Pairs trading ch6
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Pairs Trading Ch6. Pairs Selection in Equity Markets 郭孟鑫. Stationary and Nonstationary. Stationary. For all t-s, and t-s-j. Stationary and Nonstationary. Spurious regression. Downward bias Using unit root test to verify. Introduction. In chapter 5, we need three steps

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Pairs Trading Ch6

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Pairs TradingCh6

PairsSelectioninEquityMarkets

郭孟鑫


StationaryandNonstationary

  • Stationary

Forallt-s,andt-s-j


StationaryandNonstationary

  • Spuriousregression.

    • Downwardbias

  • Usingunitroottesttoverify.


Introduction

Inchapter5,weneedthreesteps

  • Identificationofthestockpairs

  • Cointegrationtesting

  • Tradingruleformulation

    Thischapterwillfocusoncointegrationtestingandfurtheranalysis.


Introduction

  • WedrawparallelsbetweenthecommontrendsmodelforcointegrationandtheideaofAPT.


CommonTrendsCointegrationModel

  • Inference1:Inacointegrationsystemwithtwotimeseries,theinnovationssequencesderivedfromthecommontrendcomponentsmustperfectlycorrelated.(correlationmustbe+1or-1)


CommonTrendsCointegrationModel


CommonTrendsCointegrationModel

  • Inference2: Thecointegrationcoefficientmaybeobtainedbyaregressionoftheinnovationsequencesofthecommontrendsagainsteachother.


Discussion

  • First,theinnovationsequencesderivedfromthecommontrendsofthetwoseriesmustbeidenticaluptoascalar.

  • Next,thespecificcomponentsofthetwoseriesmustbestationary.


CommonTrendsmodelandAPT

Isthereturnduetothenonstationarytrendcomponent.

Isthereturnduetothestationarycomponent.


CommonTrendsmodelandAPT

  • RecallingtheAPT,thestockreturnsmaybeseparatedintocommonfactorreturnsandspecificreturns.

  • Ifthetwostockssharethesameriskfactorexposureprofile,thenthecommonfactorreturnsforboththestocksmustbethesame.


CommonTrendsmodelandAPT

  • Observation1: A pairs of stocks with the same risk factor exposure profile satisfies the necessary conditions for cointegration.

    • Condition 1: The factor exposure vectors in this case are identical up to scalar.

      Stock A:

      Stock B:


CommonTrendsmodelandAPT

  • If is the factor returns vector, and and are the specific returns for the stocks A and B.

    then,

    and,


CommonTrendsmodelandAPT

  • Condition 2: Consider the linear combination of the returns.

    Where,

    If A and B are cointegrated, then the common factor return becomes zero.


CommonTrendsmodelandAPT

  • Summary

    • The common factor return of a long-short portfolio of the two stocks is zero and the integration of the specific returns of the stocks is stationary, then the two stocks are cointegration.


The Distance Measure

  • The closer the absolute value of this measure to unity, the greater will be the degree of co-movement.


Interpreting The Distance Measure

  • Calculating the Cosine of the Angle between Two Vectors.


Interpreting The Distance Measure

  • Appendix:EigenvalueDecomposition

Let

Disadiagonalmatrixwiththeeigenvalues

Uisamatrixwitheachcolumncorrespondingto

aneigenvectors.


Interpreting The Distance Measure

  • Geometric interpretation

    • Key to doing the geometric interpretation is the idea of eigenvalue decomposition.


Interpreting The Distance Measure


Reconciling Theory And Practice

  • Deviations from Ideal Conditions

    • When two stocks are not have their factor exposures perfectly aligned.

      Signal-to-Noise(weneedbigSNR)


Example

  • ConsiderthreestocksA,B,andCwithfactorexposuresinatwofactorasfollows:


Example

  • Step1:Calculatethecommonfactorvarianceandcovariance.


Example

  • Step2:Calculatethecorrelation(absolutevalueofcorrelationisthedistancemeasure)

    • Ifwehavetochooseonepairforpurposetrading,ourchoicewouldthereforebethepair(A,B)


Example

  • Step3:Calculatethecointegrationcoefficient

    • Wewilldiscussonthenextchapter


Example

  • Step4:Calculatetheresidualcommonfactorexposeinpairedportfolio.Thisistheexposurethatcausesmeandrift.


Example

  • Step5:Calculatethecommonfactorportfoliovariance/varianceofresidualexposure.


Example

  • Step6:Calculatethespecificvarianceoftheportfolio.(assumethespecificvarianceforallofthestockstobe0.0016)


Example

  • Step7:CalculatetheSNRratiowithwhitenoiseassumptionsforresidualstockreturn.


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