.d.f on [0, 1]d , with margins U([0, 1]).Theorem 1 1. If C is a copula, and F1 , ..., Fd are univariate c.d.f., then F (x1 , ..., xn ) = C(F1 (x1 ), ..., Fd (xd )) ∀(x1 , ..., xd ) ∈ Rd (1) is a multivariate c.d.f. with F ∈ F(F1 , ..., Fd ). 2. Conversely, if F ∈ F(F1 , ..., Fd ), there exists a copula C satisfying (1). This copula is usually not unique, but it is if F1 , ..., Fd are absolutely continuous, and then, −1 −1 C(u1 , ..., ud ) = F (F1 (u1 ), ..., Fd (ud )), ∀(u1 , , ..., ud ) ∈ [0, 1]d (2) −1 −1 where quantile functions F1 , ..., Fn are generalized inverse (left cont.) of Fi ’s.If X ∼ F , then U = (F1 (X1 ), · · · , Fd (Xd )) ∼ C. 3
  4. 4. Arthur CHARPENTIER - Unifying copula families and tail dependence Survival (or dual) copulasTheorem 2 1. If C is a copula, and F 1 , ..., F d are univariate s.d.f., then F (x1 , ..., xn ) = C (F 1 (x1 ), ..., F d (xd )) ∀(x1 , ..., xd ) ∈ Rd (3) is a multivariate s.d.f. with F ∈ F(F1 , ..., Fd ). 2. Conversely, if F ∈ F(F1 , ..., Fd ), there exists a copula C satisfying (3). This copula is usually not unique, but it is if F1 , ..., Fd are absolutely continuous, and then, −1 −1 C (u1 , ..., ud ) = F (F 1 (u1 ), ..., F d (ud )), ∀(u1 , , ..., ud ) ∈ [0, 1]d (4) −1 −1 where quantile functions F1 , ..., Fn are generalized inverse (left cont.) of Fi ’s.If X ∼ F , then U = (F1 (X1 ), · · · , Fd (Xd )) ∼ C and 1 − U ∼ C . 4
  5. 5. Arthur CHARPENTIER - Unifying copula families and tail dependence Benchmark copulasDefinition 2The independent copula C ⊥ is defined as d C ⊥ (u1 , ..., un ) = u1 × · · · × ud = ui . i=1Definition 3The comonotonic copula C + (the Fréchet-Hoeffding upper bound of the set of copulas)is the copuladefined as C + (u1 , ..., ud ) = min{u1 , ..., ud }. 5
  6. 6. Arthur CHARPENTIER - Unifying copula families and tail dependence Spherical distributions q q q 2 q q q q qDefinition 4 q q q q q q q q q q q q q q q q q q q q 1 q q q qq qq q qq q q q qq q q q q q qq qq qq qq qq q q q q q q q q qq qq q q q q q q q qq q qq qqq q q q q q q q q q q q qq q q q q qq q q q q q q q q qq q q qq q q qq q qRandom vector X as a spherical distribution if q qq q q q q q qqq q q q q q q q q q q q q qq 0 q q q q qq q q q q qq q q q q q q q q q q qq q q qq q qq q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qqq q q q qq q q q qq q q q q qqqq q qq q q q qq qq −1 qq q q q q q q q q qq qqq qqq qq q qq q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q X =R·U q q q −2 q q q q q −2 −1 0 1 2where R is a positive random variable and U is uniformly dis-tributed on the unit sphere of Rd , with R ⊥ U . ⊥ q q q 0.02 2 q q q q 0.04 q q q q q q q q q q q q q q q q 0.08 q q q q q 1 q q q q q q q q q q q q q q q q q q q q q q q 0.14 q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q 0 q q q q q q q qq q q q q q q q q q qq q qq q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q 0.12 q q q qq q q q q qq q q q −1 q q q q q q qq q q q q q q q q q q q qE.g. X ∼ N (0, I). q q q q q q 0.06 q q q q q q q q −2 q q q q q −2 −1 0 1 2Those distribution can be non-symmetric, see Hartman & Wintner (AJM, 1940)or Cambanis, Huang & Simons (JMVA, 1979)) 6
  7. 7. Arthur CHARPENTIER - Unifying copula families and tail dependence Elliptical distributionsDefinition 5 2 q q q q q q q q q q q qq q q q q q 1 q q q q q q q qRandom vector X as a elliptical distribution if q q q q qq q qq q qqq q q qq qq q qq q qq q q qq q q q q q qqq q q q qq qq q qqq q q q q q q q qq qq q q q q qq q q qq q q q q qq q q q q q q q q q qq qq q q q q q qq qq q qq q qq q q q q q q q q q q q q qq q qq q qq q q q q q q q 0 q q q q qqq q q q q q q q q q q q q qq q qq q q q q q q q q q qq q q q q q qq q q qq q q q q q q q q q q qq q q q q q q qq q q q qq q q q q qq q qq q q qq q q q qq q q q q q q q q qq qq q qq qq qq q q q qq qqq q q q q qq q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q qq qq q q qq q qqq q q qq q q q q q q −1 q q q q q q q q q q q q q q q X =µ+R·A·U q q q q q q q q q −2 −2 −1 0 1 2where R is a positive random variable and U is uniformly dis-tributed on the unit sphere of Rd , with R ⊥ U . , and where A ⊥ 2satisfies AA = Σ. q q q q q q 0.06 q 0.04 qq 0.02 q q q q q q q q q 1 q q q q q q q q q q q q q q q 0.12 q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q qq q q q q q q q q q q q q q q q q q 0 q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q qq q q q q q q 0.14 qq qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q −1 q q q q q q q q q q 0.08q q q q q q q q q q q q qE.g. X ∼ N (µ, Σ). q −2 −2 −1 0 1 2Elliptical distribution are popular in finance, see e.g. Jondeau, Poon & Rockinger(FMPM, 2008) 7
  8. 8. Arthur CHARPENTIER - Unifying copula families and tail dependence Archimedean copulaDefinition 6If d ≥ 2, an Archimedean generator is a function φ : [0, 1] → [0, ∞) such that φ−1 isd-completely monotone (i.e. ψ is d-completely monotone if ψ is continuous and∀k = 0, 1, ..., d, (−1)k dk ψ(t)/dtk ≥ 0).Definition 7Copula C is an Archimedean copula is, for some generator φ, C(u1 , ..., ud ) = φ−1 (φ(u1 ) + ... + φ(ud )), ∀u1 , ..., ud ∈ [0, 1].Exemple1φ(t) = − log(t) yields the independent copula C ⊥ . 8

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