A ROBUST NON-LINEAR MULTIVARIATE KALMAN FILTER FOR ARBITRAGE IDENTIFICATION IN HIGH FREQUENCY DATA P. J. BOLLAND AND J. T. CONNOR London Busness School Deparmen of Decson Scence Sussex Place, Regens Park London NW 4SA Phone -(+44) 7-262-5050 FAX - (+44) 7-724-7875 E-Mal - pbolland@lbs.lon.ac.uk E-Mal - jconnor@lbs.lon.ac.uk ABSTRACT We presen a mehodology for modellng real world hgh frequency fnancal daa. The mehodology copes wh he errac arrval of daa and s robus o addve oulers n he daa se. Arbrage prcng relaonshps are formulaed no a lnear sae space represenaon. Arbrage opporunes volae hese prcng relaonshps and are analogous o mulvarae addve oulers. Robus denfcaon/flerng of arbrage opporunes n he daa s accomplshed by Kalman flerng. The sae space model used o descrbe he prcng relaonshps s general enough o handle boh lnear and non-lnear models. The recursve Kalman equaons are adaped o fler ck daa, cope wh he errac arrval of observaons and produce esmaes of all he arbrage prces on every me sep. We demonsrae he mehodology wh a robus neural nework fler appled o foregn exchange rangular arbrage. Tck daa from hree markes s used: $/DM, /$, /DM 993-995. The fler produces esmaes of he arbrage prce for all exchange raes on every second, ncreasng boh he speed and effcency of arbrage denfcaon. KEYWORDS: Arbrage, Foregn Exchange, Mulvarae Kalman Fler, Neural Nework, Oulers, Robus, Tck Daa.. Inroducon Arbrage s a fundamenal mechansm for achevng effcency n he fnancal markes (Ross 976). An arbrage opporuny occurs when a prce dscrepancy exss beween wo or more hghly relaed asses. The opporuny can be exploed by buyng he under prced asse and sellng he over prced asse, producng a prof whou ncurrng any rsk. Msprcng s rapdly correced n hghly compeve markes (Frenkel and Levch 975,977), herefore arbrage raders need rapd denfcaon, fas ransacons and low ransacon coss. Many arbrage
relaonshps have been denfed n he fnancal markes. Our mehodology can be appled o any sysem of lnear arbrage prcng relaonshps. Secon. descrbes he rangular foregn exchange arbrage we use o demonsrae he mehodology. Prevous sudes of arbrage denfcaon have manly been lmed o examnng daly daa and so mgh mss many of avalable nraday opporunes. Sudes ha have examned nraday daa (Rhee and Chang 992) have been lmed o examnng only a mnue fracon of he daa because of he need for smulaneous observaons. The mehodology we presen allows arbrage opporunes o be denfed wh rregular (non-smulaneous) observaons. Irregular mes seres presens a serous problem o convenonal modellng mehodologes. Several mehodologes for dealng wh errac daa have been suggesed n he leraure. Muller e al 990, sugges mehods of lnear nerpolaon beween errac observaons o oban a regular homogenous mes seres. Oher auhors (Ghysels and Jasak 995) have favoured non-lnear me deformaon ( busness-me or ck-me ), however hs mehodology has no smple equvalen for mulvarae seres. The mehodology we presen descrbes he dynamcs of fundamenal underlyng arbrage saes whch are observed as errac nosy exchange raes. We rea he errac arrval as a mssng daa problem. The Kalman fler descrbed s dscree, as he daa s only provde n quansed me seps (.e. seconds), however he mehodology could be exended o connuous me problems wh he Kalman-Bucy fler (Medch 969). The sae space represenaon descrbed n secon 2 allows us model he sysem a he maxmum resoluon of he avalable daa (Reuers daa quoed by he second). Convenonal modellng mehodologes may also be napproprae for modellng ck daa as he dsrbuon s ofen heavy aled (Dacorogna 995). Fnancal daa, especally quoaons, are prone o daa corrupon and oulers. Chung 99, dscovered 0.25% of he MMI fuures quoes were ousde of he daly hgh and low and are herefore serous daa corrupon s. Secon 3 deals our robus mehodology whch s smlar o ha descrbed by Masrelez and Marn 975 and Marn and Vandaele 983. The sae space represenaon s capable of ncorporang boh lnear and nonlnear models. The esmaon of he models s performed usng an E.M. algorhm descrbed n secon 4, whch was nroduced by Dempser, Lard and Rubn 977 o esmae parameers of models when some of he daa s mssng.
The mehodology we presen s suable for real world fnancal daa and ncreases boh he speed and effcency of arbrage denfcaon. We demonsrae he fler on $/, $/DM, /DM daa from 993-995, he resuls of he esmaon and flerng are shown n secon 5.. Foregn Exchange Arbrage We examne foregn exchange rangulaon s for arbrage opporunes (alhough he same mehodology can be appled o many varees of arbrage). In he absence of ransacons coss and bd-ask spread he followng equlbrum relaonshps mus hold for currency raes, EX(0,) EX(,2) EX(2,0) = EX(0,) EX(,2) EX(2,3) EX(3,0) = EX(0,) EX(,2) EX(2,3)......EX(m,0) =, () where EX(,j) represens he spo rae for currency j when expressed n uns of currency. If he equlbrum relaonshps n Eq.() hold, hen a sngle counres m exchange raes can be used o produce esmaes of all he cross raes, EX(, j) = EX(0, j) / EX(0,), n hs paper he US Dollar s used as he base currency. Takng logarhms of he rangular relaonshps, allows he cross raes o be expressed as log(ex(,2)) = log(ex(0,2)) - log(ex(0,)) log(ex(, j)) = log(ex(0, j)) - log(ex(0,)) log(ex(m -, m)) = log(ex(0,m)) - log(ex(0,m -)). (2) If he addve relaonshps of Eq.(2) are volaed, an arbrage opporuny exss where rskless, profable ransacons can occur. Volaons of he rangular relaonshps are analogous o an ouler n he daa se, he larger he msprcng he larger he ouler. When marke frcon' s are ncluded (ransacon coss and bd-ask spread) slgh msprcng can occur whn small bands around he arbrage prce. In he followng secon he rangular currency relaonshps are encoded whn a
sae space form and a mulvarae Kalman fler s used o denfy any sgnfcan volaons of Eq.(2). 2. Space Represenaon of FX Relaonshps The mehodology we presen below descrbes how oulers can be robusly denfed/flered n mulvarae non-lnear daa. In hs applcaon he oulers ha he Kalman fler denfes are suaons n whch an arbrage opporunes exs. The Kalman fler has been adaped o fler ck daa and o updae he esmaes of he exchange raes every me sep. The Kalman fler used s general enough o handle boh lnear and non-lnear models. For non-lnear models a pon-wse lnearzaon s performed o predc he Kalman fler s sae changes, and o updae he recursve esmaes of he error predcon covarance (Connor, Marn, Alas 994). The parameers of he models used n he Kalman fler are robusly esmaed from cleaned daa, descrbed n secon 4. The observaon vecor z n he sae space model represens he logarhm of each exchange rae observed. If all possble exchange raes (cks) are observed n a gven second hen, z s gven by z z z z z z, (3) ( m m m m T z = ( 0, ) (, 0, 2 ) (,, 0, ) (,, 2 ) (, ) (, ),,,, ) (, j where z ) = log(ex(, j)). Usually only a subse of Eq.(3) are observed. The elemens of z come n wo prncple groups : The log of he m exchange raes for he base currency (0,j), ( 0, ) ( 0, 2) ( 0, m) z, z,, z = Log ( Base Raes). (4) The log of he correspondng cross raes (,j), (, 2) (, m) ( 2, 3) ( m, m) z,, z, z,, z = Log ( Cross Raes). (5) The exchange rae msprcng problem can be formulaed no a famlar sae space model, x = f ( x ) + e, (6) z = Hx + v. (7)
The sae vecor x n Eq.(6) represens he log of he arbrage value of he m exchange raes for he base currency ($) as well as he auo-regressve srucure of he sysem. The sae ranson vecor f ( x ) n Eq.(6) represens he sysem dynamcs ha may be lnear or non-lnear (n he case of a lnear sysem f ( x ) s smply he sae ranson marx Φ ). The observaon marx, H n Eq.(7), exracs he base raes and uses he logarhmc arbrage equaons o esmaes he cross raes The sysem descrbed n Eq.(6) and Eq.(7) have wo ypes of drvng nose, e he sae nose and v he observaon nose. The sae nose e represens he varaon due o he exchange raes underlyng arbrage dynamcs. The observaon nose v has wo componens v = u + w, he frs componen u, represens he varaon caused by he ransacon coss and bd ask spread, whch allow he prce o move freely whn unprofable bounds. The second componen w, represens he addve oulers whn he daa (wheher hey are daa corrupon's or marke msprce anomales). The sae ranson vecor f ( x ) n Eq.(6), can be descrbed as a non-lnear mulvarae auo-regressve (NMAR) process for each of he base currency s exchange raes. The mulvarae auoregressve process, NMAR( p, p 2,, p m ) s defned ( ) ( ) ( ) by, 2 2 m m x = f ( x,, x, x,, x,, x,, x ) + ε ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) p ( 2 ) p ( m) p 2 2 2 2 m m x = f ( x,, x, x,, x,, x,, x ) + ε ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2) ( ) p ( 2 ) p ( m) p ( m) ( m) ( ) ( ) ( 2) ( 2) ( m) ( m) ( m) x = f ( x,, x, x,, x,, x,, x ) + ε, ( ) p ( 2) p ( m) p ( ) ( 2) ( m) where x, x,, x are he log exchange raes for he base currency, and p () s he number of auoregressve erms for he h exchange rae. In Eq.(8) f ( ) s denoe non-lnear funcons governng each ndvdual exchange rae. The sae vecor x, he sae ranson vecor f ( x ) and he dsurbance vecor e n Eq.(6) are defned as follows: (8) x x x x x x m, (9) ( ) ( ) ( 2) ( 2) ( m) ( m) T x = (,, ( ), p +,, ( 2),, p +,, ( ) ) p + ( ) ( ) ( ) f ( x ) = ( f ( x ), x,, x, ( ) p + ( 2) ( 2) ( 2) f ( x ), x,, x,, ( 2) p + ( m) ( m) ( m) T, f ( x ), x,, x m ), ( ) p + (0)
( ) ( 2) ( m) T e 0 0 0 = ( ε,,, ε,,, ε,, ), () ( j where ε ), he random sae nose assocaed wh he j h exchange rae, appears n he + p j = ( ) poson n he dsurbance vecor e (where p ( 0) = 0 ). The observaon marx H n Eq.(7) exracs he base raes and he cross raes from he sae vecor x. Each of he rows of H relae o a specfc exchange rae, he rows are defned as follows, ( T m T T m T m m T H = [ h 0, ) (, ) (, ) (, ) (, ),, h 0, h 2,, h,, h ]. (2) For base currency exchange rae (0,j) : ( 0, j ) h [ 0 0 0 0 0 0 0 0 ], (3) = ( ) ( j ) ( j ) ( m) x p x p x p x p so each h ( 0, j) exracs he base currency rae x j from x. For cross currency exchange raes (,j) : (, j) h = [ 0 0 ( ) 0 0 ( ) 0 ( j ) 0 0 ( m) ], (4) (, j) so each h h (, j ) j x = x x ). x p x p x p x p esmaes he cross rae (,j) usng he addve log relaonshps (.e. For regular (evenly spaced) me seres all he raes would be observed on every me sep. Tck daa, however, requres a mehodology capable of modellng rregular me seres. On any gven second only currences for whch a rade (ck) has occurred ener he observaon vecor and only he rows of he observaon marx whch correspond o an acual observaon are used o updae he flerng equaons. The observaon nose vecor v, n Eq.(7) also reconfgures s dmensonaly o correspond o he acual rades ha occur. Ths gves rse o several possble suaons : No observaons on any marke, = [ NULL], H = [ NULL], v = [ NULL]. z
The observaon vecor z, he observaon marx H and he observaon nose v are se o null. One or more markes produce observaons, ( 0, ) ( j, k ) e.g. base rae z and cross rae z are raded. z z = ( 0, ) h ( z j, k ), H = ( 0, ) j k v, v (, ) = h v ( 0, ) ( j, k ). All he markes produce observaons n one second, ( 0, ) z ( 0, ) h ( 0, m) z ( 0, m) h (, 2) (, 2) z z h =, H =, v (, m) (, m) z h ( z m, m ) ( m, m) h = v v v ( 0, ) ( 0, m) (, 2) v v (, m) ( m, m ). Expandng and conracng he observaon equaon n hs way, allows he sae space model o cope wh he errac arrval of ck daa and mmedaely ncorporae all new nformaon o updae he sae esmaes for all exchange raes. The mehodology produces an esmae of he saes (exchange raes), an esmae of he assocaed predcon error covarance, as well as he predcons of fuure saes a every second, regardless of any ck beng observed. 3. Robus Kalman Fler The underlyng saes x n Eq.(6), are unknown, hey are esmaed by a robus Kalman fler (Kalman 96). Usng robus mehodologes proecs he modellng procedure from serous performance degradaon caused by ll condoned daa. The recursve non-gaussan Kalman fler equaons as descrbed by Masrelez 975 and Marn and Vandaele 983 are dealed below. The robus one sep ahead predced sae vecor x and he predced observaon vecor z are gven by, x ( ~ = f x ), (5) z = H x. (6)
where ~ x s he flered sae vecor. The dsrbuon erms e and v n Eq.(6) and Eq.(7) are assumed o be zero mean, serally ndependen and muually ndependen, however no assumpons abou her dsrbuons are made. The covarance marces of e and v are denoed by Q = E( e e ) and R = E( v v ) respecvely. The modellng mehodology we employ assumes ha he nose covarance marces reman consan over me. For fnancal daa hs assumpon may be nvald (Ruz 994). The mehodology we presen can be exend o ncorporae sochasc volaly, see Harvey Ruz and Shephard 992. In an effor o lm he mpac of sochasc volaly he esmaon of Q and R was made usng a rollng wndow, see secon 4. The robusly flered esmae of he sae vecor ~ x and he predcon error covarance marx M are defned by he followng recursve relaonshps, ~ x = x + M H g ( z ), (7) M + = Φ P Φ + Q, (8) P = M M H G z H M, (9) ( ) where z = z z s he nnovaons vecor (he observed resdual), g ( z ) s he score funcon of he nnovaons wh componens, { g ( z )} { z Z } p = [ p{ z Z }], (20) ( z ) and G ( z ) s defned as he dfferenal of he score funcon, wh elemens, { ( z )} G j { g ( z )} = ( z ) j. (2) For a non-lnear sae space model he sae ranson marx Φ n Eq.(8) s esmaed by a pon-wse lnearzaon of he non-lnear model. The elemens of Φ are he paral dervaves of f evaluaed abou he robusly flered esmaes of he sae vecor ~ x, f ( x ) Φ j =, x ~ j = x x. (22)
In he sandard Kalman fler he densy funcon for he nnovaons s assumed o be Gaussan. In order o oban robusness we assume z has a symmerc heavyaled densy funcon. The score funcon g ( z ) for a Gaussan nnovaon process s lnear. For a heavy aled densy g ( z ) s gven by a non-lnear gan funcon ha lms he nfluence of large nnovaons. There s some laude gven n choce of g ( z ) n he above equaons. The score funcons for he Gaussan dsrbuon, Huber s leas nformave dsrbuon (Huber 98) and he Hampel re-descendng funcon are shown n fgure (for he one-dmensonal case). g(innovaons) Score Funcons 5 4 3 2 0-5 -4-3 -2 - - 0 2 3 4 5-2 -3-4 -5 Innovaons Gaussan Huber Hampel Fgure : Score Funcons: Gaussan, Huber, Hampel. For he case of Gaussan nnovaons he score funcon g ( z ) s gven by, g ( z ) = ( H M H + R ) z. (23) The dervaons of g ( z ) for he n-dmensonal Huber and Hampel denses s gven n Bolland and Connor 995. The sze of he nnovaons s he crcal value whch deermnes wheher he observaon s an ouler (arbrage opporuny). The magnude of he ouler s 2 defned by r = ( z z ) ( z z ) where s he covarance marx of he nnovaons. The measuremen r allows us o se a defnon for an ouler, so ha only msprcng of suffcen magnude o allow for profable rades are denfed. 4. Model Specfcaon and Esmaon
In order o produce he robus Kalman fler, esmaes of f ( x ), Q and R need o be obaned. The sae ranson funcon f ( x ) can be approxmaed wh many dfferen modellng mehodologes. There s a large body of emprcal evdence o sugges ha he domnan srucure n f ( x ) wll be a mean reverng process (Fama 965). The mean reverson could be an arfcal arefac of he daa. Roll 984, showed ha bd-ask bounce nduced srong negave auo-correlaon n fnancal daa. Tme seres of marke prces conan boh bd and ask prces so f no new nformaon arrves he rue value remans consan, any observed varaon s caused by he dfference n bd and ask prce. Bouncng beween bd and ask prces gves rse o a srong negave auo-correlaon shown n fgure 2. Ask Prce Value Bd Prce - + Fgure 2: Bd-Ask Bounce. To lm he mpac of he bd-ask bounce, md-prces were modelled. The mean reverson could also be an arefac of he prce quansaon (prces quoed n dscree uns). The saes of he sysem descrbed are he arbrage values of he exchange raes. The frs dfferences of he sae were aken o produce a saonary seres. Predcng sae changes raher han here levels requres a slgh re-formulaon of he f ( x ) s n Eq.(0). The sae ransons are formed by wo componens, a ( ) random walk componen, x (he prevous sae), plus he sae changes ( ) ( x x 2 ), so Eq.(0) becomes, d ( ) ( ) ( ) ( ) f ( x ) = ( x + d ( x x ), x,, x, 2 ( ) p + ( 2) ( 2) ( 2) ( 2) x + d ( x x ), x,, x ( 2 ),, 2 p + ( m) ( m) ( m) ( m) T, x + d ( x x ), x,, x ( m) ), 2 p + (24) ( where d ) ( x x 2 ) represens he NMAR srucure of he sae changes.
Tck daa for hree currences $/DM, /$, /DM (993-995) was used o demonsrae he mehodology. The fundamenal saes are herefore he arbrage values of $/DM, /$, so x = (log($ / DM ),log($ / )). The saes were esmaed by several dfferen models: Nave random walk model (no fler), ~ x ~ x = Random walk Kalman fler, ~ x = ~ x + M H g ( z ) Lnear MAR(,), ~ x = f ( ~ x ) + M H g ( z ) Neural Nework NMAR(n,n), ~ x ( ~,, ~ = f x x n ) + M H g ( z ) To produce an esmaon se for an MAR(n,n) we requre daa examples wh n consecuve sae changes across all saes. Due o he errac naure of he ck daa consrucng an esmaon daa se whou mssng observaons for wh more han one lag proved dffcul. An nal lnear MAR(,) model was esmaed from he daa where cks occurred on consecuve seconds for boh he /$ and /DM exchange raes. Usng he MAR(,) as an esmae of f ( x ), he robus Kalman fler was appled o he whole daa se produced a flered daa se. Ths daa se conaned he flered values of he acual observed sae changes and he esmaed sae changes produced by he smple lnear model. An MAR(n,n) esmaon se could now be consruced from he flered daa, wh he esmaed sae changes fllng he problemac mssng observaons. To avod he problems of nonsaonary n he dynamcs over he whole daa se (wo years), he parameers were esmaed on a rollng wndow of one radng week, and re-esmaed daly. 4. Esmaon of Model Parameers An esmaon maxmsaon (EM) algorhm s employed o esmae he neural nework parameers, denoed λ, of Eq.(0) or equvalenly Eq.(24). The EM algorhm, see Dempser, Lard, and Rubn (977), s he sandard approach when esmang model parameers wh mssng daa. The EM algorhm has been used n he neural nework communy before, see for example Jordan and Jacobs, and Connor, Marn, and Alas (994). The E-Sep, gven n secon 4., esmaes he mssng daa. Wh he esmaed mssng daa assumed o be rue, he parameers are hen chosen by way of
maxmsng he lkelhood. Ths procedure s erave wh new parameer esmaes gvng rse o new esmaes of mssng daa whch n urn gve rse newer parameer esmaes. 4. E-Sep Durng he esmaon sep, he mssng daa, namely he x, v, and e, of Eq.(6) and Eq.(7) mus be esmaed. Ths s accomplshed usng he robus Kalman fler of secon 3. The esmaed mssng daa s denoed ~ x, ~ v, and ~ e. 4.2 M-Sep The robus lkelhood for he sysem defned by Eq.(6) and Eq.(7) s gven by where p{ z Z } N { z Z } l( λ ) = p = (25), s a funcon of r 2, he magnude of he nnovaons 2 r = z j z j Z j z j z j Z ( ~ (, )) ( ~ T λ (, λ )) and = H M H + R s he covarance marx. Ths has been derved by De Jong 988 for he case where nal sae esmaes and nose varances are consdered. The log lkelhood s defned by N { z Z } L( λ ) = log( p ) j= The parameers o maxmse he lkelhood, arg max N j λ = log( p{ z Z }). (27) λ = Smple graden descen s used o maxmse (27), hs s done by gnorng he dependence of on λ N L λ p{ } λ ( ) log( z Z ) ( ~,, ~ p, ) = H x x x (28) λ z λ = Nong ha { g( z )} { z Z } (26) log( p ) = from Eq.(20) and usng Eq.(7), ( z ) g( z ) H M ~ x x ( ~ x,, ~ = x, λ ) (29) ( ) p whch can be used wh Eq.(28) o ge he graden n erms of clean daa L ( λ ) N λ ( ~ p p λ ) λ ( ~,, ~, ) ( ~,, ~, ) = x x x x M x x x λ = whch s equvalen o dong back-propagaon on clean daa. (30)
Ouler Observaons 2 When an ouler, z, s observed r wll be very large causng he erm 2 g. ( r ) lm he conrbuon of z o he lkelhood n (26). As menoned n secon 3, he predcon varance reflecng greaer uncerany n he fundamenal raes. Fuure conrbuons o he lkelhood wll be effeced by hs ncreased uncerany. Unvarae Case If only a sngle quoe s observed a a gven me, he assocaed graden s gven by L ( λ) z ( λ) = 2( z z ( λ)) szz (3)) λ λ whch f z corresponds o a fundamenal dollar denomnaed rae s he same as found n he unvarae case frs explored n Connor, Marn and Alas (994). Cross Currency If he quoe corresponds o a cross currency, relaed o he dollar denomnaed raes (, j) ( ) ( j) by z = x x, he graden wll be composed of conrbuons from all he predcors relaed hrough H M ( k ) L ( λ) (, j) (, j) x ( λ) = 2( z z ( λ)) ( M k,, M k, j, ). (32) λ λ k Several Quoes of he Same Currency If several quoes are observed for one of he underlyng dollar denomnaed raes, he learnng algorhm smplfes grealy. The predcon covarance marx and correspondng nverse are gven by: a b b b c d d d b a b b d c d d = =, (33) b b a b d d c d b b b a d d d c where c and d are derved from ac + ( N ) bd = and ad + bc + ( N 2) bd = 0 whch allows he reducon of Eq.(33) o he smpler form N L ( λ) ( ) x ( λ) = ( x x ( λ)) k N λ 2 (34) N λ = o
where k = c + ( N ) d. Snce several quoes are avalable, he addve nose s N smoohed ou and one has more confdence n he average of he quoes, N ( ) x, han any of he quoes would be gven alone. Ths added confdence s N = expressed n erms of he sronger graden n () where k N > k for N >. No Observaons Due o he errac naure of ck daa, ofen here wll be no observaons durng a gven perod. For hs mssng daa, here wll be no conrbuon o he lkelhood gven n (26). Bu as n he case wh exreme oulers, he uncerany n fuure predcons, wll grow and effec he lkelhood of fuure observaons. 4.2 Esmaon of Nose Varances An erave procedure was appled o produce esmaes of Q and R. The observaon nose covarance R, has wo componens, u he marke frcon's (bd-ask spread, ransacon coss ec.) and w he addve oulers (prcng anomales). As flerng procedure nerpolaes hrough addve oulers, he esmae of R s only dependen on he frs componen u, R = E( u u ). Inal esmae of observaon nose covarance R, and he sae nose covarance Q, were produced by maxmsng he lkelhood of he MAR(,) sae space model. These esmaes of R and Q were refned durng he neural nework esmaon process by repeaed applcaon of maxmum lkelhood. 5. Resuls The ck daa was obaned from Reuers, Aprl 993-Aprl 995, he md-prce was modelled. The nal lnear MAR(,) model produced had a sae change ransons marx gven by, ( ) ( ) ( ) x.. x x f ( x ) = ( ) ( ) ( ) + 0 5065 0 0358 x... 2 2 2 2 0 0564 0 436 x x 2 (35) The nal daa se from whch he MAR(,) parameer s were esmaed, was consruced from regons of consecuve cks (500 regons of consecuve pons coverng approxmaely wo weeks radng were used). The srengh of he dagonal erms demonsrae he meanng reverng. Ths model was used n he Kalman fler
o produce a flered daa se whou mssng observaons, from whch subsequen models were produced. Table show he flerng resuls for he neural nework Kalman fler and he nave random walk hypohess. The Kalman fler produces superor resuls for boh he mean squared error (MSE) and he robus medan absolue devaons (MAD). Table : Model Comparson. Model MSE MAD RW Model 2.69 (0-4 ) 0.0084 (No Fler) Neural Nework Kalman Fler.5 (0-4 ) 0.0075 Fgures 3 and 4 demonsrae he Kalman fler denfyng oulers. In Fgure 3 he flered esmaed saes for he $/DM exchange rae are represened by he sold lne, he acual observed rades by crcles, and rades occurrng on he oher exchange raes by vercal lnes. The flered saes represen he esmae of he rue arbrage value (no he bd or ask values). The effec of marke frcon s has been ncorporaed no he esmae R, so flered saes are always whn he acual bd and ask values observed. The Kalman fler denfes an ouler and uses pure predcon for he esmae of he $/DM rae a me 5.07.4 and does no follow he spurous prce movemen. The msprcng s flered and classfed as an ouler ( r = 4. 65) by he robus algorhm presened here. Fgure 3 also demonsraes how new nformaon occurrng on any of he exchange raes s mmedaely ncorporaed n he robus esmaes of all he saes. A me 5.07.27, he vercal doed lne ndcaes he observaon of a DM/ rade, he esmae of he $/ sae s nsanly updae o ncorporae he affec of he rse n he DM/.
.6625.6620 Robus Idenfcaon of Arbrage Opporunes $/DM.665.660.6605.6600.6595 r =4.65 0 0 20 30 40 50 60 Tme Fgure 3: Idenfcaon of Marke Msprcng. $/DM Tcks $/DM Saes BP/$ Tcks DM/BP Tcks Table 2: Tck Daa (bold) and Esmaed Raes (normal). $/DM Tcks /$ Tcks /Dm Tcks $/DM Esmae /$ Esmae /DM Esmae SD (0-5 ) Ln($/D) SD (0-5 ) Ln( /$).660 2.435.6608.46637 2.43444 4.33 8.38.6602.46624 2.43439 4.2 7.9.6604.6603.46627 2.43447 4.35 8.8.66030.46626 2.43443 4.2 7.7.4662.6603.46622 2.43439 4.65 8.38.6603.46624 2.43440 4.34 7.7.6603.46623 2.43440 4.57 7.9.6608.66055.4667 2.43464 4.6 8.4.66042.4669 2.43449 4.2 8.3.4665 2.4356.66047.46638 2.43488 4.65 8.38 Table 2 demonsraes he ably of he Kalman fler o deal wh he errac arrval of ck daa. Bold fon exchange raes represen seconds when cks are observed, and normal fon represens he Kalman fler esmaes. When ncomplee observaon vecors are observed he Kalman fler uses he mulvarae auoregressve srucure o esmae he unseen raes. In seconds where no observaons are observed he fler uses pure predcon o esmae he mssng currency raes. The las wo columns n able 2 show he esmaed predcon sandard devaons
for he log of he saes. Agan, bold fon ndcaes he seconds where cks where observed, and normal fon ndcaes he recursve esmaes of he Kalman fler. The predcon error sandard devaons grows seadly n perods where no cks are observed, and collapses o he one second sae predcon error sandard devaon for he predcon based on new nformaon (cks). $/DM.662.6609.6606.6603.6600.6597.6594 Robus Idenfcaon of Arbrage Opporunes 5:2:58 5:3:06 5:3:5 5:3:24 5:3:32 5:3:4 5:3:49 5:3:58 5:4:07 $/DM $/DM.6585.6582.6579.6576.6573.6570.6595.6589.6583.6577.657.6565 0:23:3 0:23:48 0:24:06 0:24:23 0:24:40 0:24:58 5:4:24 5:4:4 5:4:59 5:5:6 5:5:33 5:5:50 $/DM.6574.657.6567.6564.6560 5:07:2 5:07:29 5:07:47 5:08:04 5:08:2 5:08:38 Fgure 4: Idenfcaon of Marke Msprcng.
6. Concluson We have shown he effecveness of one flerng approach for denfyng arbrage opporunes on currency ck daa. The mehodology s deally sued o he poor daa qualy of he fnancal markes (errac arrval and addve oulers). The Kalman fler produces rae esmaes every second wheher or no any cks are acually observed, hs ncreases boh he speed and effcency of denfyng arbrage opporunes. I s sraghforward o exend he above analyss o many more exchange raes and cross raes, ncreasng he possbly of fndng msprcng. In addon hs mehodology could be readly appled o all forms of arbrage whch are descrbed by smlar ses of prce relaonshps. 7. References Bolland, P. J. and Connor, J. T., "Mulvarae Non-Lnear Kalman Flers", Techncal Repor, London Busness School, June 995. Connor, J. T., Marn, R. D., and Alas, L. E. Recurren Neural Neworks and Robus Tme Seres Predcon, IEEE Transacons on Neural Neworks, pp. 240-254, March 994. Chung, P. Y., A Transacons Daa Tes of Sock Index Fuures Marke Effcency and Index Arbrage Profably, Journal of Fnance, Vol. 46, pp 79-809, Dec. 99. Dacorogna, M. M., Prce Behavour and Models for Hgh Frequency Daa n Fnance, Proceedngs of he NNCM conference, London, England, Oc. -3, 995. Dempser, A. P., Lard, N. M., and Rubn, D. B. Maxmum lkelhood from ncomplee daa va he EM algorhm. Journal of he Royal Sascal Socey, B, 39, -38. (977). De Jong, P., The Maxmum Lkelhood of a Sae Space Model, Bomerca, Vol. 75,, pp 65-69, 988. Fama, E. F., The Behavour of Sock Marke Prces, Journal of Busness, Vol. 38, pp 34-05, Jan 965.
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