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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:

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

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.

Reconciling Theory And Practice

- Deviations from Ideal Conditions
- When two stocks are not have their factor exposures perfectly aligned.
Signal-to-Noise(weneedbigSNR)

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

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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