ALMA-Arnaud-Legoux-Moving-Average原版完整文件

ALMA

ArnaudLegouxMovingAverage

November24,2009

Copyright:ArnaudLegoux/DimitriosKouzisLoukas

ALMA;InsearchfortheperfectMovingAverage-CreatedbyArnaudLegoux&DimitrisKouzis-Loukas

EverytradingsystembasedontechnicalanalysisusesMovingAverages.UsuallyMovingAverages/crossesareconsideredtobeentryorexitsignals.FurthermoreMovingAveragesareoftenresponsiblefordistinguishingrandompricemovements(jitter)fromtherealtrend.ThismeansthatasignificantamountofyourprofitsdependsonthequalityoftheMovingAveragesyouuseforyourtradingsystem.OneofthebenefitsofMovingAveragesisthattheyhavequitestandardformwhichallowsustoeasilyswitchfromonetoanother,comparethemandchoosethebestforourtradingsystem.

TherearetwokeyfeatureswearelookingforonanexcellentMovingAveragenamedsmoothnessandresponsiveness.Smoothnessisimportantbecauseitallowsustotakedecisionsaccordingtotruetrendsinsteadofrandomnoise.Responsivenessontheotherhandisimportantinordertotakedecisionstimely.DecidingthatatrendistruewithbiglatencywastespreciousprofitPIPs.NoweveryMovingAverageisaDiscreteTimeFilterandassuchitisruledbytheUncertaintyPrinciplewhichmeansthatsmoothnessandresponsivenessareconflictingrequirements.Everybodywhohasbasicexperiencewithanymovingaveragee.g.SMAhasknowsfirsthandthata9-daySMAismuchmoreresponsiveandmuchlesssmooththana21-daySMA.Itlookslikewecan/thavebothatthesametimeandthatholdstrueifwerestrictourselvesinasingletypeofMovingAverage.OntheotherhanddifferenttypesofMovingAverageshavedifferentperformancewhenitcomestosmoothnessandresponsiveness.

Table1.FormulasandKernelsofdifferentMovingAverages

InFigure1wecanseethreeofthemostpopularmovingaveragesthatareusedineverydaypracticefromhundredsoftraders;SMA,EMAandHMA.WecanalsoseeALMA,themovingaverageweproposeandwhichoutperformseveryothermovingaveragebothintermsofsmoothnessandresponsiveness.SMAandEMAareverywellknownandpopularbecauseoftheirsimplicity.AswecanseeinTable1,theirformulasareverysimple.Theyarebothweightedsumsofthepriceofthedayforanumberofdaysinthepastwhichiscalledwindow.ForexampleanSMAwithWindow9willaddthepricesp(i)ofthe9previousdaysandthendivideitby9andgiveasactuallytheaverageofthe9previousdays.TheEMAwoulddoexactlythesamebutitwouldaddthepriceofthelastdaytwiceanddivideby10daysemphasizingthiswaylastday'spricethusincreasingtheresponsivenessslightly.WecanseethatinFigure1.IfwecomparetheTurquoise(EMA)andtheOrange(SMA)linewecanclearlyseethatEMAoutperformsSMA.BothSMAandEMAaresimpleandpowerfulbutbecausetheyworkonawindowgivingmoreorlessequalvalueonthepriceofeverydaytheirvaluerepresentsthepriceaboutWINDOWSIZE/2daysinthepast.Thatmeansthatifwerune.g.a21-daymovingaveragewegettodayaverypreciseestimationonwhatwastheprice10daysago!ThismeansmanyPIPslostifwetradethetrendbutevenworseitmeansslowexitsthatmightbeverypainfulifweareshortingortradingreversals.DespitethesewellknownweaknessesofSMA/EMAs,theyareverypopularandwidelyusedgivinglargeprofitstotraders.ThisposesthequestionwhatcouldsomebodyachievebyusingbetterMovingAverages.

ThenextlevelofMovingAveragesisHMA(HullMovingAverage).HMAisanextremelygoodfilterwhichisverydifficulttobeatbothintermsofsmoothnessandresponsiveness.AswecanseeinFigure1HMA(green)outperformsSMAandEMA,fitsthepriceverywell,ignorestherandommovementsofthepriceandatthesametimegivesverysmoothandnaturalresults.InTable2wecanseetheabstractformulaofHMA.AndHMAistheWMA(WeightedMovingAverage-animprovedversionofSimpleMovingAverageSMA)ofthedifferenceofanothertwoWMAs.ThefactthatHMAisadifference,makesitveryfastandresponsiveastheequivalenthigh-passfiltersinsignalprocessing.Nowthemaindrawbackofhigh-passfiltersandofcourseHMAistheovershoot

effectsastheonewecanseemarkedwith3inFigure1.Dependingonourtradingsystem,thoseovershootsmightbefromindifferenttodestructivebutcertainlyitistheAchilles'heelofHullMovingAverage.

Table2.FormulasandKernelsofdifferentMovingAverages

WeproposeALMA,aMovingAveragewhichperformsbetterthantheHMA.ALMAisinspiredbytheGaussianFiltersanditattacksafundamentalassumptionoftheMovingAverageswedescribedbefore.AswecanseeinFigure2MovingAverageslikeEMAandHMAassumethatthepriceclosesttothecurrentdayissomewhatmoreinformationrich(valuable)thanthepreviousonesandasaresulttheygiveequalorlargerweighttothelastfewdaysintheircalculations.Theyanswerthequestion“whatwillbetheweatherliketomorrow"with"highlylikelythesameastoday".

一一一一

Xn

..

u

IIIday4 day3 day2 day1

Figure2.Theinformationvalueofeachday

Whatthisassumptionignoresisthissecondcomponent;Theclosestto"now"weget,thehighertheuncertaintyabouttheprice.Whatdoesthismean?

$0.40

$0.35

$0.30

$0.25

$0.20

Figure3.Exampleofastockprice

Let'sassumethatFigure3givestheclosingpriceofsomestockforthelast4days.Theanswerofwhatwasthe“realprice”-theonewewouldexpectourMovingAveragetogive-onday3isclear.Somethingaround$0.30.Whyisthatsoclear?Becausewehaveday4andday2givinguscertaintyaboutthe“realprice“ofday3.Ifnowwegettoday1,thenwhatshouldanidealMovingAveragegive?Theansweriswedon'tknowbecausewedon'tknowthefuture.

Figure4.Exampleoftwodifferentscenariosfortomorrow

AswecanseeinFigure4iftomorrow'spriceisaround$0.21wewouldlikeourMovingAveragetohavetodayavalueof$0.26.Iftomorrow'spriceisaround$0.37wewouldliketoday/svaluetobe$0.32.Thismeansthatincontrasttoday3wherewehavebigconfidenceonwhatthe“realprice"is,forday1wecanhavenoconfidence.Itcouldbeequallywell$0.26or$0.32.ThismeansthataswecanseeinFigure5ourconfidenceonthe“realprice“getlowerthecloserwegettothecurrentday.

1n

■■

u

111

day4 day3 day2 day1

Figure5.Theconfidenceforthe“realprice“foreachday

Thesearethetwoconcepts,"'informationvalue"and'Informationconfidence“thatALMAcombinesansweringthequestion“whatwillbetheweatherliketomorrow"with"highlylikelythesamelikeyesterdayandthedaybeforeyesterday**-butnotnecessarilyastoday.

∣MAType ∣∣Kernel∣

ALIVIA(SIZE,a,OtTSet)

NORMAP@)e/i=i

0

?

1r

.??

U

1111

day4 day3 day2 day1

-A

Table3.FormulasandKernelsofdifferentMovingAverages

TheformulaofALMAcanbeseeninTable3.ItusesaGaussiandistributionshiftedwithanoffsetsothatit'snotevenlycenteredonthewindowbutbiasedtowardsthemorerecentdays.Theoffsetisadjustablesowecantradeoffsmoothnessandresponsiveness.Thesecondparameteristhesigma(a)parameterwhichchangestheshapeofthefiltermakingitmorewide(larg