Modelling Multivariate Skewness in Financial Returns: a SGARCH Approach

Transcription

1 Modelling Mulivariae Skewness in Financial Reurns: a SGARCH Approach Nicola Loperfido nicola.loperfido@uniurb.i Universià degli Sudi di Urbino Carlo Bo Via Saffi 42, 6029 Urbino (PU), ITALY (join work wih Giovanni De Luca)

2 Inroducion Third cumulan Financial daa SGARCH model Theoreical resuls

3 Third cumulan κ 3 { } ( ) ( ) ( ) T z = E z µ z µ ( z µ ) z = random vecor µ = expeced value of z = Kronecker produc

4 Linear ransformaions κ ( Az) = ( A A) κ ( z) A T 3 3

5 Disinc elemens The hird cumulan of a d-dimensional random vecor z conains a mos d(d)(d 2)/6 disinc elemens.

6 Block represenaion κ 3 M ( ) d d z =... M R R M d E[(Z i -µ i )(Z j -µ j )(Z k -µ k )] is in he i-h row and in he j-h column of he k-h marix M k h

7 Cenral symmery µ κ3 ( z) O z u z =

8 Skew-normal disribuion: definiion x SN d (ξ,ω,α) f(x;ξ,ω,α) = 2φ d (x-ξ;ω)φ[α T (x-ξ)]

9 Skew-normal disribuion: hird cumulan z SN d ( ξ, Ω, α ) δ = Ωα T α Ωα κ π π ( ) T z = δ δ δ

10 Empirical evidence Log-reurns of small financial markes end o be negaively skewed T ( x m T ) T = 3 < 0

11 Problem How does negaive skewness generalize o he mulivariae case?

12 Daa: descripion Source: Morgan Sanley Capial Inernaional, Inc. Time: June 25, June 23, 2008 Counries: France, Neherland, Spain

13 Noaion P i, = price in he i-h marke a ime X i, = 00(logP i, -logp i,- )= log-reurn of he i-h marke a ime x = (X,,,X p, ) T = vecor of log reurns a ime

14 k Third sample cumulan 3 n = n ( ) ( ) ( ) T X x m x m ( x m) i= i i i X x i m = daa marix = i - h column of = sample mean X T

15 Daa: hird sample cumulan

16 Sign es Rejecs he null hypoesis of symmery if he relaive frequency of negaive values is far away from ½ I suppors he hypoesis of negaive skewness of mulivariae financial reurns

17 Modelling Issue In he general case he hird cumulan includes d(d)(d2)/6 disinc elemen How can he hird cumulan be modelled using a limied number of parameers?

18 Volailiy feedback Bad News Reurns decrease Volailiy increases Reurns decrease Effec of bad news is enhanced Good News Reurns increase Volailiy increases Reurns decrease Effec of good news is miigaed

19 Error Srucure = = Ψ T shock endogenous shock exogenous, 0 N ~ d ζ ε

20 Volailiy Srucure ( ) ( ) ( ) 2, 2, , 2,, 2,,, 0, 2,,..., d q d q j j q j q i i i i k p q d q q j j q k k j q i i k i k k i k k diag D σ σ η δ ε η δ δ η η ω σ ω ζ σ ω ω σ = = = = = = =

21 Feedback Srucure Ε σ X i, i, ε ( ) 0.5 = βiε γ i ε γ i 2 / π direc effec feedback effec

22 The model european reurns = direc feedback endogenous effec effec shock volailiy facor

23 Mulivariae SGARCH: good news Ψ Ψ = = Ψ Ω = Ω δ δ δ α δ δ γ β δ α π γ,, 2 ~ T T d SN B x D

24 Mulivariae SGARCH: bad news Ψ Ψ = = Ψ Ω = Ω δ δ δ α δ δ γ β δ α π γ,, 2 ~ T T d SN G x D

25 Uncondiional disribuion D x 0.5 SN d ( γ 2 / π, Ω, α ) 0.5 SN ( γ 2 / π, Ω, α ) d

26 General properies: sign All elemens of κ 3 (x ) are eiher negaive or null.

27 General properies: rank The rank of κ 3 (x ) is never greaer han 2

28 General properies: ancillariy κ 3 (x ) does no depend on correlaions beween endogenous shocks a ime.

29 Special case: univariae d = κ 2 π ( ) x = γ 2γ 3β γ 3 4 π

30 Special case: symmery γ = 3 ( ) O x 0 κ = d

31 Special case: proporionaliy γ i [ k ( x )] = cβi i =,, d rank 3

32 Financial daa Third cumulan Mulivariae skewness SGARCH model Theoreical resuls

33 Research opics Sylized facs Alernaive measures Mulivariae kurosis

34 Essenial References Azzalini, A. and Dalla Valle, A. (996). The mulivariae skew-normal disribuion. Biomerika Azzalini, A. and Capianio, A. (999). Saisical applicaions of he mulivariae skewnormal disribuions. J. R. Sais. Soc. B 6, De Luca, G., Loperfido, N. (2004). A Skew-in-Mean GARCH Model for Financial Reurns. In Skew-Ellipical Disribuions and Their Applicaions: A Journey Beyond Normaliy, CRC/Chapman & Hal, De Luca, G., Loperfido, N. (200). Modeling Mulivariae Skewness in Financial Reurns: a SGARCH Approach. Submied for publicaion. De Luca, G., Genon, M. G.,and Loperfido, N.: A Mulivariae Skew-Garch Model, in: Advances in Economerics: Economeric Analysis of Economic and Financial Time Series, Par A (Special volume in honor of Rober Engle and Clive Granger, he 2003 winners of he Nobel Prize in Economics), Terrell, D. (Ed.), Elsevier,(2006) Franceschini, C., Loperfido, N., (2009). A Skewed GARCH-Type Model for Mulivariae Financial Time Series. In Mahemaical and Saisical Mehods for Acuarial Sciences and Finance XII, Corazza M. and Pizzi C. (Eds.), ISBN: French, K. R., Schwer, G. W. and Sambaugh, R. F. (987): Expeced sock reurns and volailiy. Journal of Financial Economics, 9, Kollo, T. and von Rosen, D.: Advanced Mulivariae Saisics wih Marices. Springer, Dordrech, The Neherlands (2005). Peirò A.: Skewness in Financial Reurns. Journal of Banking and Finance, (999) 2, Rydberg, T.H.(2000): ''Realisic Saisical Modelling of Financial Daa''. Inernaional Saisical Review, 68,3,