STOCHASTIC PROCESSES WITH SHORT MEMORY

Transcription

1 TOCHATIC PROCEE WITH HORT MEMORY Dmiry Zhain Deparmen o Higher Mahemaic and Mahemaical Phyic Tomk Polyechnical Univeriy Tomk Lenina avenue Ruia The mahemaical model o a linear yem wih he hor memory aou own ochaic ehavior i propoed. I i aumed ha he yem i under a coninual inluence o independen ochaic impule. In a hor memory approimaion he epreion o he ochaic proce i ound. An applicaion o he model propoed o capial marke procee i eamined. The approach allow orm a ochaic dierenial or procee concerned. The analog o he lack-chole equaion or ae deal on a marke wih he memory i epreed. JEL: C6 D4 G Keyword: ochaic procee auoregreive model addre: zhain@mph.phd.pu.edu.ru

2 . Inroducion The heory o ochaic procee ook a wide applicaion or modeling o he realiy i.g. Tijm 994. u i hould e noed ha he mo developed par o he ochaic procee heory i he heory o Markov proce Roger &William. Noe ha he Markov proce i a ochaic proce uch ha he uure proce realizaion depend olely on curren oday proce value and doen reer o any previou realizaion. Acually he realiy i oo complicaed ha here are a numer o procee uch ha he uure depend rongly on he hiory paed. For inance people ha ake deciion on capial marke are eniive o he hiory o ecuriie in ome nonrivial way. I i eay o imagine ha any real marke i a ime-depended nonlinear yem ee Gare 996 Mandelro 963 and Peer 994 o any capial marke proce hould e decried in appropriae way. I ollow ha a capial marke proce i an auoregreive one. All hee ac caue a wide applicaion o model wih variale volailiy like ARCH e.g. Roer Engle GARCH ee ollerlev 986; ollerlev e al. 993 EWAC ee Luca 99 and o on. A i ollow rom work o Naukonen a preen ime here doen ei any model wih variale volailiy ha eceeding oher. A choe o he model i caued moly y peciiciy o he ak olved. In preen paper we conider a linear yem ha i under coninual inluence o independen ochaic impule. We aume ha an evoluion o he linear yem depend on he memory eec o own previou deviaion rom mean. We ound he

3 epreion or he ochaic proce ha generaed y uch linear yem and conider an applicaion o he model propoed o capial marke procee.. Formulaion o he model Le u conider a linear yem a ime. We aume ha aring rom momen he yem i under inluence o ochaic impule y. Thu we have y d. Now uppoe ha impule have orm η δ where δ i Dirac dela k uncion; η are independen ochaic variale wih ollowing mean and variance: M { η } a k k { η } k k D. Here a k and k are numerical equence uch ha all k >. I i clear ha linear yem could e decried in erm o ochaic proce η. 3 k y aumpion ha ime inerval eween impule en o zero we ge he linear yem ha i under coninual inluence o independen ininieimal peruraion. Then we oain: k dη 4 where dη are independen ochaic variale wih ollowing mean and variance { d η } a d D{ d } d M η. 5 3

4 Hence ochaic proce 4 i characerized y mean and variance M { } a d D{ } d. 6 Wihou lo o generaliy i can e aumed ha proce 4 can e wrien in orm: a d dw 7 where he ochaic proce dw i deined in he ollowing way { dη } dη M dw. The inegrand o he ir iem in righ ide o epreion 7 we will name a deerminiic iem. A aove he inegrand o he econd iem in righ ide o epreion 7 we will name a ochaic iem. Furher we aume ha he dynamic o linear yem i eniive o deviaion o ochaic proce 7 rom own mean a momen paed. uch eec we can call a memory eec in hi cae we will ell aou proce memory a well. We can aume wihou lo o generaliy ha uch kind o memory depend on value o deviaion. Moreover here ei a peciic ime ha deine a ize or deph o memory o he pa. The inluence o deviaion can e decried wih help o weigh uncion where i he curren ime i ome ime paed <. In he general cae weigh uncion i deined rom peciiciy o he ak olving. u we can wrie down ome general condiion or hi uncion: 4

5 5 lim 8 The oal conriuion o memory eec i he ime-averaged value ha can e accouned a addiive erm o ochaic proce 7. o we ge { } d dw d a M. 9 The reader will have no diiculy in howing ha he value o memory eec decreae a well a oervaion period increae. I ollow eaily ha he mean o proce 9 i equal o { } d a M. Hence or proce 9 we oain d dw dw d a Φ where { } d M Φ. The erm d Φ decrie an inluence o memory eec o a econd order i.e. decrie he deviaion rom mean ha are generaed y previou memory eec. Taking ino accoun ha he memory ize i aumed o e le hen

6 oervaion ime we can uppoe ha he conriuion o econd order memory eec i negligile. I ollow ha or he linear yem wih or memory ize we oain: a d dw d dw. 3 y changing he order o inegraion in 3 we ge a d d dw Procee o a capial marke I i oen aumed ha ae price mu move randomly ecaue o eicien marke hypohei. Le u remind ha he mo commonly ued mahemaical repreenaion o ae price i a preenaion in erm o diuive procee. Hence we can wrie down a ochaic dierenial or ae price ee lack F. and chole 973 Hull 996 Wilmo 993: δ a δ δw. 5 Here a and are piecewie-mooh uncion δ W i a Wiener proce. I hould e noed ha he preenaion o a marke proce a a linear yem ha i under coninual inluence o independen ochaic impule doen ake ino accoun a poile eniiviie o he hiory o uch procee. u i i eay o imagine ha hi memory ake place ecaue o people ha deal on he marke keep in mind previou ae price. A wide ue o he ARCH and imilar o ARCH model alo conirm hi aumpion. 6

7 Wihou lo o generaliy we can aume ha he random walk o he ae price i caued y normal random change o eernal and unepeced inormaion. Thi mean ha ae price can e mahemaically preened a h ep. 6 Here h ome ochaic proce which deerminiic and ochaic erm we will noe y A and accordingly. y previou aemen we will aume ha he value o each realizaion o he proce h a ime i deined y um o deerminiic ochaic erm and y he addiive erm ha decrie he poile memory aou previouly paed deviaion rom he mean. o we ge h A d d dw. 7 To analyze a dynamic o uch capial marke proce i i ueul o rewrie proce 7 in erm o ochaic dierenial δ h h δ h. ince 7 i he diuion proce he ollowing ochaic dierenial ake place δh A δ d d δw.8 Le u noe ha epreion 8 i wriing down a he limi o equidiance ime grid when diance eween ime poin end o zero. In pie o he ac ha i i impoile o correcly predic uure ae o he marke we can evaluae uure value o he mean A and volailiy uing ond price and implied volailiy urace ee Wilmo 993 Hull 996. Le e he volailiy o proce 8 7

8 8 d d. 9 I i oviou ha a ecome mall enough he econd iem in inegrand 9 ecome negligile aer han ir iem o he inegrand. o an aympoic ehavior o volailiy 9 or mall i given y d. Now i we recall 6 we ge W A δ δ δ. 3 Epreion 3 deine a diuion proce wih or memory aou ochaic deviaion rom mean. Uing 3 i i eay o wrie down he lack-chole equaion or ae deal on he marke wih or memory eec. Thereore we have rv V V V. 4 Here VV i an opion price r i a rik-ree inere rae. Le u conider he cae when uncion функция i given y ollowing epreion ep 5 Then or volailiy 9 we ge d π Er ep 6

9 where Er i he well known error uncion. I can eaily e checked ha a he econd iem in he epreion 6 i equal o zero and he ochaic dierenial ecome decrie a dynamic o he ochaic proce wih volailiy. The increaing o caue he increaing o volailiy o proce h. Thi volailiy aympoically end o volailiy o eernal impule or large oervaion period. 4. ome concluion and remark In he preen paper we howed ha any linear yem ha i under coninual inluence o independen eernal impule and wih hor memory eec ha i caued y previou deviaion rom mean a ime paed can e decried wih help o ochaic proce wih variale volailiy. Le u noe ha uch kind model are wide ued o inveigae an auoregreion o ochaic ime erie. In conra o ARCH and imilar o ARCH model we didn make any addiional uggeion aou ehavior o ochaic proce. We howed ha he mehod propoed allow decrie procee o capial marke and i in accordance wih no arirage policy. Uing he oained reul we wroe down he lack-chole equaion or a opion price. I i eay o ee ha or vanilla opion and or weigh uncion in orm 5 he opion price increae ogeher wih volailiy 6. Reerence: lack F. and chole M The Pricing o Opion and Corporae Liailiie. Journal o Poliical Economy 973 vol. 8 p

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