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A new tool for optimal frequency selection to estimate Integrated
Variance
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Florence, March 12-13, 2013
Giulio Lorenzini, University of Florence. [email protected]
Goals and Motivations
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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.
Given n observations of the log-price Xt
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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
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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.
Yt is the observed log-priceXt is called efficient log-price (SM)ε is called the microstructure noise (measurement error)
Classical Assumptions on the MICROSTRUCTURE NOISE:• εti i.i.d centered and independent on X• Var(εti) < ∞, independent on h
This new model reproduces the SIGNATURE PLOT of RV behaviour much better
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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)
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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?
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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
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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
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Threshold estimator (Mancini, 2009)
In the absence of noise
Thershold estimator of IV (Mancini 2009)
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In order to keep out the contribution of jumps: Paul Lévy Law
In the presence of the microstructure noise:
IDEA:
∆iε keeps large for all i when h tends to 0 ∆iY 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]
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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 noiseMancini, 2012
We are able to build a test statistics for the relevance of the noise…
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IN PRACTICE: financial data are always affected by some microstructurenoises
Noise present
Noise absent
OUR USE OF THE TEST:
If : We judge the noise negligible
RELIABLE ESTIMATOR OF
Estimation of …
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Estimation of• Non-parametric: kernel-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 variance
Uniform: Gaussian:
ISSUES: distorsion in finite sample due to jumps and choice of noise law requested
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• 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
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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)
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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)
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Comparison beetwen the three criteriaSV + PJ
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Comparison beetwen the three criteriaG + CGMY (1)
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Comparison beetwen the three criteriaG + CGMY (2)
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Comparison beetwen the three criteriaGaussian
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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
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2nd January 2001
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9th January 2001
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12th July 2002
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24th March 2004 (794 Millions USD Fee from EU)
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Gaussian model + Gaussian noise with variance
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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
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THANK YOU!!
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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%
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• RV: problema della componente dei jumps
Moto browniano
1. Dividere [0,T] in n intervalli di ampiezza δ
2. Simulare n variabili Gaussiane standard N1…Nn
3.
a=0 σ=4 a=0 σ=0.4
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Varianza stocastica
1. Dividere [0,T] in n intervalli di ampiezza δ
2. Simulare n variabili Gaussiane standard W1…Wn
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ρ=-0.7a=0 σ stocastica
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Compound Poisson
1. Dividere [0,T] in n intervalli di ampiezza δ
2. Simulare n variabili di Poisson Pi .. Pn indipendenti e con parametro δλ
3. Sommare variabili indipendenti
4.
λ = 5 ν = 0.6 λ = 500 ν = 0.6