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Maximum likelihood estimation of mean reverting …

Maximumlikelihood estimationofmeanrevertingprocessesJos e CarlosGarc a frequently usedmodelsin instance,somecommodity prices(or theirlogarithms)are frequently believed to revert to somelevel parameterknowledge,basedon economicanalysisof the forcesat play, is perhapsthe mostmeaningfulchannelfor model ,databasedmethods are oftenneededto complement andvalidatea meanreverting(OUMR)model is a Gaussianmodel well suitedfor maximumlikelihood (ML)methods. Alternative methods includeleastsquares(LS)regressionof discreteautoregressive versionsof the OUMR model andmethods of moments (MM).Each methodhas advantagesand instance,LS methods may not always yielda reasonableparameterset (see Chapter3 of Dixitand Pindyck[2]) and methods of moments lack the desirableoptimality propertiesof ML or LS Maximum - likelihood (ML)methodologyfor parameterestimationof1-dimensionalOrnste in-Uhlenbeck (OR) ,our methodol-ogy ultimatelyrelieson a one-dimensionalsearch which greatlyfacilitatesestimationand easilyaccommodtesmissingor unevenlyspaced(time-wise) meanreverting(OUMR)processgiven by thestochasticdifferentialequation(SDE)d x(t) = ( x x(t))dt+ dB(t);x(0)=x0

Maximum likelihood estimation of mean reverting processes Jos e Carlos Garc a Franco Onward, Inc. jcpollo@onwardinc.com Abstract Mean reverting processes are frequently used models in real options.

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Transcription of Maximum likelihood estimation of mean reverting …

1 Maximumlikelihood estimationofmeanrevertingprocessesJos e CarlosGarc a frequently usedmodelsin instance,somecommodity prices(or theirlogarithms)are frequently believed to revert to somelevel parameterknowledge,basedon economicanalysisof the forcesat play, is perhapsthe mostmeaningfulchannelfor model ,databasedmethods are oftenneededto complement andvalidatea meanreverting(OUMR)model is a Gaussianmodel well suitedfor maximumlikelihood (ML)methods. Alternative methods includeleastsquares(LS)regressionof discreteautoregressive versionsof the OUMR model andmethods of moments (MM).Each methodhas advantagesand instance,LS methods may not always yielda reasonableparameterset (see Chapter3 of Dixitand Pindyck[2]) and methods of moments lack the desirableoptimality propertiesof ML or LS Maximum - likelihood (ML)methodologyfor parameterestimationof1-dimensionalOrnste in-Uhlenbeck (OR) ,our methodol-ogy ultimatelyrelieson a one-dimensionalsearch which greatlyfacilitatesestimationand easilyaccommodtesmissingor unevenlyspaced(time-wise) meanreverting(OUMR)processgiven by thestochasticdifferentialequation(SDE)d x(t) = ( x x(t))dt+ dB(t);x(0)=x0(1)forconstants x, andx0andwhereB(t) is thismodeltheprocessx(t) fluctuatesrandomly, buttendsto revertto somefundamentallevel x.

2 Thebehaviorof this reversion dependsonboththeshorttermstandarddeviati on andthespeedof reversionparameter . showsa samplepathfor120monthsof a meanrevertingprocessstartingata levelx(0)= 12,thattendstoreverttoa level x= 15,witha speedof reversion = 4anda shorttermstandarddeviation = 5 (onethirdof thelevel of reversion).Thesolidlineshowsthelevel of be evident is that,as opposedto randomwalks(withdrift),theprocessdoes notexhibitanexplosive behavior,butrathertendsto ,it may be shownthatthelong-termvarianceof thepro-cesshasa oftendesirablefortheanalysisof economicvariablesthathave afundamentalreasonto fluctuatearounda given example,thepriceof somecommoditiesor themarginalcostcurve fortheproductionof somegood. However,fittingor calibrationof suchmodelsis noteasyto comeby.

3 Whilealltheparametersmay have someintuitive meaningto theanalyst,measuringthemis quiteanotherstory. In thebestof casesthereis somefundamentalknowledgethatleadsto fixinga parameter,thisis hopefullythecaseforthereversionlevel x, yet,it is unlikelyto have expertknowledgeof allparametersandwe areforcedto coursethatsuch datais willillustratea maximumlikelihood (ML)estimationprocedureforfindingthepara metersof ,in orderto do this,we mustfirstdeterminethedistributionof theprocessx(t).Theprocessx(t) is a gaussianprocesswhich is wellsuitedformaximumlikelihood thesectionthatfollowswe willderive thedistributionofx(t) by solvingtheSDE(1).1 Thedistributionof the OR processTheOUmeanrevertingmodeldescribedi n (1)is a gaussianmodelin thesensethat,givenX0,thetimetvalueof theprocessX(t) is normallydistributedwithE[x(t)|x0] = x+ (x0 x) exp[ t] andVar[x(t)|x0] = 22 (1 exp[ 2 t]).

4 AppendixA explainsthisbasedonthesolutionof theSDE(1).Figure2 showsa forecastin theformof 10-50-90confidenceintervalscorresponding to theprocesswe previouslyusedas a constant, isdemonstratedby theflatnessof theconfidenceintervalsas we forecastfartherinto (which is equalto themedianintheOUmeanrevertingmodel)tends to thelevel of Maximum likelihood estimationForti 1< ti, thexti 1conditionaldensityfiofxtiis given :OUMR10-50-90confidence (xti; x, , ) =(2 ) 12( 22 (1 e 2 (ti ti 1))) 12 (2)exp[ (xti x (xti 1 x)e (ti ti 1))22 22 (1 e 2 (ti ti 1))].(3)Givenn+ 1 observationsx={xt0, .. , xtn}of theprocessx, thelog- likelihood function1correspondingto (??) is given byL(x; x, , )= n2log[ 22 ] 12n i=1log[1 e 2 (ti ti 1)] 2n i=1(xti x (xti 1 x)e (ti ti 1))21 e 2 (ti ti 1).

5 (4)Themaximumlikelihood estimates(MLE) x, and maximizethelog- likelihood functionandcanbe foundby is to relyonthefirstorderconditionswhich requiresthesolutionof a non-linearsystemof ,optimizationor we willattempttoobtainananalyticalternative forMLestimation, isbasedontheapproach foundin Barz[1].However,we doallow forarbitrarilyspacedobservations(timewis e)andavoidsomesimplifyingassumptionsmade in thatwork(forpracticalpurposes). estimationrequirethegradient of thelog- likelihood to be equalto ,themaximumlikelihood estimators x, and satisfythefirstorderconditions:1 Withconstant L(x; x, , ) x x=0 L(x; x, , ) =0 L(x; x, , ) =0 Thesolutionto thisnon-linearsystemof equationsmay be foundusinga variety of ,in thenextsectionwe willillustrateanapproach thatsimplifiesthenumericalsearch by exploitingsomeconvenient analyticalmanipulationsof hybridapproachWe firstturnourattentionto thefirstelement of thegradient.

6 We have that L(x; x, , ) x= 2n i=1xti x (xti 1 x)e (ti ti 1)1 +e (ti ti 1)Undertheassumptionthat and arenon-zero,thefirstorderconditionsimply x=f( ) =n i=1xti xti 1e (ti ti 1)1 +e (ti ti 1)(n i=11 e (ti ti 1)1 +e (ti ti 1)) 1.(5)Thederivative of thelog- likelihood functionwithrespectto is L(x; x, , ) = n +2 3n i=1(xti x (xti 1 x)e (ti ti 1))21 e 2 (ti ti 1),which togetherwiththefirstorderconditionsimpli es =g( x, ) = 2 nn i=1(xti x (xti 1 x)e (ti ti 1))21 e 2 (ti ti 1).(6)Expressions(5)and(6)definefunction sthatrelatethemaximumlikelihood have xas a functionfof and as a functiongof and orderto solve forthemaximumlikelihood estimates,we couldsolve thesystemof non-linearequationsgivenby x=f( ), =g( x, ) andthefirstordercondition L(x; x, , )/ | = ,theexpressionfor L(x; x, , )/ is algebraicallycomplexandwouldnotleadto aclosedformsolution,requiringa simplerapproach is to substitutethefunctions x=f( ) and =g( x, ) directlyinto thelikelihood functionandmaximizewithrespectto.

7 Soourproblembecomesmin V( )(7)whereV( )= n2log[g(f( ), )22 ] 12n i=1log[1 e 2 (ti ti 1)] g(f( ), )2n i=1(xti f( ) (xti 1 f( ))e (ti ti 1))21 e 2 (ti ti 1). = x= = : Anexampleof is nothardto show thatthesolutionto theproblem(7)yieldsthemaximumlikelihoode stimator . Oncewe have obtained we caneasilyfind x=f( ) and =g( x, ). Theadvantageofthisapproach is thattheproblem(7)requiresa onedimensionalsearch andrequirestheevaluationof ExampleConsidera familyof weeklyobservations(samples)fromanOrnstei n-Uhlenbeck meanrevertingprocesswithparameters x= 16, = and = 4 startingatX(0)= is known(1)thattheMLE sconvergeto thetrueparameteras thesamplesizeincreasesand(2)thattheMLE ,in practicewe donotenjoy theconvergencebenefitsgivenby getanideaof how theMLE sbehave underdifferent samplesizes,a simulationexperiment was conductedwherewe estimatedthemeanandvarianceof canbeginto appreciatetheaccuracyof themethod as well as theasymptoticbehaviorof themaximumlikelihood ConclusionIn thisnotewe developed a practicalsolutionforthemaximumlikelihood estimationof an Ornstein-Uhlenbeck ourapproach is thatby leveragingonsomemanipulationof thefirstorderconditions,we canreduceMLestimationto a onedimensionaloptimizationproblemwhich cangenerallybe solvedin a matterof theproblemto onedimensionalsofacilitatesthelocalizati onof a.

8 It is worthmentioningthatthemethod rivalsalternative methodssuchas regressionof a discreteversionof theOUmeanrevertingmodelor moment , we notethatthemethod presentedin thisnotetriviallyaccommodatesfundamental knowledgeof any of theprocessparametersby simplysubstitutingtheknownparameter(s) instance,if xis known,we forgetaboutthefunction x=f( )andsimplyplugin theknown xinto theotherequationsas theMLE [1]Barz,G.(1999)StochasticFinancialModel sforElectricityDerivatives, ,Department of Engineering-EconomicSystemsandOperations Research, StanfordUniversity,Stanford,CA.[2]Dixit, (1994)InvestmentUnderUncertainty, PrincetonUniversityPress,Princeton,NJ.[3 ]Greene,WilliamH.(1997)EconometricAnalys is, ,PrenticeHall,Upper SaddleRiver,NJ.[4]Luenberger, (1998)InvestmentScience, OxfordUniversity Press,NewYork,NY.

9 [5] ksendal,B.(1995)StochasticDifferentialEq uations:An IntroductionwithApplications,4thed.,Spri nger-Verlag,NewYork, Solvingthe Ornstein-Uhlenbeck SDEC onsidera meanrevertingOrnstein-Uhlenbeck processwhich is described by thefollowingstochas-ticdifferentialequat ion(SDE)d x(t) = ( x x(t))dt+ d B(t);x(0)=x0(8)Thesolutionof theORSDEis standardin (e tx(t)) =x(t) e tdt+e tdx(t),andthereforewe havee tdx(t) =d(e tx(t)) x(t) e tdt.(9)Multiplyingbothsidesof (8)bye t, we gete tdx(t) =e t ( x x(t))dt+e t dB(t),(10)which togetherwith(9)impliesd(e tx(t)) = e t x dt+e t dB(t).(11)Therefore,we cannow solve for(11)ase tx(t) =x0+ t0 e s x ds+ t0e s dBs,(12)or equivalentlyx(t) =x0e t+ t0 e (t s) x ds+ t0e (t s) dBs.(13)Thefirstintegralontheright handsideevaluatesto x(1 e t) andsinceBtis Brownianmotion,thesecondintegralis normallydistributedwithmeanzeroandvarian ceE[( t0e (t s) dBs)2].

10 ByItoisometry2we haveE[( t0e (t s) dBs)2]= t0(e (t s) )2ds= t0e 2 (t s) 2ds= 22 (1 e 2 t).Hence,Xtis normallydistributedwithE[Xt|X0] = x+(X0 x)e tandVar[Xt|X0] = 22 (1 e 2 t).2 See ksendal[5]fordetails.


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