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

2. Goals and Motivations 2 Most of the models of financial asset A quantity of high interest INTEGRATED VARIANCE It gives a measure of how the asset is risky: σ modulates the impact of W on the log- price model. Our GOAL is measuring the reliability of an estimator of IV in a realistic framework.

3. Given n observations of the log-price X t 3 In the absence of the Jumps: In the presence of the Jumps: Huge amount of literature aims to disentangle IV. The most efficient technique is the Threshold method. EFFICIENT

4. 4 A semimartingale (SM) model X does not fit data observed at an Ultra High Frequency (e.g 1 Sec.) Infact, the plot of RV as a function of h (SGINATURE PLOT), on empirical data, seems to explode when h tends to zero. Y t is the observed log-price X t is called efficient log-price (SM) ε is called the microstructure noise (measurement error) Classical Assumptions on the MICROSTRUCTURE NOISE: ε ti i.i.d centered and independent on X Var(ε t i ) < ∞, independent on h This new model reproduces the SIGNATURE PLOT of RV behaviour much better

5. 5 IF NO Jumps, NO noise: RV consistent and efficient for IV IF NO Jumps but in the PRESENCE noise:  For very small h, RV explodes  BIAS due to the presence of the noise.  If h is large  BIAS due to estimation error. RV(Y)

6. 6 PROBLEM Given: AN ASSET AN OBSERVATION FREQUENCY h Is the microstructure noise relevant? Can we rely on an estimator of IV designed in the absence of the noise?

7. 7 LITTERATURE SIGNATURE PLOT of RV (Hansen & Lunde, 2005): used to select an optimal h IV 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. We propose a new tool: a test based on Threshold IV

8. 8 Outline of the rest of the talk 1.Threshold estimation (Mancini, 2009) 2.Test for the relevance of the noise (Mancini, 2012) 3.Implementation of the test on simulated data: reliability check 4.Implementation of the test in empirical data 5.Conclusions

9. 9 Threshold estimator (Mancini, 2009) In the absence of noise Thershold estimator of IV (Mancini 2009)

10. 10 In order to keep out the contribution of jumps: Paul Lévy Law

11. In the presence of the microstructure noise: IDEA: ∆ i ε keeps large for all i when h tends to 0  ∆ i Y keeps large for all i  all increments exceed the threshold. Mancini, 2012 The key to establish whether the noise influences our measure of IV at fixed h, is to check if the Threshold estimator is significantly close to zero. Hyp. [True under classical assumptions]

12. 12 Let us assume: i.i.d centered, with finite variance and independent on X with law density g with, as Test for the relevance of the noise Mancini, 2012 We are able to build a test statistics for the relevance of the noise…

13. 13 IN PRACTICE: financial data are always affected by some microstructure noises Noise present Noise absent OUR USE OF THE TEST : If : We judge the noise negligible RELIABLE ESTIMATOR OF Estimation of …

14. 14 Estimation of Non-parametric: kernel-type estimatorkernel-type estimator K: triangular, Gaussian and uniform  best choice: Gaussian In practice: on the simulated models we have an estimation error ≈ 6% Parametric: RV method under assumption of Gaussian or uniform noise with varianceRV method Uniform:Gaussian: ISSUES : distorsion in finite sample due to jumps and choice of noise law requested

15. 15 Test implemented with the Threshold Minimum h = 1’’, while maximum h = 1 hour. n = 23400 observations in a day (252 days, 6.5 hours) Gaussian noise with two possible choices of its variance: Implementation of the test on simulated data Low level of noise: High level of noise

16. 16 1.MODEL Stochastic Volatility + Possion Jumps (SV + PJ) 2. MODEL Gauss + CGMY (G + CGMY) With: C = 280.11, G = 102.84, M = 102.53, Y = 0.1191, σ = 0.4 Parameters estimated for MSFT asset in CGMY, 2012 (Huang & Tauchen, 2005)

17. 17 Reliability check N=1000 paths of X. For each of them we compute S_h, then The Noise Variance is fixed and h assumes value of 1, 2, 5, 60, 120, 300 seconds (e.g h = 1  n = 23400; h = 300  n = 78)

18. 18

19. 19 Comparison beetwen the three criteria SV + PJ

20. 20 Comparison beetwen the three criteria G + CGMY (1)

21. 21 Comparison beetwen the three criteria G + CGMY (2) 21

22. 22 Comparison beetwen the three criteria Gaussian

23. 23 Implementation of the test in empirical data Observed Price of Microsoft (MSFT) Traded on NASDAQ A lot of daily transactions ( >> 23400) From 02–Jan–2001 to 31–Dec–2005 For each studied day: Plot of the prices Plot of log-returns Signature plot of RV our Test Estimation of the microstructure noise variance

24. 24 2 nd January 2001

25. 25 9 th January 2001

26. 26 12 th July 2002

27. 27 24th March 2004 (794 Millions USD Fee from EU)

28. 28 Gaussian model + Gaussian noise with variance

29. 29 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 Aït Sahalia criteria. We implemented the test on empirical data (MSFT asset). Some correlations between the increments of the log-price and of the noise should be allowed in the model for MSFT log-price BID/ASK analysis On - going

30. 30 THANK YOU!!

31. 31

32. 32 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 La stima peggiora con il diminuire della varianza di rumore di microstr. 0.4%6.1%91.7%

33. 33 RV: problema della componente dei jumps

34. Moto browniano 1. Dividere [0,T] in n intervalli di ampiezza δ 2. Simulare n variabili Gaussiane standard N 1 …N n 3. a=0 σ=4a=0 σ=0.4 34

35. 35 Varianza stocastica 1.Dividere [0,T] in n intervalli di ampiezza δ 2.Simulare n variabili Gaussiane standard W 1 …W n 3. ρ=-0.7a=0 σ stocastica 

36. 36 Compound Poisson 1.Dividere [0,T] in n intervalli di ampiezza δ 2.Simulare n variabili di Poisson P i.. P n indipendenti e con parametro δλ 3.Sommare variabili indipendenti 4. λ = 5 ν = 0.6λ = 500 ν = 0.6 