Pure Jump Lévy Processes and Self-decomposability in Financial Modelling

Transcription

1 Pure Jump Lévy Processes and Self-decomposabiliy in Financial Modelling Ömer Önalan Faculy of Business Adminisraion and Economics, Marmara Universiy, Beykoz/Isanbul/ Turkey omeronalan@marmara.edu.r

2 Ouline Inroducion Levy Processes Self-decomposable Laws Ornsein-Uhlenbeck Processes Normal Inverse Gaussian Disribuion Modelling of Reurns Applicaion of Real Daa Conclusion

3 1. Inroducion The basic models of modern finance are based on he assumpion of normaliy of asse reurns. Bu a lo of empirical sudies are shown ha he assumpion which observaions are normally disribued is a poor approximaion in he real world. Because he reurns have feaures such as jumps, semi-heavy ails and asymmery. In radiional diffusion models, price movemens are very small in shor period of ime. Bu in real markes, prices may be show big jumps in shor ime periods. 3

4 Inroducion When price process model include he jumps, he perfec hedging is imposable. In his case, marke paricipans can no be hedge risks by using only underlying asses. For hese reasons, diffusion models used in finance is no a sufficien model. A good model mus be allow for disconinuiies and jumps in price process. Lévy processes is a valuable ool in financial modeling, hey provide a good fi wih reel daa. 4

5 Inroducion In probabiliy heory, a Lévy process, named afer he French mahemaician Paul Levy, is any coninuous-ime sochasic process ha sars a 0, admis cadlag modificaion and has "saionary independen incremens. The mos well-known examples are he Wiener process and he Poisson process. Lévy process is a simple Markov process wih jumps ha allow us o capure asse reurns wihou he necessiy of inroducing exreme parameer values 5

6 Inroducion Lévy models are no adequaely fiing implied volailiy surfaces of equiy opions across boh srike and mauriy. The incremens of addiive process presen a flexible model. These processes may be obained wih self-decomposable disribuions. 6

7 Inroducion A law in class of self-decomposable laws can be decomposable ino he sum of a scaled down version hemselves and an independen erm. The class of self-decomposable disribuions is obained as a limi laws of a sequence ha independen and suiably normalized 7

8 Inroducion The properies of he reurn disribuions depend on lengh of reurn inerval. Log reurns is aken monhly are reasonably represened by a normal disribuion. If one be dealing wih ick daa, hen reurn disribuions may have heavy ails. The aim of hese paper, o review o pure jump Lévy process wih self-decomposable disribuions in financial modeling and esing for he presence of a Brownian moion componen and discriminaing beween finie or infinie variaion 8

9 . LÉVY PROCESSES ( ) 0 Definiion.1 A cadlag sochasic process on a filered probabiliy spaces ( Ω, I, ( I ) 0, P) Lévy processes if he following condiions holds = is defined is called a i) 0 = 0 ii) has independen incremens of he pas i.e. independen of { } u ; u s for 0 s s iii) has saionary incremens i.e s has he same disribuion wih s 9

10 Lévy Processes iv) is sochasically coninuous, ( ε ) 0 lim P = ε > 0 k + k Process has cadlag pahs, i.e, is a.s. righ-coninuous wih lef limis. 10

11 Basic Examples Brownian moion B ~ N( μ, σ ) Poisson Process ~ Pois( λ ) λ ( 0, ) N 11

12 Independen and Saionary Incremens A coninuous-ime sochasic process assigns a random variable o each poin 0 in ime. I is a random funcion of. The incremens of such a process are he differences s beween is values a differen imes < s. The incremens of a process is called independen if incremens s and u v are independen random variables,whenever he wo ime inervals do no overlap. The incremens is called saionary, if he probabiliy disribuion of any incremen s depends only on he lengh s of he ime inerval; in he oher words, incremens wih equally long ime inervals are idenically disribued. 1

13 Independen and Saionary Incremens Lévy processes can be viewed as coninuous ime random walks. Lévy process form building blocks for boh Markov processes and semi-maringales There is one o one relaionship beween Lévy process and infiniely divisible disribuions. 13

14 Independen and Saionary Incremens ( ) 0 Theorem.1 Le be a Lévy process. Then has = an infiniely divisible disribuion F for every. Conversely if is an infiniely divisible disribuion, F hen here exis a = ( ) Lévy process, such ha disribuion of is 0 1 given F ( ) 0 The disribuion of a Lévy process = is compleely deermined by any of is marginal disribuions 14

15 Independen and Saionary Incremens The probabiliy disribuions of he incremens of any Lévy process are infiniely divisible There is a Lévy process for each infiniely divisible probabiliy disribuion. A coninuous process wih independen incremens is Gaussian. A linear combinaion of independen Levy processes is also a Levy process. 15

16 .1 Infiniely Divisibiliy ϕ ( u) Le be a characerisic funcion of a random variable. We say ha he law of random variable is infiniely divisible, if we can always find anoher characerisics funcion. u such ha, ( u ) [ ϕ ( )] n n u ϕ = ϕ n ( ) ( ) If ~ ϕ u is infiniely divisible, hen for all n Ν here exis i.i.d random variables ( 1 n ) ( 1 n ) 1,..., n 16

17 Infiniely Divisibiliy ( ) All disribued as u and such ha, ϕ n d = ( 1 n ) ( 1 n ) n In he oher words, random variable always is decomposable ino he sum of an arbirary finie number of i.i.d. random variables 17

18 . Jump Diffusion Model These models have jumps and random evoluion beween he jump imes, where W sandard Brownian moion, compound Poisson process. = μ + σ W + N i = 1 = Y N i= 1 [ ] iu u σ + ( iux exp iu μ λ e 1) F ( dx) E e i Y i 18

19 .3 The Probabilisic Properies of Lévy Processes Le be a Lévy process, we consider following characerisic funcion, ( u) = exp( ) ϕ iu [ ] For ime inerval 0, Δ = i i 1 = by he assumpion of independen and saionary incremens; n = ( ) ( ) 1 0 n n 1 d = n n 19

20 The Probabilisic Properies of Lévy Processes ϕ ( u) = E exp ( iu ) = E exp ( iun ) = ( E exp ( iu ) n n n for n =, ( ) ( ( )) ψ ( u u E exp iu e ) ϕ = = 1 ψ ( u ) = log E exp( iu ) 1 0

21 The Probabilisic Properies of Lévy Processes The every Lévy process can be represened in he following form, = μ + σ W + Z where Z is a jump process wih infiniely many jumps. 1

22 Lévy-Khinchine Formula ( ) o The characerisic funcion of a Lévy process = is given by he, Lévy- Khinchine formula, E e σ u = exp iub + ( ) iux e 1 iux1 v( dx) iu 1 { x } ( ) where, b,σ,ν called generaing riple, 0 ν does no have mass on

23 Lévy-Khinchine Formula ν ({ 0 }) = 0 and saisfies he following inegrabiliy condiion; R min ( x,1) ν ( dx ) < 3

24 Lévy-Io represenaion Lévy-Io represenaion describes he pah srucure of a Lévy process. A Lévy process can be represened in he following way, = μ + σw + Δ s1 s ε 0 s s s { } { } { } ( ) Δ 1 + lim Δ s1 ε Δ 1 x1 ε x 1 ν dx b ( W ) 0 0, σ 0, is a sandard Brownian moion. 4

25 Δ s = s s is called Lévy measure of Lévy measure : denoes he jumps a imes. ν { } ( ) Thus, ν dx is he inensiy of jumps of size x. We assumed ha he pahs of Lévy process is defined over a finie inervals 0,. [ ] [ ] As a consequence he sum of he jumps in ime inerval 0, wih absolue jump size bigger han 1 is a finie sum for each pah. 5

26 Lévy measure ( A) = E( card{ s [ 0,1] : Δ 0, Δ A} ) ν A Β( R) In he oher words, ν A is he average number of jumps of process [ ] ( ) in ime inerval 0,1 whose sizes fall in A s In general in a Lévy process, he frequency of he big jumps deermines exisence of momens of process. The fine srucure of he pahs of he process can be read of he frequency he small jumps. s 6

27 .4 Finie Aciviy A sochasic process has finie aciviy if all pahs of process have a finie number of jumps on every finie inerval i.e., ν ( R) ν ( dx) < = R If all pahs of process has an infinie number of jumps on every finie inerval i.e., ν ( R) = Lévy process has infinie aciviy. i.e. An infinie mass accumulaes around he origin. In a such case, small jumps of occur infiniely. 7

28 .5 Finie Variaion ( ) 0 Le be a Lévy process. = = 0 dx σ { } ( ) i) If and x 1 x ν < and process have finie variaion i.e., Δ s 1 < min( x,1) ν ( dx) < s i f and only if R ii) If σ 0 or hen process 1 dx { x 1 } x ν ( ) = have infinie variaion. 8

29 Subordinaors A Lévy process Z is called a subordinaor, if i has a.s. nondecreasing pahs. such a process canno have a diffusion componen. Given a Lévy process and an independen subordinaor Z, he process Y = Z is called a subordinaed Levy process. 9

30 3. SELF-DECOMPOSABLE LAWS There is a very close connecion he laws of class L and selfdecomposable laws. For hese reason, firs ime, we describe self-decomposable laws and relaed. 3.1 Laws of Class L The name of class L firs ime used by Khinchine, any random variable in class L is infiniely divisible ([30],Theorem 9.3), [6]. The infiniely divisible laws are he limi laws of riangle arrays, where arrays of independen random variables which individually negligible. Infinie divisibiliy is preserved under affine ransformaions. 30

31 ( ; n = 1,,... ) Y n 3.1 Laws of Class L Le is a sequence of independen random variables and denoes heir sum. Suppose ha, here exis cenering consans and scaling consans > 0 such ha he disribuion of b S + converges o he disribuion of some random n n a n b n n S n = Y i i= 1 a n R variable 31

32 3.1 Laws of Class L Then we say ha, random variable is a member of class L. As explained above, we shorly can say ha, if a random variable has same disribuion as he limi of some sequence of normalized sums of independen random variables, random variable has a disribuion of class L. 3

33 Self-decomposabiliy Definiion3.1 (Self-decomposabiliy) We suppose ha ϕ( u) is he characerisic funcion of a law. We say ha his law is self-decomposable and wrie if for c 0, 1 and ( u) ( cu) ϕ ( u) ϕ =ϕ ϕ SD ( ) u R c ϕ ( u) ( u) where is a characerisic funcion. is uniquely c deermined. For example normal case, ϕ c 33

34 For example normal case, Self-decomposabiliy ϕ ( u) = exp ( c) u c 1 We can resae his definiion for random variables as follows, 1 The disribuion of a random variable is self-decomposable, for any consan c ( 0,1) we can a find independen random variable such ha, c 34

35 Self-decomposabiliy d where = denoes equaliy in disribuion d = c + According o ([30], heorem 15.3) a random variable has a disribuion of class L if and only if he law of he is self-decomposable. The class of self-decomposable disribuions is a proper subse of infiniely divisible disribuions. A random variable or is probabiliy μ or probabiliy densiy f is self-decomposable,if he corresponding characeriic funcion is in he class SD. c 35

36 Self-decomposabiliy Self-decomposable laws arise as marginal laws in auoregressive ime series models c = 1 + ε The Lévy measure of he self-decomposable laws is absoluely coninuous wih following densiy form, ( ) ν dx = k( x) dx x 36

37 Self-decomposabiliy SD is closed under affine mappings, i.e., for all reals a and b if ϕ SD hen, e ibu ϕ a u SD ( ) SD If here exiss a unique Lévy process Z such ha d = e 0 s dz s 37

38 Self-decomposabiliy ϕ and ψ and Z1 Le denoe he characerisic funcions of respecively, hen ψ o dv () = ψ ( v) v log ϕ log () = exp ( logϕ() ), 0, ψ ( 0) = 1 38

39 Self-decomposabiliy ( x) k (, x) where, increasing for and decreasing for x,. ( ) The densiy of self-decomposable disribuions is unimodal. ( ) 0 = Le be a Lévy process. ( ) ( ) The process 1 self-decomposable if and only if selfdecomposable for every > 0. 39

40 Self-decomposabiliy The characerisic funcion of self-decomposable laws have following form, E u σ = R ( ) k ( x ) [ ] iu iux e exp iuμ + e 1 iux1 { x 1} < dx x μ R ( x) 0 Where,, and ( ) ( ) min 1, dx < k R x k x x 40

41 Self-decomposabiliy A self-decomposable random variable is he value a uni ime of some pure jump Lévy processes which sample pahs have bounded variaion. When he levy densiy inegraes x in he region x < 1, for μ = x < 1 x k ( x ) dx x 41

42 Self-decomposabiliy The characerisic funcion of, E [ ] ( ) iu iux k( x) e 1 exp e = x dx We can consider ha, he reurns are he sum of a suiable number of approximaely independen random variables. Furhermore he reurn disribuion is a limi disribuion. Self-decomposable disribuions can be consider as candidae for he uni period disribuion of asse reurns 4

43 Self-decomposabiliy For example, he Laplace (double exponenial) random variable has he probabiliy densiy, x ( x) = e, x R Is characerisic funcion is equal o, f 1 ϕ SD ϕ ( ) ( iux u = exp e 1) = exp u e x ( ivx e 1) 0 x e dx x dx dv v 43

44 3. Self-Decomposable Laws and Addiive Processes Addiive processes are obained from Lévy processes by relaxing he condiion of saionary of incremens A levy process is a addiive process wih requiremen ha he incremens mus be saionary. Addiive processes have markovian propery hence we can obain complee law of process, if we know he laws of incremens The rajecories of he addiive process can make jumps. A law is self-decomposable if and only if i is he law of uni ime of an addiive processes. 44

45 3.3 Self-similariy and Self-decomposabiliy ( ) 0 = A sochasic process is called self-similar for any given, if we can find such ha, a 0 d b R ( ) ( ) a 0 = b 0 Where, b can be expressed as b H = a, H 0 In he oher words, we say ha one sochasic process is selfsimilar such ha, he change an in ime scale can be compensaed by a corresponding change in he spaces scale. 45

46 Self-similariy and Self-decomposabiliy The H is called Hurs exponen. The connecion beween selfdecomposable laws and self-similar addiive process is given by [31]. A law is self-decomposable if and only if i is he law a uni ime of a self-similar addiive process. ( ) ϕ u Le be a characerisic funcion of a law. hen i can ake a characerisic funcion of Lévy process as follow, 46

47 Self-similariy and Self-decomposabiliy ( u) [ ϕ( u) ] N ϕ = Where, N is ime scale, if ϕ u is a infinie divisible, ϕ ( u) is a characerisic funcion. ( ) Now we describe a new funcion as follows, ψ k, H ( θ ) H H k h = ϕ θ ϕ θ N N 1 47

48 Self-similariy and Self-decomposabiliy I is a characerisic funcion if and only if is selfdecomposable ([30],p.99), ([7],p.1884). The saionary process and he self-similar process are relaed by using firs Lamperi represenaion 48

49 Self-similariy and Self-decomposabiliy Proposiion 3.1: Le = he a new process defined by ( ) R be a saionary process, H Z = log, > 0, Z0 = 0andH > is a self-similar process wih index H>0. 0 Conversely if he process Z is self-similar wih H>0, hen process defined by is a saionary process ( ) ( H ) R = e Z e R 49

50 Self-similariy and Self-decomposabiliy ( ) 0 In general a Lévy process = may be consruced from he a H-self-similar process in accordance wih Z = e s 1 H dz s 50

51 4. ORNSTEIN-UHLENBECK PROCESSES ( ) 0 = Ornsein-Uhlenbeck process (OU) saisfies following differenial equaion, d = λ d + dz λ = ( Z ) 0 where > 0 and Z is a Lévy process. Homogenous Lévy process Z [ ( )] < E log 1+ Z 1 has ha propery 51

52 Z Ornsein-Uhlenbeck Processes If is non-gaussian Lévy process, above differenial equaion has a unique soluion. I has following form, = e λ 0 + e λ ( s ) dz s 0 where λ R and 0 is iniial sae. The Ornsein-Uhlenbeck process is a Markov process and possesses a saionary regime under a mild log-momen condiion. 5

53 Ornsein-Uhlenbeck Processes ν 0 Le be Lévy measure such ha saisfies loginegrabiliy condiiion, x 1 log x ν 0 ( dx) < If a Lévy process saisfies above condiion, hen i has a selfdecomposable disribuion. 53

54 Ornsein-Uhlenbeck Processes O U process has he following, moving average represenaion: λ () ( s = e ) dz O U process has following correlaion funcion; s Corr [ ] λ s = e s, 54

55 Ornsein-Uhlenbeck Processes O U process can be defined as a Lamperi represenaion of he Brownian moion Z ( ) λ ( λ = e Z e ) 55

56 Ornsein-Uhlenbeck Processes ( ) 0 If an = OU process is saionary, i s characerisic funcion has following form, ϕ ( ) ( λ u = ϕ u e ) ϕ ( u) This denoes ha, he marginal disribuion of is selfdecomposable. ϕ c ( ) ( ( ) ( λ u = exp κ u κ u e ) c ( ) 56

57 Ornsein-Uhlenbeck Processes ( u) κ ( ) Where, is cumulan of and denoed by as Le Z be he background driving Lévy process of OU processes and ( u) = logϕ( u) Where v Lévy measure of Z process. κ log { x > } x ν ( dx) < 57

58 Ornsein-Uhlenbeck Processes Then he law of converge owards a selfdecomposable law as Characerisic funcion of his law is given by, ζ u = 0 ( ) ( λ s u exp φ u e ) φ 1 is he characerisic exponen of process ζ ds 58

59 Ornsein-Uhlenbeck Processes We can say ha he limi disribuion of an OU process is self-decomposable. The disribuion of a random variable is selfdecomposable if d = e 1 0 s dz s 1 ζ 59

60 Sochasic Volailiy In Black-Scholes model,volailiy is sandar deviaion of he reurn.volailiy is unobserved and has o be esimaed from pas prices. Volailiy is no consan, i can be ake as sochasic. For example, in BNS model,reurn process, d = [ ] μ + β σ d + σ dw Volailiy process may be as an OU process, σ dσ = λ σ d + dz The process is defined as a saionary markov process λ 60

61 Sochasic Volailiy Where Z is an increasing Levy process and W an independen Brownian moion ( ) 0 The price process has coninuous sample pahs. ( ) 0 The volailiy process σ is a non-gaussian Ornsein- Uhlenbeck process ( σ ) 0 The volailiy process is no direcly observable Finally,sochasic volailiy (SV) models are used o capure he impac of ime-varying volailiy on financial markes. 61

62 5. NORMAL INVERSE GAUSSIAN DISTRIBUTION Normal Inverse Gaussian Disribuion was inroduced by [5]. This disribuion is used in [9], [8], [9] o model equiy reurns. NIG disribuion has semi-heavy ails and a special case wih λ = 1 parameer in he class of Generalized Hyperbolic Disribuions (GH). GH class are infiniely divisible and self-decomposable. 6

63 Normal Inverse Gaussian Disribuion ( ) The probabiliy densiy funcion of he NIG α, β, μ, δ is defined by as follows, f NIG α δ ( x, α, β, μ, δ ) = exp δ α β + β ( x μ) π. K 1 α δ δ + + ( x μ) ( x μ) wih parameers, (, ) < < δ > 0, μ and 0 β 63

64 Normal Inverse Gaussian Disribuion K ν The funcion is modified Bessel funcion of hird kind wih index. ν K ν 1 = u ν 1 ( x) u exp x u du ( x > 0) 0 K 1 π x x ( x) =. e K ν ( x) = K ( x) ν 64

65 Normal Inverse Gaussian Disribuion The characerisic funcion of NIG disribuion, ϕ NIG = + iuμ ( u) exp δ α β δ α ( β + iu) The normal Inverse Gaussian disribuion has semi-heavy ails,i.e. f NIG ( x) ~ cons x 3 e ( α x + β x) as x 65

66 66 Normal Inverse Gaussian Disribuion The cenral momens of a random variable are [ ] 1 β α β δ μ + = E [ ] γ α δ β α β δ β α δ = + = Var

67 67 Normal Inverse Gaussian Disribuion [ ] 1 3 β α δ α β = Skew [ ] β α δ α β + + = Kurosis

68 Normal Inverse Gaussian Disribuion The Lévy measure of NIG has following form, α δ ( dx) =. exp{ β x} K ( α x )dx ν NIG 1 π x The probabiliy densiy funcion of NIG disribuion, f NIG α ( ) ( ) α + β x ~ exp δ α β e π x δ 3 x as x ± 68

69 5.1 Parameer Esimaion of NIG Disribuion Momen Esimaors These esimaes can be used as parameer esimaes which obained in real daa. Le m i, i = 1,,3,4 be sample momen of real daa. ˆ ρ = 3 m 3 m m ˆ μ ˆ β ˆ δ ˆ ρ = m 1 ˆ ( β = m m ρ ) ˆ 69

70 Parameer Esimaion of NIG Disribuion ˆ δ = m ˆ β ˆ ρ 3 + ˆ ρ ˆ ˆ α = ˆ ρ + β 70

71 71 Parameer Esimaion of NIG Disribuion ( ) 1 + = α β α β δ μ E ( ) 3 1 = α β α δ Var ( ) = α β α β α Skew ( ) = α β α β α Kur

72 Parameer Esimaion of NIG Disribuion β 3 ( Skew ( )) α = Kur ( ) 1 + β 4 α Skew 3 ( ) Ku( ) 5 7

73 5. The Simulaion of Normal Inverse Gaussian Disribuions The algorihm can be sae as following [9], [9], [10]. Generae from Y from N(0,1) Generae from Z from IG δ, α β Se = μ + β Z + Z Y ( ) Z The simulaion of Inverse Gaussian disribuion Sample Y from N(0,1) and se V = Y Se K η V 1 = η + 4η δ V + η δ δ V 73

74 74 Parameer Esimaion of NIG Disribuion wih Se β α δ η = + + < + = K U K U K K Z η η η η η

75 Parameer Esimaion of NIG Disribuion The simulaion of V Generae U1 and U uniform random variables N N 1 [ lnu ] cos ( π U ) 1 = 1 1 [ lnu ] sin ( π U ) = 1 Boh of N1 and N have sandard normal disribuion are independen. 75

76 6. THE MODELLING OF RETURNS We can consider a general exponenial Levy process model for sock prices, S = S exp ( ) 0 wih a Levy process ( : 0) The disribuion of a Levy process is uniquely deermined by any of is one dimensional marginal disribuion So we may use he disribuion of 1 The disribuion of 1 is infiniely divisible and is characerisic funcion is given by Levy-Khinchine formula. 76

77 6. THE MODELLING OF RETURNS An infiniely divisible disribuion can be esimaed from he real daa which available for he paricular asse. The log reurns of he model have independen and saionary incremens along ime inervals of lengh 1. The pah properies of he levy process carry over o S. If is a pure jump levy process, hen S is alsa a pure jump process. 77

78 6. THE MODELLING OF RETURNS We consider log price changes, = ln S+ Δ ln S reflecs he muliplicaive characer of price changes. When we use daily price daa, ypically will have he value 1. In generally, he properies of reurn disribuions depends on he lengh of he reurn inerval. Δ Δ 78

79 Δ The Modelling of Reurns For long as a resul of cenral limi heorem, he reurns can be described wih a Gaussian disribuion bu for ick daa, he normal disribuion can no be suiable In his case, as a model Generalized hyperbolic disribuions and i s subse can be used These disribuions are infiniely divisible and selfdecomposable Levy markes give good fi o reurn disribuions. 79

80 The Modelling of Reurns A non-parameric hreshold echnique is proposed o es he inegraed variance which based on discree observed prices by [13]. We can deermine wheher jump ype process is suiable or no as a model for price process, using wih his echnique. 80

81 ( ) o The Modelling of Reurns is he Lévy process generaed by he normal inverse Gaussian disribuion ha was fied o he real daa. The incremens of price process a long ime inerval 1, are disribued according o he NIG disribuion. +1 S The securiy price denoes he presen price of securiy The risk one day a head is calculaed as follow, 81

82 The Modelling of Reurns Var ( ) [ ( )] S S = S Var exp = S Var [ exp( )] 1 1 The disribuion of is he fied NIG 8

83 The Modelling of Reurns Levy processes are semimaringales. As a consequence, he following sochasic inegrals, 0 f () s d and A() s d s o are well definie for L 83

84 The Modelling of Reurns In classic Black-Scholes model, price process is soluion of below sochasic differenial equaion ds = S μ d + S σ dw, S 0 > 0 The soluion, S 1 exp μ d = S 0 σ + σ W 84

85 6.1 Tes Saisics We wan o make inference on o unobserved volailiy process from he observed price process, Blumenhal-Geoor, index is described as follows, p α inf p 0, x ν ( dx) < α [ 0,] = { x 1} This index measures aciviies of small jumps of Lévy process. For jump ype Lévy process, α = 1 For example, Normal inverse Gaussian moion has infiniely variaion and 85

86 Tes Saisics An infinie aciviy process wih Blumenhal-Geoor, index α < 1 has pahs wih finie variaion while henhesamplepahshaveinfinie variaion α > 1 An esimaor for he inegraed variance given by 0 σ ds s This esimaor is consisen in he presence of jumps and characerize is asympoic behavior via a cenral limi herem. V denoes hreshold esimaor for inegraed variance Under he very general assumpions, realised m-power variaion converge o quadraic variaion. 86

87 The esimaion of quadraic variaion(qv) δ Le denoes a ime periods beween observaions QV process is defined as following, [ δ ] ( δ ) = δ i δ ( i 1) lim δ 0 i= 1 ( ) ( ) σ ds δ We used daily reurns o measure incremens of he quadraic variaion p 0 s 87

88 6.1.1 Tes for he Presence of a Coninuous Maringale Componen of a Levy Model Tes procedure as follows; Choice a coefficien We choose a hreshold β [ 0.5,1 ] β r ( k) = k Se Δ i Y = Δ i + σ k Z i ( 0,1) Where as T k, Z i ~ N n = k 0 σ T is measured by as an annually, is known sandard deviaion, denoes log-reurns. Δ i 88

89 Tes for he Presence of a Coninuous Maringale Componen of a Lévy Model Null Hypohesis: H 0 : σ 0 Tes saisics is calculaed as following Vˆ k n Δ i= 1 ( iy ) 1 {( Δ Y ) r( k )} = i ˆ Qk = ( Δ iy ) 1 {( Δ Y ) r( k )} 1 3k n i= 1 4 i T k = ˆ k σ V k Qˆ n. k k N ( 0,1) 89

90 Tes for he Presence of a Coninuous Maringale Componen of a Lévy Model { T k > 1.96 } α If is near 0.05, we accep null hypohesis. P Under he alernaive hypohesis : σ 0, above es saisic diverges o +. { } α 05 1 If P > 1.96 >> 0., we rejec he null hypohesis T k H 90

91 6.1. The es wheher he jump componen has finie variaion Null Hypohesis : H : α 1 0 < Δ i M ˆ = Δ 1{ ( ) ( )} k Z, Z ~ N ( 0,1 ) i +σ Δ > r k i i i Vˆ k n Δ i= 1 ( ) i Mˆ 1 ( Δ M ) r( k ) = i Q ( ) k = ΔiM 1 ( Δ M ) r( k) { } ˆ 1 3k n i= 1 ˆ 4 { } i 91

92 The es wheher he jump componen has finie variaion T α k = ˆ k V α n. k kqˆ k N ( 0,1) { } α If P > 1.96 is near 0.05, we accep null. T k 9

93 6.1.3 The es of Presence Jumps The variance of a Lévy processes is esimaed as following form, S = n 1 i= 0 ( ) i+ 1 [ ] Where each i is a division of inerval 0, for each. and = n. Δ n Ν i = = Δ i i 1 i n i 93

94 94 The Tes of Presence Jumps ( )( ) = + = n i i i i i S = Δ Δ = n i n n i i n S

95 The Tes of Presence Jumps H : Null hypohesis: is a coninuous Lévy process 0 Z n S n S K S K 4 4 = K = ( π ) 0. 5 Z n > z α If, we rejec he null hypohesis z α where is a chosen quanile. 95

96 7. Applicaion o Real daa We consider he ime series of NIKKEI 5 in Japan reurns from o and ISE Compound 100 index in Turkey reurns from o , 75 We use daily daa and k = 1 5 = 0, and r ( k) = k We divided reurn series such ha each group have 500 daily reurn for NIKKEI 5 index Similarly, ISE Compound 100 index reurns divided sub groups which include 300 daily reurn. 96

97 Applicaion o Real daa Index Level (JPY) Year FIGURE 1. Hisory of NIKKEI 5 Index in Japan

98 Applicaion o Real daa 15% 10% 5% 0% -5% -10% -15% Reurns (daily coninuosly compounded rae %) 005 Year FIGURE. Hisory of NIKKEI 5 Index reurns in Japan

99 Applicaion o Real daa 0,5 NIKKEI 5 0, Normal disribuion PDF Daily mean: 0.01% Daily sd. Deviaion: 1.3% 0,15 0,1 0, % -1% -10% -8% -6% -4% -% 0% % 4% 6% 8% 10% 1% 14% 16% FIGURE 3. PDF of Daily Reurns of Nikkei 5 from 198/01/04 o 005/09/30 99

100 Applicaion o Real daa 15% 10% 5% 0% -5% % -15% FIGURE 4. Hisory of ISE Composie 100 Index reurn in Turkey 100

101 Applicaion o Real daa Close FIGURE 5. Hisory of ISE Composie 100 Index in Turkey

102 Applicaion o Real daa 14,00% 1,00% 10,00% 8,00% Normal Frequency 6,00% 4,00%,00% 0,00% -7% -6% -5% -4% -3% -% -1% 0% 1% % 3% 4% 5% 6% 7% FIGURE 6. Frequency of ISE Composie 1000 Index in Turkey 10

103 Applicaion o Real daa ISE ,095 0, , , ,07818 Vˆk Qˆ k 0, ,1095 0, , , T k ( j) 0, , ,5644 9, , Table 1. The esing for he presence of a Brownian componen for ISE Composie 100 index price process 103

104 Applicaion o Real daa ISE Vˆk 0, , , ,03769 Qˆ k T k ( j) 0, , ,89E-06 0,018 4,334696, , ,95618 Table. The esing finie variaion of jump componen of ISE Composie 100 index price process 104

105 Applicaion o Real daa NIKKEI 5 Vˆk Qˆ k T k ( j) , ,3476 0, , , ,7848 0, ,0776 0, , , , ,6515 5, ,68 16, , ,37838 Table 3. The esing for he presence of a Brownian componen for NIKKEI 5 Index reurns 105

106 Applicaion o Real daa NIKKEI 5 Vˆk Qˆ k T k ( j) , ,0595 0, ,0498 0, , ,09E-09 0, , , , ,39E-08 0, , , , , ,01413 Table 4. The esing finie variaion of jump componen of NIKKEI 5 Index reurns 106

107 Applicaion o Real daa INDE Z n NIKKIE 5 179,0877 ISE Composie ,6736 Table 5. The esing he presence of jumps under he hypohesis ha price process is coninuous Lévy process. 107

108 Applicaion o Real daa Index NIKKEI 5 ISE Composie 100 αˆ βˆ μˆ δˆ 0, , , , , ,0078 0, ,00038 Table 6. Esimaed parameers of he normal inverse Gaussian disribuion 108

109 8. CONCLUSION The laws of self-decomposable disribuions class are be consiued as a limi laws of Lévy driven Ornsein- Uhlenbeck process These laws can be relaed wih general addiive process. The Lévy processes has a flexible srucure o model in real phenomena in finance such as heavy-ails, jumps and volailiy smile In his sudy, we focus on Levy process and i s incremens laws. Self-decomposable laws are sub class of infiniely divisible laws 109

110 CONCLUSION The family of self-decomposable laws always has boh of selfsimilariy and saionary of incremens. We esed wheher he pure jump Levy process is a suiable model or no for financial reurn series For his es, we used a hreshold esimaor of quadraic variaion by proposed by Con and Mancini We applied his es saisics o wo inernaional index reurn series. 110

111 CONCLUSION Our empirical analysis show ha, Nikkei 5 index model mus include Brownian moion componen The jumps of Nikkei 5 reurn index have infinie variaion ISE 100 reurn index model mus include Brownian moion componen The jumps of ISE 100 reurn index have infinie variaion 111

112 CONCLUSION A self-decomposable random variable has he same disribuion of as a scaled version of iself and an independen residual random variable. Self-decomposable processes are Levy processes wih jump arrival raes ha are decreasing in he jump size A self-decomposable random variable also has a disribuion of class L which means i can moivaed as a limi law wih more general scaling han he Gaussian limi law. 11

113 CONCLUSION This means ha self-decomposable processes can be moivaed as limi laws where he independen infleunces being summed are of differen orders of magniude Thus hey are appropriae building blocks for sochasic processes used o model financial markes. Finally, we can say ha, a good fi model for real reurn daa have o include boh of a Brownian componen and a infinie variaion componen. 113