Discovery of multi-spread portfolio strategies for weakly-cointegrated instruments using boosting-based optimization

Transcription

1 Discovery of muli-spread porfolio sraegies for weakly-coiegraed isrumes usig boosig-based opimizaio Valeriy V. Gavrishchaka Head of Quaiaive Research, Alexadra Ivesme Maageme, ew York, 007 Absrac Icreasig complexiy of he moder marke dyamics requires ew quaiaive frameworks for he discovery of sable porfolio sraegies. Impora requiremes iclude he abiliy of he coupled ad self-cosise opimizaio of he dyamic sraegies ad asse allocaios as well as robus buil-i mechaisms for he sraegy complexiy corol o esure accepable ou-of-sample performace. Recely iroduced boosig-based opimizaio aurally icorporaes all hese feaures. Origially, he framework was described as a geeric ool for he discovery of compac porfolio sraegies from a give pool of exisig fiacial isrumes ad base radig sraegies. Here I oulie he impora geeralizaio of his framework ha allows simulaeous discovery of ew syheic isrumes represeed as geeralized spreads of exisig fiacial isrumes ad dyamic radig sraegies for each such spread. Deailed argumes ad real-marke example clarify he essece of his ew framework as a powerful geeralizaio of he exiig pairs radig sraegies ad coiegraio-based echiques. Keywords: Boosig, Esemble Learig, Coiegraio, Pairs/Spread Tradig, Marke-eural Sraegies, Porfolio Opimizaio.. Iroducio Icreasig complexiy of he moder marke dyamics ad fiacial isrumes requires ew quaiaive frameworks for he discovery of he robus porfolio sraegies. Impora requiremes for such a sysem iclude he abiliy of he coupled ad self-cosise opimizaio of he dyamic sraegies ad asse allocaios as well as ierpreabiliy of he obaied porfolio sraegy. I coras o he classical radig sraegy ad porfolio opimizaio frameworks, i should also provide robus buil-i mechaisms for he sraegy complexiy corol ha esures accepable ou-of-sample performace. Recely iroduced boosig-based opimizaio framework aurally icorporaes all hese feaures [5-7]. Prelimiary resuls of is applicaio o he real marke daa cofirmed he pracical value of he proposed framework. Origially, he framework was described as a geeric ool for he auomaed discovery of he compac porfolio sraegy from a give pool of exisig fiacial isrumes ad base radig sraegies. Icreasig umber of he iellige marke paricipas improves marke efficiecy ad makes i more difficul o discover sable porfolio sraegies ha operae wih exisig fiacial isrumes. ovel coiegraio-based ad relaed approaches for he discovery of he log/shor marke-eural porfolios ad pairs radig sraegies could offer more sable soluios [-3]. These ools help o discover ew syheic isrumes (expressed as spreads of exisig isrumes ha have more predicable ime series wih a ypical mea-reverig feaure ad o direcly observed by he maoriy of oher marke paricipas. However, as hese echiques become more widespread, ypical pairs or ses of coiegraed isrumes become well-kow. This makes i much more difficul o exploi deviaios (spreads from such co-depedecies o a regular basis. Moreover, rigorous coiegraio relaios ca be ofe violaed makig sraegies relayig o he simple meareverig dyamics of he paricular spread usable. Coiegraio-based echiques aloe do o provide ay geeric soluios for spread radig i he more commo ad less ideal siuaios where oly weak/parial or complex ime-varyig co-depedecies exis. The pool of such weakly coiegraed isrumes could be very large. This could offer a lo of uexplored poeial for he discovery of sable marke-eural porfolio sraegies. Ideed, ay pair/se of isrumes ha have oe or more commo ecoomic/marke facors i heir approximae facor model ca be a cadidae for such a pool. However, o discover sable porfolio sraegies based o spreads of weakly coiegraed isrumes oe should exploi o oly spreads (syheic isrumes wih simple

2 mea-reverig feaures bu also wih more geeral characerisics icludig muli-scale reds ad oher complex dyamics. The geeric framework should also be able o discover dyamic sraegies for each of hese muliple spreads ad opimally combie hem. I his work, I oulie he impora geeralizaio of he origial boosig framework ha allows simulaeous discovery of ew syheic isrumes represeed as geeralized spreads of exisig fiacial isrumes ad dyamic radig sraegies for each such spread. This framework ca be cosidered as a powerful geeralizaio of he exiig echiques for he pairs radig icludig coiegraio-based ools [-3]. Real marke example illusraig operaioal deails of he ew framework ad is poeial pracical value is also preseed. 2. Marke-eural porfolio selecio ad pairs radig: Limiaios of he exisig echiques ad muli-spread geeralizaio Tradiioal echiques for pairs radig ad hedgig are based o a relaive value aalysis of asse prices [3]. The wo or more asses ca be seleced o he basis of iuiio, fudameals, log-erm correlaios, pas experiece or oher empirical kowledge. Obvious limiaio of hese approaches is he absece of he sysemaic saisical or fudameal framework ha ca be geerically applied across he uiverse of fiacial isrumes. More sophisicaed ad sysemaic echiques for hedgig ad cosrucio of he marke-eural porfolios are based o facor models ha ca be buil usig various saisical or fudameal approaches [2,3]. However, i may cases, i is difficul o ideify a miimum se of measurable risk facors ha are required for a facor model wih accepable accuracy. ovel coiegraio echiques [-4] provide a powerful geeralizaio of he radiioal approaches by iroducig a sysemaic saisical framework for he cosrucio of syheic pairs/ses i he form of appropriae log/shor combiaios of wo or more asses. Coiegraio is esseially a ecoomeric ool o ideify possible sable relaioships bewee a se of ime series [3,4]. I is origial form coiegraio ca be used o ideify poeial hedges for a give posiio or o cosruc marke-eural buy&hold porfolios [-3]. Coiegraio ca also be used o measure he shorerm deviaios from he equilibrium [3]. This is ieresig as a poeial source of saisical arbirage sraegies. Deviaios from he log-erm fair price relaioship, ideified by coiegraio aalysis, ca be cosidered as saisical mispricigs ha should always rever owards he loger erm equilibrium [3]. I he case of saisical arbirage, coiegraio ca be cosidered as a exesio of he relaive value sraegies such as pairs radig. I he case of hedgig ad marke-eural porfolio cosrucio coiegraio ca be cosidered as exedig facormodel hedgig o iclude siuaios where he uderlyig risk facors are o measurable direcly, bu are isead maifesed implicily hrough heir effec o asse prices [3]. The mos commo mehod of esig for coiegraio [4] is based upo he cocep of a coiegraig regressio. I his approach a paricular ime series ( arge series S 0, is regressed upo he remaider of he se of ime series ( coiegraig series S, S, : 0, = +! " = S S + s # (, If series are coiegraed he saisical ess will idicae ha s is saioary ad he parameer vecor (,-α, β,..., β deermies he coiegraio relaioship. Two sadard saisical ess are recommeded by Egle ad Grager [4]. The relevace of coiegraio o hedgig is based upo he recogiio ha much of he risk or sochasic compoe i asse reurs is caused by variaios i marke ad/or ecoomic facors which have a commo effec o may asses. This viewpoi forms he basis of radiioal asse pricig models such as he CAPM (Capial Asse Pricig Model of Sharpe ad he APT (Arbirage Pricig Theory of Ross [2]. Esseially hese pricig models ca be formulaed as S i, = # i + " i, $ f, +! i, = $ % (2 This geeral formulaio relaes chages i asse prices ΔS i, o sources of sysemaic risk expressed as chages i marke ad/or ecoomic facors Δf, ogeher wih a idiosycraic asse-specific compoe ε i,. The presece of marke-wide risk facors creaes he possibiliy of hedgig hrough he cosrucio of appropriae combiaio of asses. For example, if asse price dyamics ca be described by (2 he he combied reur of he porfolio wih log posiio i asse S ad shor posiio i asse S 2 is give by % S, $ % S 2, = (# $ # 2 + &(", $ " 2, % f, (3 = + (!, $! 2, From (3 i is clear ha he proporio of variace caused by marke-wide facors will be sigificaly

3 reduced if all correspodig facor exposures are similar, i.e. # :! "! (4, 2, A commo approach o hedgig is o assume ha oe ca explicily cosruc a reasoable facor model (2 usig exisig fudameal or saisical approaches [2]. Usig hese facor models oe ca ry o selec a opimal log/shor porfolio wih miimal exposures o hese facors (i.e., marke-eural. However, whe i is o possible o ideify a reasoable se of explici facors, coiegraio provides a aleraive mehod for implici hedgig he commo uderlyig sources of risk. Give a paricular arge asse S 0, a coiegraio regressio ( is used o creae a syheic asse as a liear combiaio of asses S (= which exhibi he maximum possible log-erm correlaio wih he arge asse [-4]. The coefficies β of his liear combiaio ca be foud usig sadard ordiary leas squares (OLS regressio. The obaied syheic asse provides a opimal saisical hedge for he arge asse. Coefficies β specify capial allocaios for he hedgig asses (i uis of he arge asse capial, ad posiio ypes: shor for posiive β ad log oherwise (arge asse is log. A liear combiaio of asses acs as a proxy for he uobserved (implici commo risk facors. Alhough coiegraig regressio ca be used o esimae he fair price relaioship bewee a se of asses, i does o provide ay ools o deec deermiisic compoes i he mispricig dyamics for a possible saisical arbirage sraegy. Oly for simple mea-reverig dyamics of such mispricigs, addiioal iellige echiques are o required. I he real applicaios o sock/idex daa, coiegraio relaioship bewee logarihms of ime series is cosidered, i.e., all S should replaced by l(s i (. The essece of he pairs radig ca be described i erms of he spread s recovered from (: s 0, "! l( S, = = l( S # (5 Wihou loss of geeraliy we will cosider α=0 from ow o. The mai goal of he radiioal pairs radig is o fid a se of β ha produces spread (5 wih sable mea-reverig properies. This search ca be based o coiegraio or oher releva echiques. A sric requireme for a sable mea-reverig dyamics of he spread makes pairs radig simple oce such pair or group of isrumes is foud. For example, oe ca ake log or shor posiio (depedig o he spread sig i he syheic isrume described by (5, i.e. log/shor posiios i he uderlyig isrumes S wih capial allocaios specified by β, whe spread s becomes larger ha some criical value ad exi he posiio whe s becomes close o zero. However, he umber of udiscovered isrume pairs/ses wih simple mea-reverig spread dyamics decreases as coiegraio ad relaed echiques become widespread. This leads o more efficiecy wih respec o his ype of arbirage ad spread dyamics becomes more complex ad less sable. Also, eve he bes pairs/ses foud by coiegraio o hisorical daa ca sigificaly chage heir codepedecy i he fuure. Therefore, sraegies, ued o he sigle regime of he mea-reverig spread, could abruply break dow. To resolve hese limiaios, he more geeric spread radig sraegies, ha ca work wih differe ypes of complex spread ime series, should be cosidered. Besides spreads wih more complex meareverig behavior, oe ca also use a large class of spreads wih muli-scale reds ha ca ofe be exploied by he porfolio of dyamic sraegies [5,6]. Accordig o (3, his meas ha he more geeric goal of syheic isrume cosrucio is o o remove depedece o all commo facors specified by codiio (4, bu oly o some leas deermiisic facors. I his way, i could be possible o discover spreads wih cleaer ad more predicable reds ad oher paers ha i he origial ime series of he uderlyig isrumes. The relaxaio of requiremes for he spread dyamics will sigificaly expad he uiverse of exisig isrumes ha ca be poeial cosiues of he log/shor syheic isrumes suiable for he sable radig. Ideed, may differe weaklycoiegraed isrumes, ha have oly some similar exposures o he commo risk facors, could be icluded. I addiio, he more complex spread dyamics will esure ha several differe regimes are covered a he raiig sage. This could sigificaly improve ou-of-sample sabiliy of he sraegies compared o he radiioal pairs sraegies ued for he sric mea-reverig hisorical behavior. To illusrae poeial spread-dyamics diversiy ha ca be foud eve for wo uderlyig ime series, we use hisorical daa for S&P mid-cap (MID ad S&P500 (SPX idexes. Mid-cap idex is cosidered as a arge ime series S 0. These wo idexes demosrae quie sigifica coiegraio wih ypical mea-reverig feaure for he las few years. However, i he loger ru (> 6-7 years, heir overall codepedece is less sable ad more diverse. This suggess ha varyig oly oe parameer β, may differe ypes of spread dyamics ca be observed.

4 I he followig we will use argume of he logarihm i (5 as a spread measure: s * " $ = S0, #! S, = (7 I he limiig case of β =0, he origial arge ime series S 0 is recovered from s *. This simplifies direc comparisos bewee spread ad uderlyig ime series. I fig., he origial mid-cap ime series ad MID-SPX spread for differe values of β are ploed. I is clear ha by chagig β, differe ypes of spread dyamics ca be obaied: from complex muli-scale reds o he low-oise log-erm reds ad meareverig behavior. More deailed aalysis of he ypes ad predicabiliy of spread dyamics ca be performed usig variace raio ess [3] Fig.. MID ime series (solid lie ad MID-SPX spread ime series for β =.75 (dashed lie, firs from he boom, β =0.75 (dashed-doed lie, ad β =-0.3 (doed lie, las from he boom. Oe of he possible approaches o he geeric spread radig is o choose a sigle se of β ha creaes spread wih desired dyamics ad o search for he echical sraegies ha are capable o exploi possible deermiisic paers i he spread ime series. As show i previous works [5,6], boosigbased opimizaio ca be effecively used for he discovery of sable porfolio of sraegies wih adapive capial for a sigle isrume. The same framework ca be used o exploi muli-scale reds ad/or oscillaory paers i spread ime series. However, diversiy of he spread dyamics suggess ha more geeric ad more powerful approach should iclude combiaio of dyamic radig sraegies applied o muliple spreads specified by differe ses of β (i.e., muliple syheic isrumes. The fial porfolio sraegy would sill be specified as a log/shor variable-capial sraegy applied o he iiially chose uderlyig isrumes. As described i he ex secio, boosig-based opimizaio could be easily geeralized o allow simulaeous self-cosise discovery of radig sraegies ad ew syheic isrumes. 3. Boosig-based framework for he discovery ad opimizaio of he muli-spread porfolio sraegy As described i my previous works [5-7], boosig for opimizaio could be based o differe boosig frameworks. However, he geeralized AdaBoos algorihm for classificaio [8,2,3] could be a preferred choice i may applicaios due o is simpliciy, comprehesive heoreical foudaio, ad prove robus performace i a large umber of realisic classificaio problems. For our purposes i is sufficie o describe boosig algorihm oly for wo-class classificaio problem, where classifier oupus eiher + or -. Geeralized AdaBoos for wo-class classificaio cosiss of he followig seps [2]: ( w / = (8. ( ( I( y ( # =! w " h x (8.2 = = (!( w y h ( x " (8.3 = & + # & # $ + ( * = l$! '! l 2 % ' " 2 % ' ( " w (8.4 ("! y h ( x Z ( + ( = w exp / T f ( x =! h ( x /!" = T = (8.5 " (8.6 Here is a umber of raiig daa pois, x is a model/classifier ipu se of he -h daa poi ad y is he correspodig class label (i.e., - or +, I(z = 0 for z<0 ad I(z= oherwise, T is a umber of ( boosig ieraios, w is a weigh of he -h daa poi a -h ieraio, Z is weigh ormalizaio cosa a -h ieraio, h (x ->[-; +] is he bes base hypohesis/model a -h ieraio, ρ is a margi

5 corol parameer, ad f(x is a fial weighed liear combiaio of he base hypoheses. Boosig sars wih equal ad ormalized weighs for all raiig daa (sep (8.. A base classifier h (x is raied usig weighed error fucio ε (sep (8.2. If a pool of several ypes of base classifiers is used, he each of hem is raied ad he bes oe (accordig o error fucio is chose a he curre ieraio. The raiig daa weighs for he ex ieraio are compued i seps (8.3-(8.5. Accordig o (8.5, a each boosig ieraio, daa pois misclassified by he curre bes model (i.e., y h (x < 0 are pealized by he weigh icrease for he ex ieraio. I subseque ieraios, AdaBoos cosrucs progressively more difficul learig problems ha are focused o hard-o-classify paers. This process is corolled by he weighed error fucio (8.2. Seps (8.2-(8.5 are repeaed a each ieraio uil sop crieria γ < ρ (i.e., ε >= /2(-ρ or γ = (i.e., ε = 0 occurs. Sep (8.6 represes he fial combied (boosed model ha is ready o use. The model classifies ukow sample as class + whe f(x > 0 ad as - oherwise. Deails of more geeral versios of boosig algorihms are give i [2]. Boosig also offers a flexible framework for he icorporaio of oher esemble learig (model combiaio echiques. Isead of choosig he sigle bes model a each boosig ieraio, oe ca choose mii-esemble of models ha is opimal accordig o oher esemble learig echiques. For example, i is ofe useful o form a equal weigh esemble of several comparable bes models a each ieraio. I may cases, his geeralizaio improves ou-of-sample performace compared o he sadard boosed esemble. Porfolio sraegy discovery is a direc opimizaio raher ha classificaio problem. However, i was argued [5-7] ha for a large class of obecive fucios, boosig for classificaio (8.- (8.6 ca be efficiely used as a basis for he framework ha could be labeled as boosig for opimizaio or boosig-based opimizaio. Oe of he aural ad robus obecives for he radig sraegy opimizaio cosise wih marke euraliy is o require reurs (r geeraed by he sraegy o a chose ime horizo (τ o be above cerai hreshold (r c. By calculaig sraegy reurs o a series of iervals of legh τ shifed wih a sep Δτ ad ecodig hem as + (for r >= r c ad - (for r < r c, oe obais symbolically ecoded ime series (disribuio of sraegy reurs. Corary o he classificaio problems, here he purpose is o o correcly classify (bewee + ad -, bu raher o icrease he umber of + samples (i.e., he umber of cases wih supercriical reurs. This ca sill be cosidered as classificaio problem wih poeially ueve sample umber bewee wo classes. The obecive o have maximum umber of samples i + class (i.e., r >= r c ca be icorporaed io he boosig operaio (8.-(8.6 by cosiderig oupu - as misclassificaio, i.e., y =+ for all. I such seig, boosig (8.-(8.6 provides a framework for opimizaio, where maximizaio obecio fucio is a hi rae, i.e., umber of + samples divided by he oal umber of samples. Symbolic ecodig ad correspodig obecive fucio ca be based o ay complex codiio ha combies differe measures of profi maximizaio ad risk miimizaio specified by he uiliy fucio of ieres. This ca be easily achieved wih combiaio of several simple codiios. I he case of radig sraegy opimizaio, he fial usage of boosig oupu is differe from he classical case of boosig for classificaio. Isead of usig weighed liear combiaio (8.6 of he base models as a fial model for classificaio, oe uses boosig weighs o cosruc porfolio of sraegies. The iiial capial is disribued amog differe base sraegies i amous accordig o he weighs (α /Σα obaied from boosig which are already ormalized. As discussed i [5-7], boosig ca be used o discover he opimal combiaio of differe dyamic radig sraegies for a sigle fiacial isrume as well as he simulaeous combiaio of radig sraegies ad differe isrumes. I boh cases, boosig seps (8.-(8.6 are applied o a pool of base sraegies {BS i (p i }, where p i is a vecor of adusable parameers for sraegy BS i. However, i he firs case all base sraegies are applied o a ime series of a sigle isrume S 0, while se {BS i }x{s } of all possible pairs of sraegies BS i ad isrumes S should be used i he laer case. Accordig o error fucio (8.2, if he obecive is o maximize he umber of supercriical reurs o he shifed iervals, he followig opimizaio problems are solved for all base sraegies BS i (p i ad isrumes S a each boosig ieraio,: mi pi ( r ( # c r BSi ( pi S! " & ' ( w I (, $% = (9 Here, r τ is a reur produced by he sraegy BS i (p i applied o he isrume S over -h shifed ierval of legh τ ad r c is a chose hreshold value. Based o he resuls of hese miimizaio procedures for all (i, pairs, he bes pair sraegy-isrume of he curre ieraio is added o he porfolio. A possible geeralizaio is o cosider syheic isrumes (defied by he fixed β ses isead of origial isrumes S ad o apply base sraegies o

6 he correspodig spread ime series (7. However, he more geeric ad flexible approach should iclude simulaeous opimizaio of he spread ime series iself. This ca be ierpreed as a adapive discovery of ew syheic isrumes. Thus, a each boosig ieraio, a opimal vecor of β coefficies is foud by solvig he followig coupled miimizaio problem: * * ( r ( # c r BSi ( pi, s (! " & ' ( mi p w I i ( $% =, (0 I his geeralized framework, he fial oupu of he boosig opimizaio is a porfolio of differe syheic isrumes defied by heir β vecors (i.e., muliple spreads wih correspodig opimal sraegies for each of hem. I geeral, performace of his framework could be very differe from ha relyig o he same se of base sraegies ad uderlyig isrumes bu reaig hese isrumes idepedely. Ideed, search for he sable sraegies for each idividual isrume (specified by (9 could ecouer all he problems associaed wih cosaly improvig efficiecy of he exisig isrume dyamics due o heir direc availabiliy o he icreasig umber of iellige marke paricipas. I coras, he geeralized framework (0 adapively creaes ew syheic isrumes ha are o direcly exposed o oher marke players ad feaurig ime series properies exploiable by he echical radig sraegies. 4. Applicaio example The proposed framework for he discovery of muli-spread porfolio sraegies ca be used wih a wide rage of uderlyig fiacial isrumes ad base radig sraegies. For he illusraio of a ypical applicaio, wo well-kow idexes, S&P500 ad S&P mid-cap, will be used. As discussed i secio 2, he log-erm codepedecy of hese wo ime series is quie complex, so ha sable porfolio sraegy cao be easily discovered by co-iegraio ad relaed ools aloe. I is also oe of may examples where spreads wih sigificaly differe dyamics ca be geeraed by chagig us oe β coefficie. I pracical seigs, differe ypes of base echical sraegies (red-followig, oscillaor-ype, ec. ca be used. However, o sress ou flexibiliy of he boosig framework i compariso o he exisig approaches dealig solely wih simple mea-reverig spreads, oly red-followig (momeum sraegies will be icluded i he base pool. This will be he mos obvious illusraio of differece wih simple oscillaor-ype sraegies used i he case of sable mea-reverig dyamics. For clariy, a base sraegy pool is resriced o a se of simple red-followig sraegies [9]: he expoeial movig average of he daily closig prices EMA(,a for ery combied wih adapive railig sops ATS(m,α based o differe volailiy measures for exi. Ery io log (shor posiio o spread occurs whe EMA (s * >EMA - (s * (EMA (s * < EMA - (s *, i.e., he curre value of EMA is greaer (smaller ha he previous day value. Here s * is spread give by (7, is a umber of pois o be averaged, ad a is a smoohig cosa. Posiio ery occurs a he ex radig day. More sophisicaed low-pass filers [9] could be added i addiio o or isead of EMA i real applicaios. As meioed i secio 2, erace io log spread posiio meas akig simulaeous log ad shor posiios i he uderlyig isrumes accordig o heir β values. Posiio side for a paricular isrume (i.e., log or shor is deermied by he sig of is β. The capial allocaio C for each isrume is specified by he absolue value of is β : Cmax C = " ( + "! = Here is he oal umber of uderlyig isrumes ad C max is he maximum oal capial exposure (boh log ad shor. I he shor spread posiio he side of each uderlyig isrume is us he opposie. Log (shor spread posiio is closed whe iraday spread fells below sc max [-ασ(m] (umps above sc mi [+ασ(m]. Here sc max (sc mi is a maximum (miimum of he spread (compued usig closig prices wih respec o prese day, ad σ(m is a daily reur sadard deviaio of he spread or oher volailiy measure compued usig m las pois. Trailig sops based o global ad local (i.e., he mos rece maximum (miimum calculaios used for he exi spread level ca demosrae complimeary performace. I his example, boh exi ypes have bee used i he base sraegy pool. I is ofe useful o iclude muliple railig sops wih differe volailiy measures io a base sraegy pool. For example, volailiy measures based o he exreme (bar values (i.e., ope, high, low, ad close prices [0 ad refereces herei] usually provide sigificaly more accurae esimaio of he curre volailiy, especially whe ime ierval m is small (impora i he regimes wih fas volailiy chages. I he preseed example, four differe volailiy measures have bee used: simple sadard deviaio

7 based o close prices as well as Parkiso, Garma & Klass, ad Rogers & Sachell esimaors [20]. For each ype of volailiy measure (esimaor ad exi level (global/local, wo base sraegies [EMA(,a, ATS(m,α] are icluded i he pool. Oe is log-oly (igores shor ery sigals ad he oher is shor-oly (igores log ery sigals. Thus, he pool of oal 6 base sraegies has bee used i he preseed example. Daily sraegies cosidered here are very olera o a wide rage of rasacio coss ha deped o he ype of radig vehicles used (e.g., ETFs, fuures, or mixure. For simpliciy of preseaio, i he repored resuls such coss are igored. The mi/max values of iraday spread used for esig of sop-loss execuio are approximaed based o daily bar daa. Ulike sigle isrume radig, a rigorous modelig of pairs radig requires iraday daa. Adapive boosig (8.-(8.6, (0 wih ρ=0. ad Τ=7 has bee applied o he described base sraegy pool ad wo uderlyig isrumes: MID as arge ad SPX wih variable β. Biary hi rae obecive fucio wih r c =5% (aualized for he horizo of τ=63 days ad Δτ=20 days is used. Traiig daa period is 997/05/5-2005/05/5. A every boosig ieraio, each sraegy [EMA(,a, ATS(m,α] from he base sraegy pool is opimized wih respec o (,a,m,α,β parameer se usig global opimizaio algorihm based o simulaed aealig combied wih dowhill simplex mehod []. A each boosig ieraio, i addiio o he bes model, oher models ha are iferior o he bes model by o more ha 2% are also reaied wih equal weighs. The resul of boosig-based opimizaio is porfolio of 7 complimeary muli-scale sraegies [EMA(,a, ATS(m,α] operaig o 7 differe spread ime series calculaed from he uderlyig MID ad SPX daa wih 7 differe β values. The rages of parameers are he followig: =..30, a= , m=4..87, α= , ad β= Disribuio of aualized reurs of he boosed porfolio of MID-SPX muli-spread radig sraegies for he horizo of τ=63 busiess days is ploed i fig.2 (solid lie. Here, a period from 997/0/0 o 2006/0/0 is covered. Fig.2 preses boh i-sample (raiig daa ad up o.5 years of ou-of-sample daa. Sabiliy of he obaied porfolio sraegy eve for such shor-erm horizo is obvious. Reur disribuio improves for larger ime horizos. Complexiy of he uderlyig idexes is evide from he disribuio of he reurs produced by he buy&hold sraegies: MID-log (dashed lie ad ½ MID-log, ½ SPX-shor (doed lie. I coras o he sable disribuio of oly posiive reurs geeraed by he boosed porfolio sraegy, buy&hold sraegies have comparable umber of egaive ad posiive reurs Fig.2. Disribuio of he aualized reurs (% for 63 days horizo: MID-SPX muli-spread porfolio sraegy (solid lie, MID log buy&hold (dashed lie, ½ MID log ad ½ SPX shor buy&hold (doed lie. Fially, we illusrae he mechaism resposible for sabiliy ad robusess of he boosed porfolio sraegy. As obvious from he very boosig operaio, obaied sigle sraegies are locally ucorrelaed or egaively correlaed ad capable o suppor each oher hrough may differe marke regimes. Ofe he global performace (i.e., a average over all marke regimes of sigle members of he boosed porfolio is o impressive. However, sice hey are locally complemeary o each oher, he global performace of he fial porfolio could be very sable ad sigificaly superior o he bes sigle sraegy Fig.3. Disribuio of he aualized reurs (% for 63 days horizo: MID-SPX muli-spread porfolio sraegy (solid lie ad each idividual sraegy from he porfolio (doed lies.

8 This is illusraed i fig.3, where disribuios of he aualized reurs o he 63 day period for all 7 sraegies from he boosed porfolio are show by doed lies. The correspodig disribuio of reurs produced by he boosed porfolio is show by solid lie. The qualiaive ump from he sub-opimal performace of sigle sraegies o he impressive sabiliy of heir combiaio is very clear. Reurs of hree sraegies wih comparable weighs from he curre porfolio are ploed i fig.4 chroologically. These sraegies exploi spread reds o differe ime scales (=70,, 50 ad operae o differe spread ime series (β=0.8, -0., (solid, dashed, ad doed lies. I is clear from fig.4 ha hese sraegies are complimeary o each oher. This is he uderlyig source of he boosed porfolio robusess ad sabiliy Fig.4. Shifed aualized reurs (% of 3 idividual sraegies from he MID-SPX muli-spread boosed porfolio (i chroological order. 5. Coclusios Limiaios of he moder echiques for he ideificaio of he marke-eural porfolios ad dyamic spread ( pairs radig sraegies have bee discussed. A geeric boosig-based framework for he discovery of he sable muli-spread porfolio sraegy, ha ca address may uresolved issues, has bee proposed. The framework ca be efficiely used for he simulaeous discovery of ew syheic isrumes specified as spreads of exisig isrumes ad opimal radig sraegies for each such spread. Preseed argumes ad real-marke example clarify he essece of he ew framework as a powerful geeralizaio of he exiig pairs radig echiques ad coiegraio ools. Exisig approaches are applicable oly o a small ad cosaly decreasig subse of exisig isrumes wih a simple mea-reverig spread dyamics. I coras, he proposed framework ca be used wih much larger se of weakly-coiegraed isrumes feaurig complex spread dyamics. 6. Refereces [] C. Alexader, ad A. Dimiriu, "The Coiegraio Alpha: Ehaced Idex Trackig ad Log-Shor Equiy Marke eural Sraegies", ISMA Fiace Discussio Paper o , 2002 [2] C. Alexader, Marke Models: A Guide o Fiacial Daa Aalysis, Wiley, 200 [3] A.. Burgess, Usig coiegraio o hedge ad rade ieraioal equiies, i Applied quaiaive mehods for radig ad ivesme, edied by C. Duis, J. Laws, ad P. aim, Wiley, 2003 [4] R.F. Egle, ad C.W.J. Grager, Coiegraio ad error-correcio: Represeaio, Esimaio ad Tesig, Ecoomerica, 55, 25, 987 [5] V.V. Gavrishchaka, Boosig frameworks i fiacial applicaios: From volailiy forecasig o porfolio sraegy opimizaio, CIEF, 2005 [6] V.V. Gavrishchaka, Boosig-based framework for porfolio sraegy discovery ad opimizaio, ew Mahemaics ad aural Compuaio, vol. 2, 2006 (o appear [7] V.V. Gavrishchaka, Boosig-Based Frameworks i Fiacial Modelig: Applicaio o Symbolic Volailiy Forecasig, Advaces i Ecoomerics, 20B, 23, 2006 [8] T. Hasie, R. Tibshirai, ad J. Friedma, The Elemes of Saisical Learig: Daa Miig, Iferece, ad Predicio, Spriger, 200 [9] J.O. Kaz, ad D.L. McCormick, The ecyclopedia of radig sraegies, McGraw- Hill, 2000 [0] K. Li ad D. Weibaum, The empirical performace of he aleraive exreme value volailiy esimaors, Workig Paper, Ser School of Busiess, ew York, 2000 [] W.H. Press, S.A. Teukolsky, W.T. Veerlig, ad B.P. Flaery, umerical Recipes i C: The Ar of Scieific Compuig, Cambridge Uiversiy Press, 992 [2] G. Rasch, Robus boosig via covex opimizaio: Theory ad applicaios, PhD hesis, Posdam Uiversiy, 200 [3] R.E. Schapire, The desig ad aalysis of efficie learig algorihms, PhD hesis, MIT Press, 992

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