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6 0 0 2 Hidden Forces andFluctuations fromMoving Averages: n a ATestStudy J 2 1 V.Alfia,b ,F.Coccettid,M.Marottae,L.Pietroneroa,c,e,M.Takayasuf aUniversità“LaSapienza”, Dip. di Fisica, 00185, Rome, Italy. h] bUniversità“Roma Tre”,Dip.di Fisica, 00146, Rome, Italy. p cIstitutodei Sistemi Complessi - CNR,ViaFosso delCavaliere 100, 00133 Rome, Italy. - dMuseo Storico dellaFisica e CentroStudi eRicerche “EnricoFermi“,ViaPanisperna, Rome, Italy. c eApplied Financial Science, NewYork,USA. o fDepartment of Computational Intelligence andSystemsScience,TokyoInstitute of Technology, s MailBox G3-524259 Nagatuta-cho, Yokohama 226-8502, Japan. . s c i s y Abstract h p Thepossibilitythatpricedynamicsisaffectedbyitsdistancefromamovingaveragehasbeenrecentlyintroducedas [ newstatisticaltool.Thepurposeistoidentifythetendencyofthepricedynamicstobeattractiveorrepulsivewith 1 respecttoitsownmovingaverage.Weconsideranumberoftestsforvariousmodelswhichclarifytheadvantages v andlimitationsofthisnewapproach.Theanalysisleadstotheidentificationofaneffectivepotentialwithrespect 9 to the moving average. Its specific implementation requires a detailed consideration of various effects which can 8 alterthestatisticalmethodsused.However,thestudyofvariousmodelsystemsshowsthatthisapproachisindeed 0 suitable todetect hiddenforces in themarket which go beyondusualcorrelations and volatility clustering. 1 0 6 Keywords: Complexsystems, Timeseries analysis,Effective potential, Financial data 0 PACS: 89.75.-k,89.65+Gh, 89.65.-s / s c i s 1. Introduction seems to have various advantages with respect to y theusualHurstexponentindescribingthe fluctu- h p Theconceptofmovingaverageisverypopularin ationsofhighfrequenciesstock-prices. v: empiricaltradingalgorithms[1]but,uptonow,it Amorespecificanalysisofthesefluctuationscan i hasreceivedlittle attentionfromascientificpoint befoundintworecentpapers[6,7]whichattempt X of view [2, 3, 4]. Recently we have proposed that to determine the tendency of the price to be at- r a new definition of roughness can be introduced tracted or repelled from its own moving average a by considering fluctuations from moving averages (Fig. 1). This is completely different from the use with different time scales [5]. This new definition of moving averages in finance, in which empirical rulesandpredictionsaredefinedintermsofapri- ∗ Correspondingauthor oriconcepts[1].Theideaisinsteadtointroducea Email address: [email protected] (V. statisticalframeworkwhichisabletoextractthese Alfi). Preprintsubmitted toElsevier Science 2February2008 tendencies fromthe pricedynamics. movingaverage.In bothapproaches[6,7]it is as- sumedtobe quadratic: 1155 2 1100 φ P(t)−PM(t) = P(t)−PM(t) . (3) 55 (cid:16) (cid:17) (cid:16) (cid:17) Despite this similar starting point the two stud- )) 00 ies proceed along rather different perspectives. In P(tP(t Ref.[6]thethreeessentialparametersofthemodel --55 (b;M;σ) are consideredas constants with respect --1100 tot.Then,byanalyzingthepricefluctuationsover a suitable time intervalandfor a long time series, --1155 the valuesofthe three parametersareidentified. 00 110000 220000tt 330000 440000 550000 In Ref. [7] instead the analysis is performed by lookingdirectlyattherelationbetweenP(t+1)− Fig.1.Exampleofamodelofpricedynamics(inthiscase P(t)andP(t)−PM(t).Thispermitstoderivethe a simple random walk) together with its moving average formofthepotentialandtoidentifytheparameter definedastheaverageovertheprevious50points.Theidea isthatthedistanceofthepricefromitsmovingaveragecan b(t) and its time variation. For the US$/Yen ex- lead to repulsive (blue arrows) or attractive (red arrows) changeratesthepotentialisfoundtobequadratic effective forces. anditispossibletorescaleitwiththeterm1/(M− 1)observingagooddatacollapse.Thiswouldim- plythatitisnotnecessarytospecifythetimescale ofthe movingaverage. 2. The Effective Potential Model The basic idea is to describe price dynamics in 3. TestStudies terms of an active random walk (RW) which is influencedbyitsownmovingaverage.Thisinduces Given these different perspectives, which arise complex long range correlations which cannot be fromthesamebasicmodel,wedecidedtoperform determinedbytheusualcorrelationfunctions and aseriesoftests ofthis approachwhichwe present that can be explored by this new approach [6, 7]. inthispaper.Webelievethatthesetestscaneluci- The basic ansatz is that price dynamics P(t) can datevariouspropertiesandlimitationsofthe new be described in terms of a stochastic equation of approachandrepresentausefulinformationforits type: future developmentsandapplications. P (t+1)−P(t)= In Fig. 1 we show a simple RW and a moving averagewhichrepresentsitsownsmoothedprofile. d =−b(t) Φ P(t)−PM(t) + The analysis is performed by plotting the values d P(t)−PM(t) (cid:16) (cid:17) of P(t+1)−P(t) as a function of P(t)−PM(t) +σ(t)ω(t(cid:16)) (cid:17) (1) andderivingthepotentialbyintegratingfromthe center[7].ThesimpleRWleadstoaflatpotential where ω(t) corresponds to a random noise with (no force) as expected (Fig. 2). Than we can take unitaryvarianceand thesmoothedprofile(previousmovingaverage)as adatasetbyitselfandrepeattheanalysisbycom- M 1 paringit to a new,smoother moving average(not PM(t)≡ P(t−k) (2) M shown). As one can see in Fig. 2 this leads to an k=1 X apparentrepulsivepotentialwhichshouldbe con- is the movingaverageoverthe previousM steps. sideredasspurious.Thisisduetothefactthatthe ThepotentialΦtogetherwiththepre-factorb(t) smoothedcurveimpliessomepositivecorrelations describetheinteractionbetweenthepriceandthe as shown in Fig. 3. Therefore in this framework 2 11 22 00 00 alal alal ntinti--11 ntinti--22 ee ee otot--22 otot--44 PP PP simple RW --33 ssmmooootthheedd RRWW --66 Attractive Potential RReeppuullssiivvee PPootteennttiiaall --88 --44 --2200 --1155 --1100 --55 00 55 1100 1155 2200 --1155 --1100 --55 00 55 1100 1155 PP((tt))--PPMM((tt)) PP((tt))--PPMM((tt)) Fig.2.Effectivepotentialforarandomwalk(flatline)and Fig. 4. Effective potential reconstructed from a series of asmoothedrandomwalk(convexparabola).Theapparent data obtained by a dynamics corresponding to Eqs.(1, 2) repulsive potential corresponding to the smoothed RW is forthe twocases ofattractive andrepulsivepotentials. In spurious and due to the correlations corresponding to the this case M =30 and b=±1. The units of the potential smoothingprocedure.Theunitsofthepotentialaredefined aredefined byEq.(1). byEq.(1). at very short times which should be filtered with 11 simple RW suitablemethodsinordertoperformthepotential ssmmooootthheedd RRWW analysis[7]. 00..88 00..66 )) 111222 ττ (( ρρ 00..44 111000 00..22 888 ))) 00 P(tP(tP(t 666 00 1100 2200 ττ 3300 4400 5500 ∆∆∆ 444 Fig.3. Autocorrelation (ρ) ofthe priceincrements forthe 222 simpleRWanditssmoothedprofile.Onecanseethatthe 000 smoothing procedure induces positive correlations up to 000 222000000 444000000 ttt 666000000 888000000 111000000000 the smoothing length (inthis case 10steps). Fig. 5. Absolute price variations for different time positivecorrelationsleadtoadestabilizingpoten- steps(τ = 1 (yellow); τ = 5 (blue); τ = 10 (red)) corre- tial with respect to the moving average. The op- spondingto the dynamics of Eqs.(1, 2). positewouldhappenfornegativecorrelations(zig- zagbehavior). Wenowconsiderthemodelofthequadraticpo- The interesting question is however if one can tential as in Refs. [6, 7]. The effective potential is identifyanontrivialsituationintermsoftheeffec- easily reconstructed as shown in Fig. 4. We also tivepotentialbutinabsenceofsimplecorrelations. show in Fig. 5 the behavior of the absolute price This would be the new, interesting situation and variationsfordifferenttimesteps.Thecorrelation thecorrespondingforcescanbeconsideredashid- function for the price and volatility are shown in den,inthesensethattheydonothaveanyeffectin Fig. 6 which clarifies that, in this case, no simple the usual correlation functions. Real stock-prices correlation is present, nor is there any volatility data clearly do not show any appreciable correla- clustering effect. This is an interesting result be- tion, otherwise they would violate the simple ar- cause it shows that the new method is able to de- bitrage hypothesis. In the exchange rates instead tecthiddenforceswhichhavenoeffectintheusual there is a zig-zag behavior (negative correlation) correlationsofpricesorvolatility. 3 11 0 PPrriiccee vvaarriiaattiioonnss 00..88 Price absolute variations -0.5 al-1 00..66 nti ττ()() 00..44 ote-1.5 ρρ P-2 00..22 RW with asymmetric drift -2.5 00 -3 -15 -10 -5 0 5 10 15 00 1100 2200 ττ 3300 4400 5500 P(t)-P (t) M Fig. 6. The correlation analysis of price variations shows Fig. 8. Effective potential corresponding to the dynamics no correlations between price differences and no volatility of Eq.(4) with ǫ1 =0.05 and ǫ2 =0.10. One can see that clustering effect. This implies that the presence of attrac- in this case the distribution is asymmetric and it extends tiveorrepulsiveforceswithrespecttothemovingaverages moreinthe directionforwhich the instabilityis stronger. isnot detectable withthe usual statistical indicators. In this model the effective force only depends on the sign ofP(t)−PM(t)andnotonitsspecificvalue.Theunitsof 4. ProbabilisticModels thepotential aredefined by Eq.(1). p(↑)=1/2+ǫ1 for P(t)−PM(t)>0 (4) 2 (p(↓)=1/2−ǫ1 1 al 0 p(↑)=1/2−ǫ2 for P(t)−PM(t)<0 nti (p(↓)=1/2+ǫ2 . e-1 ot P-2 This implies a tendency of destabilization (repul- RW with drift sionfromPM(t))whosestrengthisonlydependent -3 onthe signof P(t)−PM(t). In principle the situ- -4-15 -10 -5 0 5 10 15 ationcanbeasymmetricwithǫ1 6=ǫ2.Thepoten- P(t)-P (t) tialanalysisforthiscaseleadstoapiecewiselinear M potentialinwhichthe slopesarerelatedtoǫ1 and Fig. 7. Effective potential corresponding to a RW with a ǫ2 (Fig.8).One canalsosee that oneline extends constantdriftwhichalterstheprobabilityforastepupor down. The units of the potential aredefined by Eq.(1). 11 PPrriiccee vvaarriiaattiioonnss We now consider some variations to the RW 00..88 Price absolute variations which depend on P(t) − PM(t). We modify the 00..66 probabilityofacertainstepratherthanthesizeof )) thestepasinEq.(1).Thesimplestmodelistoadda ττ (( 00..44 ρρ constantdrift,independentonthevalueofPM(t). The effective potential corresponding to this case 00..22 is simply linear as shown in Fig. 7. One can see 00 thatinthiscasethepointwhereP(t)−PM(t)=0 isnotaspecialpointandthismodelappearstobe 00 1100 2200 ττ 3300 4400 5500 oversimplifiedwithrespecttothedatasetanalyzed upto now[6,7]. Fig.9.Correlationanalysisofpricevariationsandvolatility A more interesting model is represented by the for the model of Eq. (4). Also in this case no detectable correlations arepresent. following dynamics for a RW with only up and downsteps: morethantheotherindicatinganasymmetricdis- 4 tribution. Also in this case the correlation of the 2e-05 pricevariationsandvolatilitiesshownodetectable 1.5e-05 effectasshowninFig.9.Clearlyinthiscasetheef- freeclatitvioenpsobteentwtieaelnisPju(ts+ta1)re−pPre(ste)natnadtioPn(to)f−thPeMco(tr)- ntial 1e-05 e whose microscopic origin is instead in the modifi- ot 5e-06 P cationofthe probabilityforunitary steps. 0 -5e-06 -0.01 -0.005 0 0.005 0.01 0.015 5. Fractal Model P(t)-P (t) M Itmaybeinterestingtoconsideralsothecaseof Fig. 11. Effective potential corresponding to the fractal a fractalmodel constructedby an iterativeproce- price model. The units of the potential are defined by dure [2],Fig.10. Eq.(1). that of a reference RW (b =0). The first observa- tionisthattherepulsivepotentialmakesthedistri- bution broader (super diffusion) while the attractive potential makes it narrower (sub diffusion). ThisbehaviorwasalreadyobservedinRefs.[6,7]. 111 bbb(((ttt)))>>>000 000...111 bbb(((ttt)))==<000 000...000111 WWW 000...000000111 Fig.10. Example ofa fractal model distributionofprice 111eee---000444 The fractal model does not have a specific dy- 111eee---000555 namics but, since it is often considered as to cap- ture some properties of real prices, we consider of 111eee---000666 ---222000 ---111555 ---111000 ---555 000 555 111000 111555 222000 some interest to study if this model would corre- PPP(((ttt)))---PPP (((ttt))) MMM spond to some type of effective potential. In Fig. 11wecanseethattheeffectivepotentialisslightly Fig.12.Distributionofthefluctuations,W(P(t)−PM(t)), attractive.Giventhe symmetry ofthe modelcon- for the dynamics of Eq. (1-3) and different values of the parameter b. struction,theasymmetryobservedintheeffective potential is probably due to the backward con- structionofthe correspondingmovingaverage. Less trivial is the fact that the distributions are wellrepresentedbygaussiancurves. In Fig. 13 we show the same distributions corresponding to the probabilistic model of Eq. (4) 6. Analysisofthe Fluctuations for the case of asymmetric attractive and repul- siveeffects.Inthiscasethereisamarkeddeviation Wenowconsiderthenatureoffluctuationsfrom from the gaussian behavior and the case of repul- the moving average by analyzing the probability sive trend develops two separate peaks. It will be distributionW P(t)−PM(t) forthevariousmod- interestingtocheckthecorrespondingdistribution els. In Fig. 12(cid:16)we show the(cid:17)distributions corre- onrealstock-priceswhichweintendtoperformin spondingtothequadraticpotentialascomparedto the future. 5 [3] R.N. Mantegna, H.E. Stanley, An Introduc- 000...111 tion to Econophysics, Cambridge University Press,Cambridge,2000. 000...000111 [4] J.P. Bouchaud, Theory of Financial Risk 000...000000111 andDerivativePricing,CambridgeUniversity WWW Press,Cambridge,2003. 111eee---000444 AAAttttttrrraaaccctttiiivvveee RRRWWW [5] V. Alfi, F. Coccetti, M. Marotta, A.Petri, SSiimmppllee RRWW 111eee---000555 Repulsive RW L.Pietronero, Roughness and Finite Size Ef- fect in the NYSE Stock-Price Fluctuations, 111eee---000666 ---111555 ---111000 ---555 000 555 111000 111555 preprint2006. PPP(((ttt)))---PPP (((ttt))) MMM [6] VR. Baviera, M. Pasquini, J. Raboanary, M.Serva,MovingAveragesandPriceDynam- Fig.13.Distributionsofthefluctuations,W(P(t)−PM(t)), forthedynamicsofEq.(??).Inthiscasethedistributions ics, International Journal of Theoretical and became asymmetricdue to different values of ǫ1 andǫ2. Applied Finance,vol.5,num.6,pag.575-583, 2002. 7. Conclusionsand Perspectives [7] M. Takayasu, T. Mizuno, H. Takayasu, Potentials of Unbalanced Complex Ki- netics Observed in Market Time Series, In summary the idea to consider price dynam- http://arxiv.org/abs/physics/0509020. ics as influenced by an effective force dependent onthe distance ofprice P(t)fromits ownmoving averagePM(t) represents a new statistical tool to detecthiddenforcesinthemarket.Theimplemen- tation of the analysis can be seriously affected by the eventual presence of positive or negative correlations. However, we have shown by a series of models and tests, that this new method is able to explore complex correlations which have no effect on the usual statistical tools like the correlations ofpricevariationandthe volatilityclustering. The method provides an analysis of the senti- mentofthemarket:aggressiveforthecaseofrepul- siveforcesandconservativeforattractiveones.In thisrespectitmayrepresentabridgebetweenthe financial technical analysis and the application of statistical physics to this field. In addition it may also be useful to analyze the results of the different strategies and behaviors which arise in agent basedmodels. References [1] B.J.Murphy,TechnicalAnalysisoftheFinan- cialMarkets,PrenticeHallPress,1999. [2] B.B. Mandelbrot, Fractals and Scaling in Fi- nance,SpringerVerlag,New York,1997. 6