A Hybrid Model for Estimation of Volatility of Call Option Price Using Particle Filter

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1 IJCSI Ieraoal Joural of Compuer Scece Issues, Vol. 9, Issue 4, o, July 0 ISS (Ole: A Hybrd Model for Esmao of Volaly of Call Opo Prce Usg Parcle Fler Sul Kumar Dhal, Prof.( Dr. Srvash Prasad, Prof. (Dr. Maoraa ayak 3 Assocae Professor, Regoal College of Maageme Chadrasekharpur, Bhubaeswar, Odsa, Ida Dea( Academc,Gadh Isue of Techologcal ad Advaceme Gohrapara, Bhubaeswar, Odsa, Ida 3 Charma, Isue of Techcal Educao ad Research Gagapaaa, Bhubaeswar, Odsa, Ida Absrac I he rece years, he dsrbuo of possble fuure losses for porfolos, such as bods or loas, exhbs srogly asymmerc behavor. I hs paper, we have aalyzed he effecve porfolo rsk maageme hrough a compuaoal sae space model by usg parcle fler hrough sequeal esmao of volaly. The compuaoal model comprses wh Exeded wegh Movg Average Model ad Black Scholes-Opo Prcg model as well as GARCH deermsc volaly model. The oucome of he model esablshes he effecveess of parcle fler for esmag volaly of call opo prces for fuure porfolo reurs ad ca able o predc he vesor s facal rsk ad measures a sgfca maer. Keywords: Porfolo, facal rsk, volaly, parcle fler, call opo, pu opo.. ITRODUCTIO The volaly of a sock s defed as he measure of varao of prce of a facal srume over a me perod. Whe he me perod of eres s oe year, he he volaly s a aual volaly ad whe he me perod of eres s oe day, he he volaly s a daly volaly Whasoever, aual volaly s frequely esmaed by frs esmao daly volaly usg daly log sock reurs daa. The hree ma purposes of Esmag volaly are for rsk maageme, for asse allocao, ad for akg bes o fuure volaly. A large par of rsk maageme s measurg he poeal fuure losses of a porfolo of asses, ad order o measure hese poeal losses, esmaes mus be made of fuure volales ad correlaos The Black-Scholes paral defereal equao ad ulmaely solve he equao for a Europea call opo I he BSOPM (Black Scholes-Opo Prcg model framework, he aual volaly s ake as cosa. I employs a commo mehod whch smply calculaes he year sample sadard devao of he daly log reurs of he sock over he pas days by usg he equaos as below: S R R ( ( Where he average value of he sock reur s gve as R R Where s he umber of sock reur ad R l( S / S ( S gves us a esmae of daly volaly.. Sce year s a aual quay; we have o scale or esmae S. whch s esmaed by year s year / TD (3 Where TD s he aual umber of Tradg Days (TD To smulae reur values for esg he mehods, we wll use he sochasc dffereal equao ha correspods o geomerc Browa moo, ds d dx S where S ( s he sock prce a me, s a measure of he average rae of growh of he asse prce, d s he chage me, s he volaly, ad dx s kow as a Weer process because s a radom ormal varable wh a mea of zero ad a sadard devao of d For he umercal smulao he al asse prce was se equal o p, Where p s a umercal value. I erms of S = p, ad S s he closg prce for day. Copyrgh (c 0 Ieraoal Joural of Compuer he Scece model Issues. All Rghs Reserved. 0

2 IJCSI Ieraoal Joural of Compuer Scece Issues, Vol. 9, Issue 4, o, July 0 ISS (Ole: We wll assume ha we ca oba a closg sock prce for 365 cosecuve days S S S d S dx The sample sadard devao S, provdes a very smple ool for esmag daly volaly sce assgs, equal wegh o each daly log reur R. Apar from hs, we ca also ulze que more accurae weghed echques Lke ARCH ad GARCH models. From equao ( R - whch s defed as he couously compouded reur o he sock durg day -.. Squarg ad S of he codoal daly varace o day, usg he mos rece ervaos of u, we ca oba he equao (4 as below: ^ _ ( R R _ ^ ( R R (4 We ca employ aoher allurg echque for esmag he codoal daly varace ha volves assgg weghs o he mos rece ervaos of u as show he equao (5. ^ _ ( R R Where, (5 0, Whe >= As we agree o a po ha he mos rece ervaos mus be assg more wegh as compare o he earler ervaos. The rece ervaos lkely coas more formao abou he curre level of he codoal daly varace. The oal Wegh of all ervao mus be hudred perceage.e oe. Furher hs dea ca be exeded by addg a log ru average VL he equao (5. The log ru average mus have a wegh as specfed equao (6. ^ _ V ( R R L (6 Whe ad 0, Whe >= If we replace u equao (6 by he rue log-erm rue mea u of he reur U, he we ca able o oba a deermsc expresso for he rue codoal daly varace. Ths leads o Egle s ARCH( model, where he weghs aga sum o uy ad he expressoal represeao s as below: ^ V ( R Where L u (7 u ca be cosdered o be zero. The Bollerslev s GARCH approach whch exeds he dea of Egle s ARCH approach equao (7 by applyg rue codoal daly varace from pas days o he deermsc expresso for he rue codoal daly varace of he curre days. The GARCH ( P,Q model may be defed as he equao (8, 9, 0 z; u u Where ^ P Q VL M ; V 0; L 0; for,,..., ; 0, for,,...,m. (8 (9 (0 The erm s a zero mea radom dsurbace, or sock, he mea u of U. The equao (9 s kow as he codoal daly mea model ad equao (0 s kow as he codoal daly varace model. ca also be vewed as he codoal daly varace of he GARCH dsurbace erm value as: wh he expressoal u V (u V ( E ( V (, Copyrgh (c 0 Ieraoal Joural of Compuer Scece Issues. All Rghs Reserved.

3 IJCSI Ieraoal Joural of Compuer Scece Issues, Vol. 9, Issue 4, o, July 0 ISS (Ole: The mos popular specfcao of he codoal daly varace for he smple GARCH(P,Q model s a GARCH(, whch s represeed as: ow we roduce he followg sequeal esmao of volaly model framework wh sysem equao ad ervao equao as follows: L V, Where ; VL 0;, 0 ( Sysem Equao : [ ( ] Observao Equao: Obs C [ BSOPM ( P(( S S :, S, T The expoeally weghed movg average (EWMA model s a parcular case of GARCH(,, model whe we se γ=0, ad he equao ( The EWMA Model wh he expresso as follows: ( For modelg me varyg volaly purposes, ulzao of EWMA models ca produce more sgfcace. For esmag volaly from a well recogzed model lke GARCH(M, ca be embedded wh Black Scholes Opo Prcg Model usg Auxlary Parcle fler echques. The Prcg model for Europea Syle call opo wh o-dvde uderlyg sock prce of Black Scholes Opo Prce model has he soluo for C (Call Opo a Tme r( T C S ( d Ke ( d (3 Where r ( T l( S / Ke d T T r ( T l( S / Ke d T T here deoes he cdf of he sadard Gussa (0,. S s he uderlyg sock prce a me, r s he rsk free aual rae of eres, K s he Srke Prce of he call opo o be maure T-.. MODEL FRAME WORK Geerally represes he daly codoal varace of he uderlyg sock prce a day, whereas S s deoed as, he uderlyg sock prce a day. Le we defe c as he erved marke prce of he call opo a day. T s deoed as he maury dae of he call opo ad r as aual rsk free rae of eres. Where λ=0.94 The errors he sysem ad ervao equaos are addve whch ca be ad as such he equaos may aurally be expressed as follows. Sysem Equao : X l( l([ ( ] Observao Equao : Obs Z l( C l([ BSOPM ( P(( S S :, S, T As per he defo, he sysem error erm whch s addve o esure he o-egavy of he codoal daly varace, requred o be oegave. The ervao error erm s addve so ha he erved opo prce c s esured as o-egave ad helps o eforce Mero's frs lower boud,. The addve error erms sgfes ha he "error" geerally scales wh he sgal whch meas, o sheer bass, hgher values of boh ad c are more proe o hgher ose as compared o lower values of ad c. Sysem ose equao (0 s represeed as ad he addve ervao ose equao ( s represeed as. I he sysem equao (0, bechmark Rskmercs EWMA model s ulzed. As meoed by Sergy Ladokh hs hess wh aalyss, he EWMA model s a smple ad geeral model ad well acceped o a wde rage of sock reur daa. Whle developg hs Rskmercs model, aalyss a J.P. Morga has processed 480 facal me seres ad assocaed each seres wh a "opmal" decay facor whch mmzed he roo mea squared error (RMSE of he codoal varace forecas as specfed Copyrgh (c 0 Ieraoal Joural of Compuer Scece Issues. All Rghs Reserved.

4 IJCSI Ieraoal Joural of Compuer Scece Issues, Vol. 9, Issue 4, o, July 0 ISS (Ole: [4]. The model employs, RMSE as he forecas error measure crero. For he daly log sock reur daa, was erved ha wh a decay facor of = 0.94, ca able o yeld he opmal resuls for he gve se of me seres. Apar from ha, I was also dscovered ha hs parcular specfcao of he EWMA model cossely ca able o capure varous characerscs of daly log sock reur daa, alog wh volaly cluserg whch aoher advaage of hs model. Rskmercs EWMA model assumes ha z ~(0,, where z as per he GARCH (P,Q model. The radom sysem error bascally capures he osysemac bases of he EWMA model. I s eded o accou for hose errors of he deermsc EWMA model whch eher vares radomly or o-sysemacally over a perod of me. I he oher sde, he deermsc volaly model s less capable o capure may of he complex feaures of sock reur daa, such as he leverage effec, ec., hece hs radom error erm s requred o be cluded for smooh fucog he EWMA model. Bu praccal, heeroskedasc error s more suably cluded as opposed o a smple whe ose assumpo for. I addo, may be worhwhle o model ay sysemac elemes of he EWMA model error, hough hrows more challegg ask. Pragmacally modelers may be more eager o use more sophscaed GARCH models, lke he E-GARCH model, raher ha aempg o model he EWMA model error due o s complex characerscs. I he ervao equao (, BSOPM s used as a base model for he erved opo prce, whle allowg for a radom ervao error whch s accoued for he o-sysemac shorcomgs of he BSOPM. I s appare ha equao (3 expresses l (c as a o-lear fuco of he sae l(. For smplfcao, he assumpo s ake ha he radom error ha processes ad are bascally represeed as Gaussa whe ose. O he oher had wh he lepokuroc aure of facal daa, seems ha a more fa-aled dsrbuo such as he Sude mgh be approprae o yeld he approprae resul. Tha s why, s appare ha a parcle fler s far more approprae as compared o a Exeded Kalma fler, despe hs beg mos basc model, because he parcle fler has bee desged a dversfed maer whch ca very effecvely able o hadle olear, o-gaussa sae space models. 3.. ASSUMPTIOS UDERTAKE I THE PROPOSED MODEL As hs model s beg passed hrough a Auxlary parcle fler, defely, all of he uderlyg assumpos of he parcle fler well ulzed here also. Sx assumpos are beg ake for smooh fucog of he desged model whch are specfed as follows: A : The Sysem error erms ~ IID ( 0, deoes Gussa whe ose where deoes he sadard devao of he sysem error process. A : I he sysem equao, he Rsk mercs Exeded Weghed Movg Average geerag process Z ~ (0,, where z. Tha sgfes ha, he Rskmercs model s a specfcao of Gussa GARCH(, model. A3 : The al pdfs of he wo sae vecor varables are Gaussa l( ~ ( 0, SD l( ad l(c ~ (0, SD l(c Where SD l( ad SD l(c deoe he sadard devao of he al sae vecor varables. Here, has bee assumed ha he radg akes place a arbragefree evrome, where he marke s compleely effce wh he vew ha uusual rades do o ake place. For whch, a opo equlbrum or rue heorecal value C s assumed o be more equvale o s erved marke prce C abs. The BSOPM model error s fully ervable, eve hough reflecs he models dvergece from a heorecal opo prce C. A4: Here, s assumed ha here exss a zero correlao bewee he uderlyg sock prces S ad he BSOPM model error ( or ervao error. The crcal po s ha whe he uderlyg sock prce daa ad opo prce daa are o recorded sychroously or he acual radg of he uderlyg sock ecessae he radg of he assocaed equy opo a ha me, hs assumpo s o vald.. A5: The codoal dsrbuo of S T s logormal ad allows o be expressed close forma A6: I he las assumpo, he uderlyg sock prce S ad he rsk-free model are erved whou error. Alhough ca be assumed ha errors he erved Copyrgh (c 0 Ieraoal Joural of Compuer Scece Issues. All Rghs Reserved.

5 IJCSI Ieraoal Joural of Compuer Scece Issues, Vol. 9, Issue 4, o, July 0 ISS (Ole: values of S are already erved he ervao error. 4. MODEL SIMULATIO For model smulao, we have o frs race ou he specfc parameer values ad al values ha have bee used hs parcular smulao process so ha smulao becomes easer. As, has bee erved ha, he sadard devao SD v, of he ervao error v s defed a a very small quay.e whch emphaszes our cofdece ha he error-adused BSOPM prce c probably does o devae from he erved marke prce c by more ha roughly 0%.e. e ±3SDV = e ±0.0 (0.74,.. Therefore, a al value l ( c of.9 has ake because ca able o fds ou he approxmae dfferece of 5% bewee c ( ad c (where l (c =.8 effecvely. All he al values aa parameers are used he model s lsed he Table. u 0.0 Ial value of he daly log reur o he uderlyg sock (bewee ed of day ad ed of day 0. z.9 Ial value of he ervao a day =. SD η 0.05 Sadard devao of frs compoe of sysem ose. SD ξ 0.04 Sadard devao of secod compoe of sysem ose. SD v 0.05 Sadard devao of ervao ose. 5. AALYSIS OF THE SIMULATIO RESULT TABLE : IITIAL VALUES AD PARAMETERS FOR MODEL SIMULATIO Varable Value Meag T 56 Maury of he call opo (days. TD 56 Aual umber of radg days. K 50 Srke prce of he call opo (Rs. p 000 umber of parcles per me sep (.e. day. Fg. : Daly Log reurs ad Observed opo prce The smulaed sock prce expereces a geeral growh red over he 56 day perod alhough here are sporadc perod of decle S as well ad hey are proeced fg. o fg. 6 wh dffere arbues. r 0.0 Rsk free rae of aual eres xl Ial value of he frs compoe of he sae a day =. x.8 Ial value of he secod compoe of he sae a day =. S 55 Ial value of he uderlyg sock a day =. µ u Log-erm average of he daly log reurs o he uderlyg sock. Fg. : Daly Log reurs ad Expeced Observed value Copyrgh (c 0 Ieraoal Joural of Compuer Scece Issues. All Rghs Reserved.

6 IJCSI Ieraoal Joural of Compuer Scece Issues, Vol. 9, Issue 4, o, July 0 ISS (Ole: The Sadard devao of he erved error s foud o be que small a.05. Ths cofrms our belef ha he error adused by BSOPM prce C probably does o devae from he erved marke prce C Obs by more roughly 0%. I has bee erved ha Auxlary parcle fler s effecve rackg he dyamc of boh compoes of he sae vecor. As our smulaed sock does o experece may sharp umps s volaly; whch proves ha he me seres s farly seady comparso o ohers. Fg. 5: Sysem ose of Smulaed Model Fg. 3: Daly Log reur of Uderlyg Sock Prce Fg. 6: Observao of Smulaed Model Aoher fac has bee erved ha, he predcve dsrbuo ge progressvely wder as we move from oe sep predco o seve sep predco whch s due o he uceray of our esmaes creases wh he forecas horzo. I over 95% of he me seps, he 95% PIs for he oe-sep, wo sep, ad seve seps predcos ca able o cofe he rue erved value. 6. COCLUSIO Fg. 4: Daly Log reur of Sysem ose The proposed hybrd model whch s a complee equlbrum model of he opo prcg problem, provdes he fal formula, whch s he fuco of ervable varables whch ca effecvely ulzed for solvg problems hrough parcle fler. Raher ha assumg ha he logarhm of he sock prce follows a ormal dsrbuo, we assume ha he square roo of he sock prce follows a ormal dsrbuo. Due o s mahemacal as well as compuaoal characerscs our proposed model hs paper carres a pragmac alerave for rsk aalyss of porfolos. Copyrgh (c 0 Ieraoal Joural of Compuer Scece Issues. All Rghs Reserved.

7 IJCSI Ieraoal Joural of Compuer Scece Issues, Vol. 9, Issue 4, o, July 0 ISS (Ole: Refereces [] Lopez, J., Evaluag he Predcve Acuracy of Volaly Model, Joural of Forecasg, vol. o 0,00, pp [] Hull, J. Opos, fuure, ad oher dervave (ffh edo, ew Jersy: Prece Hall, 003. [3] Paga, A.R ad G W Schwer, Alerave Models of Codoal Sock Volales, Joural of Ecoomercs vol o 45,990 pp [4] Arulampalam, M., Maskell, S., Gordo,., ad Clapp, T., A uoral o parcle flers for ole olear/o- Gaussa Bayesa rackg, IEEE Trasacos o Sgal Processg, vol o 50,003,pp [5] J. P. Morga, Rsk Mercs echcal docume (Fourh edo, Morga Guaraee Trus Compay [6] Sergy Laokh, Bhula Sada, Volaly Modelg facal markes Maser Thess, July 009 [7] Lsmeer, T., ad Pearso,., Rsk Maageme: a roduco: a roduco o value a rsk, Illos Uversy Ecoomcs Workg Paper, 996. [8] Rog, Ady, Chag-Cheg, A Comparso of he Forecasg Volaly Performace o EWMA Famly Models, Ieraoal Research Joural of Face ad Ecoomcs, Issue(54,00 PP 0-3 [9] Kagawa, G., Moe Carlo fler ad smooher for o- Gaussa olear sae space models, Joural of compuaoal ad Graphcal Sascs, vol o 5,996, - 5. [0] Abke, P., ad ad, S., Opos ad volaly, Ecoomcs Revew, 8,996, pp -35. [] Gordo,., Salmod, D., ad smh., A., ovel approach o olear/o-gaussa Bayesa sae esmao, IEE proceedgs-f, 40,993,pp [] A.F.M. Smh, Gelfad, A.E. Bayesa sascs whou ears: a samplg-resamplg perspecve., Amerca Sasca, vol o 46,00,pp [3] Brockwell, P., ad Davs, R., Tme seres: heory ad mehods (Secod edo, ew York: Sprger. 99 [A7] [4] Harvey, A., Forecasg srucural me seres models ad he Kalma fler, Cambrdge: Cambrdge Uversy Press [5] Harvey, A.,, A ufed vew of sascal forecasg procedures, Joural of Forecasg, vol o 8, pp [6] Black, F., ad Scholes, M., The prcg of opos ad corporae lables, oural of Polcal Ecoomcs, vol o 8,pp Copyrgh (c 0 Ieraoal Joural of Compuer Scece Issues. All Rghs Reserved.