ESTIMATION OF PARAMETERS AND VERIFICATION OF STATISTICAL HYPOTHESES FOR GAUSSIAN MODELS OF STOCK PRICE

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1 Lhuaa Joual of Sascs Leuvos sasos daba 06, vol 55, o, pp , 55,, 9 0 p wwwsascsjouall ESTIMATIO OF PARAMETERS AD VERIFICATIO OF STATISTICAL YPOTESES FOR GAUSSIA MODELS OF STOCK PRICE Dmyo Maushevych, Yevhea Mucha Uvesé du Mae Addess: Laboaoe Maceau de Mahémaques, Faculé des Sceces e Techques, Uvesé du Mae, Aveue Olve Messae, Le Mas, Face Taas Shevcheo aoal Uvesy of Kyv Addess: Volodymysa s 64, Kyv, Uae E-mal: dmma99@gmalcom, yevheamucha@gmalcom Receved: Apl 06 Revsed: Ocobe 06 Publshed: ovembe 06 Absac We cosuc models of asse pces o he Uaa soc mae ad aalyse he applcably by checg appopae sascal hypoheses usg acual obseved daa We also aalyse he pesece of jumps he dyamcs of dffee asses ad esmae he us coeffce fo he logahm of he pce of he asse by wo dffee mehods Keywods: Uaa soc mae, Uaa Soc Exchage, facoal Bowa moo, esmao of us coeffce Ioduco The mode ecoomy heavly eles o vaous sascal mehods Paculaly wdespead ae he mehods of sascal aalyss of soc pces he soc maes, o pedc he behavou of socs he fuue The exchage segme of he Uaa soc mae s o suffcely aalysed compaso o he soc maes of developed coues I Uae, hee ae e opeag soc exchages, amely: PJSC Uaa Soc Exchage (Kyv) ; PJSC Eas Euopea Soc Exchage (Kyv); PJSC Pespecva Soc Exchage (Dpopeovs); PJSC Uaa Ieba Cuecy Exchage (Kyv); PJSC Uaa Ieaoal Soc Exchage (Kyv); PJSC Kyv Ieaoal Soc Exchage (Kyv); PJSC PFTS ua-exchage (Kyv); PJSC Pydpovsa Soc Exchage (Dpopeovs); PJSC Soc Exchage IEX (Kyv); PJSC Uaa soc exchage (Kyv) We aalyse daa fom he Uaa Soc Exchage, whch focuses o he classcal model of adg socs Sce 009, he Uaa Soc Exchage ades o all majo ypes of secues (socs, bods, opos, fuues) ad adg volumes have bee seadly ceasg I hs pape we vesgae mehods of paamee esmao ad vefcao of sascal hypoheses fo a Gaussa model of he Uaa Soc Exchage Idex, whch s calculaed as he aveage pce of 0 Uaa "blue chp" socs (socs of Uae's lages compaes ad leades he felds) Fo echcal aalyss we use daa fo eceved fom he webse of he Uaa Soc Exchage [3] fo he peod fom 3 Mach 04 o 8 Augus 04 wh obsevaos ae evey 5 mues (a oal of 8000 obsevaos) A sho descpo of he aalysed daa s peseed Table Table Aalysed asses Mag Fequecy of obsevaos Asse Peod UX-5m evey 5 mues Uaa Soc Exchage Idex 3/3/04 8/8/04 Lhuaa Sascal Assocao, Sascs Lhuaa Leuvos sasų sąjuga, Leuvos sasos depaameas ISS ole

2 9 Esmao of paamees ad vefcao of sascal hypoheses fo Gaussa models of soc pce The followg fgue shows dyamcs of he aalysed asse wh espec o s value a he begg of he peod aalysed Fgue Dyamcs of Uaa Soc Exchage Idex dug he aalysed peod Aalyss of he pesece of jumps he dyamcs of he Uaa Soc Exchage Idex Facal maes somemes geeae a sgfca umbe of gaps he pce of cea shaes, so-called jumps May paccal ad heoecal sudes pove he exsece of jumps ad show he sgfca mpac o facal maageme ( pacula, s-maageme ad hedgg of he pofolos of secues) The poblem of jump defcao s que complex, because oly dscee daa ae avalable o eseaches Recely, a vaey of echques ad sascal ess, whch allow he deemao of jumps he dyamcs of soc pces, was developed Some of hem we apply o he asses ha we vesgae The queso of he pesece of jumps s esseal fo he puposes of aalyss of soc dyamcs as hey ca sgfcaly dso he esuls We buld a model of he soc pce o a fxed pobably space ( Ω, F, P), whee flao { F, [ 0, T ]} coespods o he fomao ha s avalable o mae pacpas If hee ae o jumps he dyamc of soc pce S(), we sugges ca be descbed by he sadad logomal model: whee W() s d ( S( ) ) µ ( ) d + σ ( ) dw ( ) F -adaped sadad Bowa moo, whle pocesses µ ( ) ad volaly () l, () σ ae F - adaped pocesses, such ha S() s Io's pocess wh couous ajecoes If hee ae jumps he soc pce dyamc, he d l ( S() ) µ ( ) d + σ ( ) dw ( ) + Y ( ) dj ( ), () whee J () s a coug pocess depede of W( ), whch s esposble fo he appeaace of jumps Y() deemes he sze of jump a me ad does o deped o he pevous jumps o W() Obsevaos of S () ae avalable a dscee me pos 0 0 < < < T, fuhemoe Δ s cosa fo all We cosde local dyamcs of he pocess wh he "wdow" of K cosecuve obsevaos ha ae used o deeme he sascs defed below Defo Sascs L( ), whch allows us o chec f hee was a jump dug he me eval ], s defed as follows (,

3 D Maushevych ad Ye Mucha 93 ( ) L ( ) S l S ( ) ( ) ˆ σ mˆ S whee ( ) j m ˆ l ad K K + S j ( ) ( ) S j S j ˆ σ l l K + S j K j S j, ( ) ( ) [] povdes poof of he followg heoem, whch allows us o buld a sascal es fo defcao of jumps he soc pce dyamcs by dscee obsevaos α Theoem Le L () mee Defo, wh he sze of he wdow K Op ( Δ ), whee < α < 0,5 Assume ha he pocess S ( ) s descbed by equao ( ) o ( ) Le A be a se of {,, } such ha fo me eval (, ] hee ae o jumps If Δ 0, he max A S L ( ) whee ξ has he dsbuo fuco F ( x) exp( e x ) C C ξ, ( l( ) ) l( π ) + l( l( ) ), c 0,7979, S c π c( l( ) ) c( l( ) ) ad s he umbe of obsevaos Based o hs heoem we ca cosuc a sascal es o chec he hypohess 0 : dug me eval (, ] hee wee o jumps he soc pce dyamc agas he aleave : dug me eval (, ] hee was a leas oe jump he soc pce dyamc ad hs dyamc s descbed by model ( ) Le α be he sgfcace level If ( ) L S C ( l ( α )) l β, he we accep he ull hypohess, ohewse we accep he aleave hypohess The paamees used fo he puposes of ou aalyss of jumps he soc dyamcs ae gve Table Table Paamees fo aalyss of jumps he soc dyamcs UX-d K 55 α 0,95 ( ( )) β l l 0,95, C 3,844 S 0, 956 Afe all he ecessay calculaos, accodg o he es, we ca see ha hee ae jumps dyamc of he Uaa Soc Exchage Idex Moe pecsely, 49 jumps oo place dug he peod aalysed

4 94 Esmao of paamees ad vefcao of sascal hypoheses fo Gaussa models of soc pce Fgue The hsogam of me gaps bewee wo cosecuve jumps he dyamcs of he Uaa Soc Exchage As meoed eale, jumps ca have a sgfca mpac o he esuls of he aalyss of he dyamcs of he Uaa Soc Exchage Idex Thus, fo he puposes of fuhe aalyss, coespodg values wee excluded fom he soc dyamcs The followg fgue shows he dyamcs of he Uaa Soc Exchage Idex dug he aalysed peod afe he excluso of he jumps Fgue 3 The dyamcs of Uaa Soc Exchage dex dug he peod aalysed, wh jumps emoved 3 Facoal Bowa moo ad he aalysed model As pa of hs pape we cosde a sochasc model of he Uaa Soc Exchage Idex based o facoal Bowa moo, whch s descbed deal below Facoal Bowa moo s a geealzao of he Wee pocess oweve, ule he case of he Wee pocess, he cemes of Facoal Bowa moo ae coelaed (ad heefoe depede) Pocesses of hs ype wee fsly cosdeed he wo of Madelbo ad Va ess 968 Defo A adom pocess B () defed o he me eval [ 0,T ] s called a sadad facoal Bowa moo wh a us coeffce [ 0,], f: () B () s a Gaussa pocess; B 0 0; () ( ) (3) EB () 0;

5 D Maushevych ad Ye Mucha 95 (4) EB ( s) B ( ) ( s s ) +, fo ay s, [ 0, T] B : The value of he us coeffce deemes he ype of pocess ( ) If If If, he he pocess B () >, he cemes of B () <, he cemes of B () s a Wee pocess; ae posvely coelaed; ae egavely coelaed oe, ha he pocess of cemes of he facoal Bowa moo X ( ) B ( ) B ( ) called facoal Gaussa ose ad () E( B ( + ) B ( ) ) + s EX Facoal Bowa moo has a umbe of mpoa popees, mos of whch ae lsed he followg poposo B has he followg popees: Poposo Facoal Bowa moo ( ) B s self-smla, amely, ems of dsbuos B ( a) a B ( ) () Pocess () Gaussa pocesses oly facoal Bowa moo has he popey of self-smlay () B () s a pocess wh saoay cemes, e: B ( ) B ( s) B ( s) (3) Tajecoes of facoal Bowa moo B ( ) Amog all ae almos eveywhee o-dffeeable oweve, almos all ajecoes B () have a olde expoe scly less ha, e, fo each such pah hee s a cosa C, such ha fo 0 < ε < : ε B ( ) B ( s) C s (4) Fo facoal Bowa moo s possble o defe sochasc egals, also ow as facoal sochasc egals We cosde he followg logomal model of he Uaa Soc Exchage Idex: Y() Y0 exp( μ +σb ( ) ), whee Y s he value of Uaa Soc Exchage Idex a mome ; () () B s facoal Bowa moo wh a us coeffce ; μ ad σ ae coeffces of df ad volaly especvely Fs, we cosde he poblem of esmao of he us coeffce by dscee obsevaos of pocess Y a me pos: () T,,, 4 Esmao of he us coeffce 4 Defo of he us coeffce ad s evaluao usg scalable a The us coeffce s a measue of log-em memoy, whch s used he aalyss of me sees Below we povde a pecse defo Defo 3 The us coeffce s defed ems of he asympoc behavo of he escaled age as a fuco of he me spa of a me sees as follows: R( ) E ( ), S

6 96 Esmao of paamees ad vefcao of sascal hypoheses fo Gaussa models of soc pce whee R ( ) s he age ad S( ) s he vaace of he fs obsevaos Ths defo allows us o buld a smple esmao pocedue fo he us coeffce based o he X,,, m usg he followg algohm [] obseved values of he me sees { } Fo ay,,, m s ecessay o calculae escaled age of he me sees{ X,,, } Thus, cosde he auxlay me sees Z ( X µ ),,,, whee µ X s coespodg aveage The he esmao of he escaled age ca be foud as R( ) max( Z,, Z ) m( Z,, Z ) S( ) ( X µ ) ow a esmao of he us coeffce ca be foud by applyg a egesso aalyss o he followg lea model R( ) l C + *l( ) S( ) whee C s some cosa Y,,, be a me sees whch epeses soc pce a momes We apply hs mehod o acual daa Le { } The he pu daa fo he puposes of aalyss s a auxlay me sees Y + X l,,,, whch eflecs he dyamcs of chages pofably he logomal Y model Usg he R sofwae evome we made he ecessay calculaos ad he coduced egesso aalyss A scae plo of coespodg egesso model s peseed below Fgue 4 Scae Plo Fom he esuls of he aalyss we foud he esmao of us coeffce o be ˆ We also calculaed uppe ad lowe bouds of 95-pece cofdece eval fo ˆ Up , ˆ Dow

7 D Maushevych ad Ye Mucha 97 Thus we see ha he us coeffce fo he obseved values of he Uaa Soc Exchage Idex s geae ha /, dcag a posve log-em coelao bewee cemes of Uaa Soc Exchage Idex Thus, afe hgh values Uaa Soc Exchage Idex ceases ae moe lely, ad vce vesa Ths ca be paally explaed by fluece o he soc mae of vaous macoecoomc facos ad he oveall ecoomc suao (we obseve a seous log-em decle sce 008, esulg fom he log-em global ecoomc css) oweve, gve he fac ha we cosde a Uaa Soc Exchage Idex wh a hgh fequecy of obsevaos ( ou case - evey 5 mues), we ca see ha he above facos have much less mpac, ad shoem adom facos have much moe sgfca effec 4 Esmao of he us coeffce he aalysed model usg quadac vaaos Fo he aalysed model he us coeffce ca be esmaed usg he popey of self-smlay of facoal Bowa moo Fo hs pupose we use he followg heoem, whch s a cosequece of esuls of Dozz, Mshua ad Shevcheo [5] Theoem Le B () be a facoal Bowa moo wh us coeffce ha s obseved a jt momes j, j 0,, The ( B ( j ) B ( jl ) T l, l Based o hs heoem we ca pove he followg poposo Poposo Le he pocess Y() sasfy he followg model Y( ) exp ( μ +σb ( ) ) whee B () s a facoal Bowa moo wh us coeffce, μ ad σ ae cosa, ad hs jt pocess s obseved a momes j, j 0,, The Y( ) j l σ T l, l Y( jl) Poof By he defo of pocess Y(), we oba Y ( + ) l j ( ) μ + σ ( B ( ) ( ) j B j Y j j l j T T μ + μσ B ( j ) B ( j ) + σ ( B ( j ) B ( j ) T μ + Cμ T + ε + σ ( ( ) ( ) B j B j Whee he las equaly we used popey (4) of facoal Bowa moo fom Poposo I follows fom Theoem ha he las sum s σ ( ( ) ( )) j jl l The fs wo sums go o zeo fase ha B B σ T l fo ε < Thus, we have

8 98 Esmao of paamees ad vefcao of sascal hypoheses fo Gaussa models of soc pce ( j+ l ) ( ) Y l σ T Y l j l Poposo allows us o buld a smple esmao fo he us coeffce, whch s deved fom jt obsevaos of pocess Y() a pos j, j 0,, ad based o he ao of quadac vaaos ( j ) ( j ) ( j ) ( ) Y l ˆ l Y l( ) Y l Y j Obvously he cossecy of hs esmao s a dec cosequece of Poposo We also appled hs esmao pocedue o he acual values of he Uaa Soc Exchage Idex fo he peod aalysed, ad deemed ha he esmae of he us paamee s ˆ 0536 The wo dffee esmao mehods ha we cosdeed gve almos decal esuls Fo fuhe aalyss we use esmao Ĥ ow we ca cosde he poblem of esg he hypohess of whehe he obseved dyamcs of he Uaa Soc Exchage Idex coespod o he aalysed model 5 Tes fo he vefcao of whehe he obseved daa ca be descbed by he aalysed model 5 Peseao of he Wee pocess as a facoal sochasc egal Facoal Bowa moo ca be peseed he fom of a sochasc egal fom a Wee pocess ad vce vesa I pacula, [4] Ila oos, Eso Valela ad Joma Vamo cosdeed eel, facoal sochasc egals whch mae possble o cove facoal Bowa moo o a Wee pocess Ths asfomao s descbed deal below Cosde he followg fuco ( ) ( ) ws, cs s, s ( 0, ) 0, s ( 0, ) 3 whee c B, + The followg heoems ae poved [4] Theoem 3 Le B () be a facoal Bowa moo wh us coeffce The he ceeed Gaussa pocess has depede cemes ad EM 0 (, ) ( ) M wsdb s c 4 ( )

9 D Maushevych ad Ye Mucha 99 3 Γ whee c I pacula M s a magale Γ + Γ( ) Theoem 4 Le M be a sochasc egal whch was defed Theoem 3 ad defe W s dms c 0 The W s a Wee pocess Thus, he facoal Bowa moo ca be asfomed o Wee pocess hough smple egal asfomao oe ha hs popey s chaacesc fo facoal Bowa moo Thus, f he pocess W, cosuced as above, s a Wee pocess, he he pocess B ( ) s facoal Bowa moo Ths feaue allows o buld a es ha vefes whehe a me sees ca be descbed by he aalysed logomal model 5 Sascal es of coespodece o he aalysed model Ou as s o buld a es based o he obsevaos of me sees Y() a pos T 3, 0,, whch allows us o chec he followg hypohess: 0 : Y() s descbed by model Y( ) Y exp 0 ( μ+ σb ( ) ), whee B () s a facoal Bowa moo wh us coeffce, ad μ ad σ ae he coeffces of df ad volaly, especvely I ou case me sees Y() epeses he Uaa Soc Exchage Idex Cosde he followg lea asfomao of he me sees Y( ) : T T ( + ) T T M w, l Y l Y,,, 0 T T T m w,,,, 0 ad fuhe he asfomao of he obaed me sees: T ( + ) 0 L M M,,, T ( + ) 0 l m m,,, By Theoem 4, fo suffcely lage, he coespodg sums covege o egal asfomaos ad L μl + W,,, T whee W s a Wee pocess Thus, whe he ull hypohess holds, he followg egesso model s a sadad Gaussa egesso: L+ L μ( l+ l) + ε, 0,, whee ε s a sequece of depede omal vaables

10 00 Esmao of paamees ad vefcao of sascal hypoheses fo Gaussa models of soc pce Thus ode o cofm he ull hypohess, s suffce ad ecessay o cofm he hypohess of omaly of esduals he above-meoed egesso model Fo hese puposes, we wll use he Jaque Bea es (see [6]-[7]) oe ha as was show he pevous seco, f he Jaque Bea es shows ha he esduals ae omally dsbued, he Y() sasfes he aalysed model Thus, fo a lage umbe of obsevaos he pobables of Type I ad Type II eos fo ou es cocde wh he coespodg pobables of he Jaque Bea es fo esduals Afe coducg all ecessay asfomaos, we had 0 obsevaos of L ad l ad bul he coespodg egesso model I ode o apply he Jaque Bea es, we calculaed he value of he Jaque Bea sasc fo he umbe of obsevaos 0 ad he umbe of egessos : 9 ( 3) JB S + K 6 4 whee S ad K ae he coeffces of sewess ad uoss of of he esduals of he egesso model A sogam of he esduals of he egesso model s peseed he followg fgue Fgue 5 sogam of he esduals he aalysed egesso model The calculaed value of Jaque Bea sasc s JB Thus, fo sgfcace level α 005 he heshold value fo he es s ( Q χ ) ( 005) The acual value of he sasc s less ha he heshold value, ad we accep he ull hypohess Thus we ca coclude ha he poposed model ca be used o descbe he dyamcs of asses o he Uaa Soc Exchage, a leas he sho em pespecve 6 Coclusos The poblem of eseachg he Uaa soc mae has ecely become moe mpoa ad uge Compaed o he soc maes of he ecoomcally developed coues of Euope ad Ameca, whee he coespodg mahemacal appaaus has bee developg sce he mddle of he weeh ceuy, he Uaa mae s compaavely youg The pupose of hs pape s o mae he fs seps owads coducg he ecessay aalyss I hs pape we aalysed he pesece of jumps he dyamc of asses o he soc mae ad sascally cofmed he exsece The esuls ae mpoa ad ca be used fo a wde age of puposes I pacula, s ecessay o cosde he effec of jumps dug he calculao of he fa pce of devaves o he asses of he Uaa Soc Exchage Fo he puposes of fuhe aalyss, jumps wee excluded fom cosdeao

11 D Maushevych ad Ye Mucha 0 Also hs sudy we evaluaed he us coeffce fo he logahm of he pce of he aalysed asses by wo dffee mehods I pacula, we showed ha he asses o he mae mgh have log-em memoy, bu gve a lage umbe of adom facos affecg he value of a asse a evey mome, he esmaed value of he us coeffce appeas o be close eough o The poblem cosdeed hs pape s exemely eleva o mode Uaa eales, because a developed ad ope soc mae s oe of he ecessay facos fo he ecoomc developme of a couy The esuls of hs wo fom a bass fo fuhe aalyss Refeeces Suzae S Lee, Pe A Mylad Jumps Facal Maes: A ew opaamec Tes ad Jump Dyamcs Oxfod Uvesy Pess, 007 Bo Qa, Khaled Rasheed us Expoe ad Facal Mae Pedcably Facal Egeeg ad Applcaos, ovembe 8 0, 004 MIT, Cambdge, USA 3 Webse of he Uaa Soc Exchage hp://wwwuxua/ 4 Ila oos, Eso Valela ad Joma Vamo A elemeay appoach o a Gsaov fomula ad ohe aalycal esuls o facoal Bowa moos Beoull 5(4), 999, Maco Dozz, Yulya Mshua ad Geogy Shevcheo Asympoc behavo of mxed powe vaaos ad sascal esmao mxed models 6 Jaque, Calos M; Bea, Al K (980) "Effce ess fo omaly, homoscedascy ad seal depedece of egesso esduals" Ecoomcs Lees 6 (3): Jaque, Calos M; Bea, Al K (987) "A es fo omaly of obsevaos ad egesso esduals" Ieaoal Sascal Revew 55 (): 63 7 BIRŽOS KAIŲ GAUSO MODELIŲ PARAMETRŲ VERTIIMAS IR STATISTIIŲ IPOTEZIŲ TIKRIIMAS Dmyo Maushevych, Jevheja Mucha Saaua Sudaom Uaos veybų popeų os ayvų aų modela, a aamas sases hpoezes ealems duomems, amas jų amumas Įvaų veybų popeų aų damoje amas šuolų buvmas veamas veybų popeų aos logamo uso desas dvem sgas meodas Rešma žodža: Uaos veybų popeų a, Uaos veybų popeų bža, upmes Bauo judesys, uso deso vemas