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Pairs Trading Ch6

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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  1. Pairs TradingCh6 PairsSelectioninEquityMarkets 郭孟鑫

  2. StationaryandNonstationary • Stationary Forallt-s,andt-s-j

  3. StationaryandNonstationary • Spuriousregression. • Downwardbias • Usingunitroottesttoverify.

  4. Introduction Inchapter5,weneedthreesteps • Identificationofthestockpairs • Cointegrationtesting • Tradingruleformulation Thischapterwillfocusoncointegrationtestingandfurtheranalysis.

  5. Introduction • WedrawparallelsbetweenthecommontrendsmodelforcointegrationandtheideaofAPT.

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

  7. CommonTrendsCointegrationModel

  8. CommonTrendsCointegrationModel • Inference2: Thecointegrationcoefficientmaybeobtainedbyaregressionoftheinnovationsequencesofthecommontrendsagainsteachother.

  9. Discussion • First,theinnovationsequencesderivedfromthecommontrendsofthetwoseriesmustbeidenticaluptoascalar. • Next,thespecificcomponentsofthetwoseriesmustbestationary.

  10. CommonTrendsmodelandAPT Isthereturnduetothenonstationarytrendcomponent. Isthereturnduetothestationarycomponent.

  11. CommonTrendsmodelandAPT • RecallingtheAPT,thestockreturnsmaybeseparatedintocommonfactorreturnsandspecificreturns. • Ifthetwostockssharethesameriskfactorexposureprofile,thenthecommonfactorreturnsforboththestocksmustbethesame.

  12. 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:

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

  14. CommonTrendsmodelandAPT • Condition 2: Consider the linear combination of the returns. Where, If A and B are cointegrated, then the common factor return becomes zero.

  15. 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.

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

  17. Interpreting The Distance Measure • Calculating the Cosine of the Angle between Two Vectors.

  18. Interpreting The Distance Measure • Appendix:EigenvalueDecomposition Let Disadiagonalmatrixwiththeeigenvalues Uisamatrixwitheachcolumncorrespondingto aneigenvectors.

  19. Interpreting The Distance Measure • Geometric interpretation • Key to doing the geometric interpretation is the idea of eigenvalue decomposition.

  20. Interpreting The Distance Measure

  21. Reconciling Theory And Practice • Deviations from Ideal Conditions • When two stocks are not have their factor exposures perfectly aligned. Signal-to-Noise(weneedbigSNR)

  22. Example • ConsiderthreestocksA,B,andCwithfactorexposuresinatwofactorasfollows:

  23. Example • Step1:Calculatethecommonfactorvarianceandcovariance.

  24. Example • Step2:Calculatethecorrelation(absolutevalueofcorrelationisthedistancemeasure) • Ifwehavetochooseonepairforpurposetrading,ourchoicewouldthereforebethepair(A,B)

  25. Example • Step3:Calculatethecointegrationcoefficient • Wewilldiscussonthenextchapter

  26. Example • Step4:Calculatetheresidualcommonfactorexposeinpairedportfolio.Thisistheexposurethatcausesmeandrift.

  27. Example • Step5:Calculatethecommonfactorportfoliovariance/varianceofresidualexposure.

  28. Example • Step6:Calculatethespecificvarianceoftheportfolio.(assumethespecificvarianceforallofthestockstobe0.0016)

  29. Example • Step7:CalculatetheSNRratiowithwhitenoiseassumptionsforresidualstockreturn.

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