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A new tool for optimal frequency selection to estimate Integrated Variance

A new tool for optimal frequency selection to estimate Integrated Variance. Giulio Lorenzini , University of Florence . lorenzini_giulio@hotmail.com. Florence, March 12-13, 2013. Goals and Motivations. Most of the models of financial asset. A quantity of high interest.

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A new tool for optimal frequency selection to estimate Integrated Variance

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  1. A new tool for optimal frequency selection to estimate Integrated Variance Giulio Lorenzini, UniversityofFlorence. lorenzini_giulio@hotmail.com Florence, March 12-13, 2013

  2. Goals and Motivations Mostof the modelsoffinancialasset A quantityof high interest INTEGRATED VARIANCE Itgives a measureofhow the assetisrisky: σmodulates the impact of W on the log-price model. Our GOAL ismeasuring the reliabilityofanestimatorof IV in a realisticframework.

  3. Given n observationsof the log-price Xt In the absenceof the Jumps: EFFICIENT In the presenceof the Jumps: Huge amount of literature aims to disentangle IV. Themost efficient technique is the Threshold method.

  4. A semimartingale(SM) model X does not fit data observed at an Ultra High Frequency (e.g 1 Sec.) Infact, the plot ofRVas a functionof h (SGINATURE PLOT), on empirical data, seemstoexplodewhen h tendsto zero. • Ytis the observed log-price • Xtiscalledefficient log-price (SM) • εiscalled the microstructurenoise (measurementerror) • ClassicalAssumptionson the MICROSTRUCTURE NOISE: • εtii.i.dcentered and independent on X • Var(εti) < ∞, independent on h Thisnewmodelreproduces the SIGNATURE PLOT of RV behaviourmuchbetter

  5. RV(Y) IF NO Jumps, NO noise: RV consistent and efficientfor IV IF NO Jumpsbut in the PRESENCE noise: • Forverysmall h, RV explodesBIAS due to the presenceof the noise. • If h islargeBIAS due toestimationerror.

  6. PROBLEM • Given: • AN ASSET • AN OBSERVATION FREQUENCY h Is the microstructurenoiserelevant? Can werely on anestimatorof IV designed in the absenceof the noise?

  7. LITTERATURE • SIGNATURE PLOT of RV (Hansen & Lunde, 2005): used to select an optimal hIV estimation is possible only if no jumps. • Optimal h choice by minimizing mean square IV estimation error (Zhang, Mykland & Aït-Sahalia, 2005, Bandi & Russel, 2008)X has no jumps; h is optimal on average along many observed path. Wepropose a newtool: a test based on Threshold IV

  8. Outline of the rest of the talk Thresholdestimation (Mancini, 2009) Test for the relevanceof the noise (Mancini, 2012) Implementationof the test on simulated data: reliabilitycheck Implementationof the test in empirical data Conclusions

  9. Thresholdestimator(Mancini, 2009) In the absenceofnoise ThersholdestimatorofIV (Mancini 2009)

  10. In ordertokeep out the contributionofjumps: Paul LévyLaw

  11. In the presenceof the microstructurenoise: Hyp. [True under classicalassumptions] Mancini, 2012 IDEA: ∆iεkeepslargeforall i when h tendsto 0  ∆iYkeepslargeforalli allincrementsexceed the threshold. The key toestablishwhether the noiseinfluencesourmeasureof IV at fixed h, istocheckif the Thresholdestimatorissignificantlycloseto zero.

  12. Test for the relevanceof the noise Mancini, 2012 Letus assume: • i.i.dcentered, with finite variance and independent on X • withlaw density g • with , as We are abletobuild a test statisticsfor the relevanceof the noise… 12

  13. Noisepresent Noiseabsent IN PRACTICE: financial data are always affected by some microstructure noises OUR USE OF THE TEST: If : Wejudge the noisenegligible RELIABLE ESTIMATOR OF Estimationof …

  14. Estimationof • Non-parametric: kernel-typeestimator K: triangular, Gaussian and uniform best choice: Gaussian In practice: on the simulatedmodelswehaveanestimationerror ≈ 6% • Parametric:RV methodunder assumptionofGaussian or uniformnoisewithvariance Uniform: Gaussian: ISSUES: distorsion in finite sample due tojumps and choiceofnoiselawrequested

  15. Implementationof the test on simulated data • Test implementedwith the Threshold • Minimum h = 1’’, whilemaximumh = 1 hour. • n = 23400 observations in a day (252 days, 6.5 hours) • Gaussiannoisewithtwopossiblechoicesofitsvariance: Low levelofnoise: High levelofnoise

  16. 1. MODELStochasticVolatility + PossionJumps (SV + PJ) (Huang & Tauchen, 2005) 2. MODELGauss + CGMY (G + CGMY) With: C = 280.11, G = 102.84, M = 102.53, Y = 0.1191, σ = 0.4 Parametersestimatedfor MSFT asset in CGMY, 2012

  17. Reliabilitycheck N=1000 pathsof X. ForeachofthemwecomputeS_h, then The NoiseVarianceisfixed and hassumesvalueof 1, 2, 5, 60, 120, 300 seconds (e.g h = 1  n = 23400; h = 300  n = 78)

  18. Comparisonbeetwen the threecriteria • SV + PJ

  19. Comparisonbeetwen the threecriteria • G + CGMY (1)

  20. Comparisonbeetwen the threecriteria • G + CGMY (2) 21

  21. Comparisonbeetwen the threecriteria • Gaussian

  22. Implementationof the test in empirical data • Observed Price ofMicrosoft (MSFT) • Traded on NASDAQ • A lotofdailytransactions( >> 23400) • From 02–Jan–2001 to 31–Dec–2005 • Foreachstudiedday: • Plot of the prices • Plot oflog-returns • Signatureplot of RV • our Test • Estimationof the microstructurenoisevariance

  23. 2ndJanuary2001

  24. 9thJanuary2001

  25. 12thJuly 2002

  26. 24th March 2004 (794 Millions USD Feefrom EU)

  27. Gaussianmodel + Gaussiannoisewithvariance

  28. Conclusions • We found the optimal sampling frequency through a new statistic test based on the threshold estimator behavior when the observation frequency tends to zero . • We used this sampling frequency for the estimation of the integrated variance. • We compared the outcome of our study with the results obtained using Bandi & Russell and AïtSahalia criteria. • Weimplemented the test on empirical data (MSFT asset). On - going • Some correlations between the increments of the log-price and of the noise should be allowed in the model for MSFT log-price • BID/ASKanalysis

  29. THANK YOU!!

  30. Stima della densità del rumore di microstruttura Dalla proprietà i.i.d del rumore di microstruttura, la relazione tra la densità di e la densità di : • Kernel: escludo i punti sotto la soglia e medio tra 3 punti consecutivi 0.4% 6.1% 91.7% • La stima peggiora con il diminuire della varianza di rumore di microstr.

  31. RV: problema della componente dei jumps

  32. Moto browniano Dividere [0,T] in n intervalli di ampiezza δ Simulare n variabili Gaussiane standard N1…Nn a=0 σ=4 a=0 σ=0.4

  33. Varianza stocastica  Dividere [0,T] in n intervalli di ampiezza δ Simulare n variabili Gaussiane standard W1…Wn a=0 σ stocastica ρ=-0.7

  34. Compound Poisson Dividere [0,T] in n intervalli di ampiezza δ Simulare n variabili di Poisson Pi .. Pn indipendenti e con parametro δλ Sommare variabili indipendenti λ = 5 ν = 0.6 λ = 500 ν = 0.6 

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