Regular Variation and Financial Time Series Models

Transcription

1 Regular Variaion and Financial Time Series Models Richard A. Davis Colorado Sae Universiy Thomas Mikosch Universiy of Copenhagen Bojan Basrak Eurandom

2 Ouline Characerisics of some financial ime series IBM reurns Muliplicaive models for log-reurns (GARCH, SV) Regular variaion univariae case mulivariae case new characerizaion: is RV c is RV? Applicaions of regular variaion Sochasic recurrence equaions (GARCH) Poin process convergence Wrap-up Exremes and exremal index Limi behavior of sample correlaions

3 Characerisics of some financial ime series Define = ln (P ) - ln (P - ) (log reurns) heavy ailed P( > x) ~ C x α, 0 < α < 4. uncorrelaed ˆ ρ ( h) near 0 for all lags h > 0 (MGD sequence) and have slowly decaying auocorrelaions ρˆ ( ) and ˆ h ρ ( h) converge o 0 slowly as h increases. process exhibis volailiy clusering. 3

4 Log reurns for IBM /3/6-/3/00 (blue=96-98) 00*log(reurns) ime 4

5 Sample ACF IBM (a) 96-98, (b) (a) ACF of IBM (s half) (b) ACF of IBM (nd half) ACF ACF Lag Lag 5

6 Sample ACF of abs values for IBM (a) 96-98, (b) (a) ACF, Abs Values of IBM (s half) (b) ACF, Abs Values of IBM (nd half) ACF ACF Lag Lag 6

7 Hill s plo of ail index for IBM (96-98, ) Hill Hill m m 7

8 Muliplicaive models for log(reurns) Basic model = ln (P ) - ln (P - ) (log reurns) = σ Z, where {Z } is IID wih mean 0, variance (if exiss). (e.g. N(0,) or a -disribuion wih ν df.) {σ } is he volailiy process σ and Z are independen. Properies: E = 0, Cov(, +h ) = 0, h>0 (uncorrelaed if Var( ) < ) condiional heeroscedasic (condiion on σ ). 8

9 Muliplicaive models for log(reurns)-con = σ Z (observaion eqn in sae-space formulaion) Two classes of models for volailiy: (i) GARCH(p,q) process (General AuoRegressive Condiional Heeroscedasic-observaion-driven specificaion) σ = α 0 + α - + L+ α p -p +β σ - + L+β q σ -q. Special case: ARCH(): = ( α = α 0 Z = A +α B - +α )Z 0 Z (sochasic recurrence eqn) h ρ ( h) = α, if α < /3. 9

10 Muliplicaive models for log(reurns)-con GARCH(,): = σ Z, σ = α 0 + α - + α - +β σ -. Then Y = (, )' follows he SRE given by σ - σ - αz = Z - - +β α σ α 0 0 Quesions: Exisence of a unique saionary soluion o he SRE? Regular variaion of he join disribuions? 0

11 Muliplicaive models for log(reurns)-con = σ Z (observaion eqn in sae-space formulaion) (ii) sochasic volailiy process (parameer-driven specificaion) logσ = ψ jε j, ψ j j= j= <,{ ε }~ IIDN(0, σ ) ρ 4 ( h) = Cor( σ, σ h)/ EZ + Quesion: Join disribuions of process regularly varying if disr of Z is regularly varying?

12 Regular variaion univariae case Def: The random variable is regularly varying wih index α if P( > x)/p( >) x α and P(> )/P( >) p, or, equivalenly, if P(> x)/p( >) px α and P(< x)/p( >) qx α, where 0 p and p+q=. Equivalence: is RV(α) if and only if P( ) /P( >) v µ( ) ( v vague convergence of measures on R\{0}). In his case, µ(dx) = (pα x α I(x>0) + qα (-x) -α I(x<0)) dx Noe: µ(a) = -α µ(a) for every and A bounded away from 0.

13 Regular variaion univariae case (con) Anoher formulaion (polar coordinaes): Define he ± valued rv θ, P(θ = ) = p, P(θ = ) = p = q. Then is RV(α) if and only if or P( x, / S) P( > ) > α x P( θ S) P( x,/ ) P( > ) > α v x P( θ ) ( v vague convergence of measures on S 0 = {-,}). 3

14 Regular variaion mulivariae case Mulivariae regular variaion of =(,..., m ): There exiss a random vecor θ S m- such ha Equivalence: P( > x, / )/P( >) v x α P( θ ) ( v vague convergence on S m-, uni sphere in R m ). P( θ ) is called he specral measure α is he index of. P( ) v µ ( ) P ( > ) µ is a measure on R m which saisfies for x > 0 and A bounded away from 0, µ(xb) = x α µ(xa). 4

15 Regular variaion mulivariae case (con) Examples:. If > 0 and > 0 are iid RV(α), hen = (, ) is mulivariae regularly varying wih index α and specral disribuion P( θ =(0,) ) = P( θ =(,0) ) =.5 (mass on axes). Inerpreaion: Unlikely ha and are very large a he same ime. Figure: plo of (, ) for realizaion of 0,000. x_ x_ 5

16 . If = > 0, hen = (, ) is mulivariae regularly varying wih index α and specral disribuion P( θ = (/, / ) ) =. 3. AR(): = Z, {Z }~IID symmeric sable (.8) { ±(,.9)/sqr(.8), W.P Disr of θ: ±(0,), W.P..00 Figure: plo of (, + ) for realizaion of 0,000. x_{+} x_ 6

17 Applicaions of mulivariae regular variaion Domain of aracion for sums of iid random vecors (Rvaceva, 96). Tha is, when does he parial sum a n = converge for some consans a n? n Specral measure of mulivariae sable vecors. Domain of aracion for componenwise maxima of iid random vecors (Resnick, 987). Limi behavior of a n n = Weak convergence of poin processes wih iid poins. Soluion o sochasic recurrence equaions, Y = A Y - + B Weak convergence of sample auocovariances. 7

18 Operaions on regularly varying vecors producs Producs (Breiman 965). Suppose, Y > 0 are independen wih ~RV(α) and EY α+ε < for some ε > 0. Then Y ~ RV(α) wih P(Y > x) ~ EY α P( > x). Mulivariae version. Suppose he random vecor is regularly varying and A is a marix independen of wih Then 0 < E A α+ε <. A is regularly varying wih index α. 8

19 Applicaions of mulivariae regular variaion (con) Linear combinaions: ~RV(α) all linear combinaions of are regularly varying i.e., here exis α and slowly varying fcn L(.), s.. P(c T > )/( -α L()) w(c), exiss for all real-valued c, where w(c) = α w(c). Use vague convergence wih A c ={y: c T y > }, i.e., T P( A c ) P ( c > ) = µ (A ) : w( ), c = c α L ( ) P( > ) A c where -α L() = P( > ). 9

20 Converse? Applicaions of mulivariae regular variaion (con) ~RV(α) all linear combinaions of are regularly varying? There exis α and slowly varying fcn L(.), s.. (LC) P(c T > )/( -α L()) w(c), exiss for all real-valued c. Theorem (Basrak, Davis, Mikosch, `0). Le be a random vecor.. If saisfies (LC) wih α non-ineger, hen is RV(α).. If > 0 saisfies (LC) for non-negaive c and α is non-ineger, hen is RV(α). 3. If > 0 saisfies (LC) wih α an odd ineger, hen is RV(α). 0

21 Applicaions of mulivariae regular variaion (con) Idea of argumen: Define he measures m ( )= P( )/ ( -α L()) By assumpion we know ha for fixed c, m (A c ) µ(a c ). {m } is igh: For B bded away from 0, sup m (B) <. Do subsequenial limis of {m } coincide? If m ' v µ and m '' v µ, hen µ (A c ) = µ (A c ) for all c 0. A c Problem: Need µ = µ bu only have equaliy on A c, no a π-sysem. In general, equaliy need no hold (see Ex in Meerschaer & Scheffler (00)).

22 Applicaions of heorem. Kesen (973). Under general condiions, (LC) holds wih L()= for sochasic recurrence equaions of he form Y = A Y - + B, (A, B ) ~ IID, A d d random marices, B random d-vecors. I follows ha he disribuions of Y, and in fac all of he finie dim l disrs of Y are regularly varying (if α is non-even).. GARCH processes. Since squares of a GARCH process can be embedded in a SRE, he finie dimensional disribuions of a GARCH are regularly varying.

23 Examples Example of ARCH(): =(α 0 +α -) / Z, {Z }~IID. α found by solving E α Z α/ =. α α Disr of θ: P(θ ) = E{ (B,Z) α I(arg((B,Z)) )}/ E (B,Z) α where P(B = ) = P(B = -) =.5 3

24 Examples (con) Example of ARCH(): α 0 =, α =, α=, =(α 0 +α -) / Z, {Z }~IID Figures: plos of (, + ) and esimaed disribuion of θ for realizaion of 0,000. x_{+} x_ hea 4

25 Applicaions of heorem (con) Example: SV model = σ Z Suppose Z ~ RV(α) and logσ = j= ψ ε j j, j= ψ j <,{ ε }~ IIDN(0, σ ). Then Z n =(Z,,Z n ) is regulary varying wih index α and so is n = (,, n ) = diag(σ,, σ n ) Z n wih specral disribuion concenraed on (±,0), (0, ±). Figure: plo of (, + ) for realizaion of 0,000. x_ x_ 5

26 6 Poin process applicaion Theorem Le { } be an iid sequence of random vecors saisfying of he 3 condiions in he heorem. Then if and only if for every c 0 where {a n } saisfies np( > a n ), and N is a Poisson process wih inensiy measure µ. {P i } are Poisson ps corresponding o he radial par, i.e., has inensiy measure α x α (dx). {θ i } are iid wih he specral disribuion given by he RV, : : ' / ', = = ε = ε = j P d n a n i i n N N θ c c c c, : : / = = ε = ε = j P d n a n i i n N N θ

27 Poin process convergence Theorem (Davis & Hsing `95, Davis & Mikosch `97). Le { } be a saionary sequence of random m-vecors. Suppose (i) finie dimensional disribuions are joinly regularly varying (le (θ k,..., θ k ) be he vecor in S (k+)m- in he definiion). (ii) mixing condiion A (a n ) or srong mixing. (iii) Then limlimsupp( > an y 0 k n k r exiss. If γ > 0, hen n > a n y) = 0. k ( k) α ( k) ( k) α γ = lime( θ0 θ j ) + / E θ0 k j= N n : = n = ε a d N : = / n Pi ij i= j= ε Q, (exremal index) 7

28 Poin process convergence(con) (P i ) are poins of a Poisson process on (0, ) wih inensiy funcion ν(dy)=γαy α dy. j= εq, i, are iid poin process wih disribuion Q, and Q is he weak limi of ij k k k ( k) α ( k) ( k) α ( k) 0 θ j ) + I ( ε ( k )/ E( θ θ θ j + j= 0 ) j= k lim E( θ ) Remarks:. GARCH and SV processes saisfy he condiions of he heorem.. Limi disribuion for sample exremes and sample ACF follows from his heorem. 8

29 Exremes for GARCH and SV processes Seup = σ Z, {Z } ~ IID (0,) is RV (α) Choose {b n } s.. np( > b n ) Then P n ( b x) exp{ x α n }. Then, wih M n = max{,..., n }, (i) GARCH: P( b γ is exremal index ( 0 < γ < ). (ii) SV model: P( b M x) exp{ γx α n n M x) exp{ x }, α n n }, exremal index γ = no clusering. 9

30 (i) GARCH: P( b (ii) SV model: P( b Exremes for GARCH and SV processes (con) M x) exp{ γx α n n M x) exp{ x α n n } } Remarks abou exremal index. (i) (ii) γ < implies clusering of exceedances Numerical example. Suppose c is a hreshold such ha Then, if γ =.5, (iii) /γ is he mean cluser size of exceedances. (iv) Use γ o discriminae beween GARCH and SV models. (v) P n ( bn P( b n M n c) ~.95 c) ~ (.95) =.975 Even for he ligh-ailed SV model (i.e., {Z } ~IID N(0,), he exremal index is (see Breid and Davis `98 ).5 30

31 Exremes for GARCH and SV processes (con) ** * * *** *** ime 3

32 Summary of resuls for ACF of GARCH(p,q) and SV models GARCH(p,q) α (0,): d ( ρˆ ( h)) h=, K, m ( Vh / V0 ) h=, K, m, α (,4): α (4, ): / α d ( n ρ ( h) ) γ (0)( V ). ˆ h=, K, m h h=, K, m / d ( n ρ ( h) ) (0)( G ). ˆ γ h=, K, m h h=, K, m Remark: Similar resuls hold for he sample ACF based on and. 3

33 Summary of resuls for ACF of GARCH(p,q) and SV models (con) SV Model α (0,): / α d h+ α h ( n / ln n) ρˆ ( h). σ σ σ α S S 0 α (, ): / d ( n ρ ( h) ) (0)( G ). ˆ γ h=, K, m h h=, K, m 33

34 Sample ACF for GARCH and SV Models (000 reps) (a) GARCH(,) Model, n= (b) SV Model, n=

35 Sample ACF for Squares of GARCH (000 reps) (a) GARCH(,) Model, n= b) GARCH(,) Model, n=

36 Sample ACF for Squares of SV (000 reps) (c) SV Model, n= (d) SV Model, n=

37 .40 Amazon reurns May 6, 997 o June 6, 004. Series Residual ACF: Abs values Residual ACF: Squares

38 Wrap-up Regular variaion is a flexible ool for modeling boh dependence and ail heaviness. Useful for esablishing poin process convergence of heavy-ailed ime series. Exremal index γ < for GARCH and γ =for SV. Unresolved issues relaed o RV (LC) α = n? here is an example for which, > 0, and (c, ) and (c, ) have he same limis for all c > 0. α = n and > 0 (no rue in general). 38