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Bootstrap in Finance

Bootstrap in Finance. Esther Ruiz and Maria Rosa Nieto (A. Rodríguez, J. Romo and L. Pascual) Department of Statistics UNIVERSIDAD CARLOS III DE MADRID Workshop Modelling and Numerical Techniques in Quantitative Finance A Coruña 15 de octubre de 2009. Motivation

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Bootstrap in Finance

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  1. Bootstrap in Finance Esther Ruiz and Maria Rosa Nieto (A. Rodríguez, J. Romo and L. Pascual) Department of Statistics UNIVERSIDAD CARLOS III DE MADRID WorkshopModelling and NumericalTechniques in QuantitativeFinance A Coruña 15 de octubre de 2009

  2. Motivation • Bootstrappingtoobtainpredictiondensities of futurereturns and volatilities • GARCH • Stochasticvolatility • Bootstrappingtomeasurerisk: VaR and ExpectedShortfall • Conclusions

  3. 1. Motivation • High frequency time series of returns are characterized by volatility clustering: • Excess kurtosis • Significant autocorrelations of absolute returns (not independent)

  4. Financialmodelsbasedoninferenceonthedynamicbehaviour of returns and/orpredictions of momentsassociatedwiththedensity of futurereturnswhichassumeindependent and/orGaussianobservations are inadequate. • Bootstrapmethos are attractive in thiscontextbecausethey do notassumeany particular distribution.

  5. However, bootstrapprocedurescannotbebasedonresamplingdirectlyfromobservedreturnsas they are notindepedent. • Assume a parametricspecificationof thedependence and bootstrapfromthecorrespondingresiduals. • Generalizethebootstrapproceduresto cope withdependentobservations • Li and maddala (1996) and Berkowitz and Kilian (2000) show thattheparametricapproachcouldbepreferable in manyapplications.

  6. Consequently, wefocusontwo of themost popular and simplermodelstorepresentthedynamicproperties of financialreturns: GARCH(1,1) ARSV(1)

  7. There are twomainrelatedareas of application of bootstrapmethods • Inference: Obtainingthesampledistribution of a particular estimatoror a statisticfortesting, forexample, theautoregressivedynamics in theconditional mean and variance, unitroots in the mean, fractionalintegration in volatility, inferencefor trading rules: Ruiz and Pascual (2002, JES) • Prediction: Prediction of futuredensities of returns and volatilities: VaRand ES

  8. Bootstrapmethodsallowtoobtainpredictiondensities (intervals) of futurereturnswithoutdistributionalassumptionsontheprediction error distribution and incorporatingtheparameteruncertainty • Thombs and Schucany (1990, JASA): backwardrepresentation • Cao et al. (1997, CSSC): conditionalonestimatedparameters • Pascual et al. (2004, JTSA): incorporateparameterundertaintywithoutbackwardrepresentation

  9. Example AR(1) TheMinimum MSE predictor of yT+kbasedontheinformationavailable at time T isgivenbyitsconditional mean where In practice, theparameters are substitutedbyconsistentestimates. Therefore, thepredictions are givenby Predictions are made conditional on the available data

  10. Thombs and Schucany (1990) where are bootstrapreplicates of thestandardizedresiduals and are obtainedfrombootstrapreplicates of the series basedonthebackwardrepresentation where ShouldwefixyTwhenbootstrappingtheparameters??? NO

  11. Pascual et al. (2004, JTSA) propose a bootstrapproceduretoobtainpredictionintervals in ARIMA modelsthat do not requiere thebackwardrepresentation. Therefore, thisprocedureissimpler and more general, as it can cope withmodelsforwhichthebackwardrepresentationdoesnotexist as, forexample, GARCH.

  12. 2. Bootstrapforecast of futurereturns and volatilities Weconsidertheprediction of futurereturns and volatilitiesgeneratedby GARCH and ARSV(1) models 2.1 GARCH 2.2 ARSV Bothmodelsprovidepredictionintervalswhich are narrow in quite times and wide in volatileperiods.

  13. 2.1 GARCH (1,1): Pascual et al. (2005, CSDA) Consideragainthe GARCH(1,1) modelgivenby Therefore, AssumingconditionalNormalityof returns: · one-stepaheadpredictionerrors are Normal. · predictionerrorsfortwoor more stepsahead are not Normal. · one-step-aheadvolatilitiesonlyhaveassociatedparameteruncertainty. · volatilities more thanone-stepaheadalsohaveuncertaintyaboutfutureerrors.

  14. Bootstrapprocedure • Estimateparameters and obtainstandardizedresiduals • Obtainbootstrapreplicates of the series and estimatetheparameters. • Bootstrapforecasts of futurereturns and volatilities Usingthebootstrapestimates of theparameterswiththe original observations (conditional)

  15. 2.2 ARSV(1) models The ARSV(1) model can belinearizedbytakinglogs of squares Bootstrapmethodsforunobservedcomponentmodels are muchlessdeveloped. Previousprocedurescannotbeimplementedduetothepresence of severaldisturbances. In thiscontext, theinterestisnotonlytoconstructdensities of futurevalues of theobserved variables butalso of theunobservedcomponents.

  16. TheKalmanfilterprovides • one-step-ahead (updated and smoothed) predictions of the series togetherwiththeir MSE • estimates of thelatentcomponents and their MSE • Bootstrapprocedures can beimplementedtoobtaindensities of • Estimates of theparameters • Predictiondensities of futureobservations: Wall and Stoffer (2002, JTSA), Rodríguez and Ruiz (2009, JTSA) • Predictiondensitiesof underlyingunobservedcomponents:Pferfferman and Tiller (2005, JTSA), Rodríguez and Ruiz (2009, manuscript)

  17. Rodríguez and Ruiz (2009, JTSA) propose a bootstrapproceduretoobtainpredictionintervals of futureobservations in unobservedcomponentmodelsthatincorporatetheparameteruncertaintywithoutusingthebackwardrepresentation.

  18. The proposed procedure consists on the following steps: • Estimate the parameters by QML, and obtain the standardized innovations, • Obtain a sequence of bootstrap replicates of the standardized innovations, • Obtain a bootstrap replicate of the series using the IF with the estimated parameters Estimate the parameters, obtaining and

  19. 4) Obtain the conditional bootstrap predictions

  20. However, as wementionedbefore, whenmodellingvolatility, theobjectiveisnotonlytopredictthedensity of futurereturnsbutalsotopredictfuturevolatilities. Therefore, weneedtoobtainpredictionintervalsfortheunobservedcomponents. • At themoment, Rodríguez and Ruiz (2009, manuscript) propose a proceduretoobtainthe MSE of theunobservedcomponents.

  21. Consider, for example, the random walk plus noise model: In this case, the prediction intervals are given by Estimated parameters Normality assumption

  22. Random walk with q=0.5 Estimates of the level and 95% confidence intervals: In red with estimated parameters and in black with known parameters.

  23. Our procedure is based on the following decomposition of the MSE proposed by Hamilton (1986) • He proposes to generate replicates of the parameters from the asymptotic distribution and then to estimate the MSE by The filter is run with original observations

  24. Wepropose a non-parametricbootstrap in whichthebootstrapreplicates of the series are obtainedfromtheinnovationformafterresamplingfromtheinnovations.

  25. 3. Var and ES In thecontext of financialriskmanagement, one of the central issues of densityforecastingistotrack certainaspects of thedensities as, forexample, VaR and ES. Considerthe GARCH(1,1) model In thiscontexttheVaR and ES are givenby

  26. In practice, assumingthatthemodelisknown, boththeparameters and thedistribution of theerrors are unknown. Therefore, weobtaintheestimates Bootstrapprocedureshavebeenproposedtoobtainpointestimates of theVaRbycomputingthecorrespondingquantileof thebootstrapdistribution of returns (Ruiz and Pascual, 2004, JES)

  27. Bootstrapprocedures can alsobeimplementedtoobtainestimates of VaR and ES togetherwiththeir MSE. Christoffersen and GonÇalves (2005, J Risk) proposeto compute bootstrapreplicates of theVaRby where

  28. Instead of usingtheresidualsobtained in each of thebootstrapreplicates of the original series, Nieto and Ruiz (2009, manuscript) proposetoestimate q0.01by a secondbootstrapstep. Foreachbootstrapreplicate of the series of returns, weobtain n randomdrawsfromtheempiricaldistribution of the original standardizedresiduals, Then, theconstant q0.01 can beestimatedbyany of thethreealternativeestimatorsdescribedbefore. In thisway, weavoidtheestimation error involved in theresiduals

  29. Monte Carlo experiments: Coverages

  30. Conclusions and furtherresearch • Fewanalyticalresultsonthestatisticalproperties of bootstrapprocedureswhenappliedtoheterocedastic time series • Furtherimprovements in: • bootstrapestimation of quantiles and expectationsto compute theVaR and ES • construction of predictionintervalsforunobservedcomponents (stochasticvolatility) • Multivariateextensions: Engsted and Tanggaard (2001, JEF)

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