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Fourier Spot Volatility Estimator: Asymptotic Normality and Efficiency with Liquid and Illiquid High-Frequency Data.

Mancino ME, Recchioni MC - PLoS ONE (2015)

Bottom Line: The recent availability of high frequency data has permitted more efficient ways of computing volatility.We address this issue by using the Fourier estimator of instantaneous volatility introduced in Malliavin and Mancino 2002.We prove a central limit theorem for this estimator with optimal rate and asymptotic variance.

View Article: PubMed Central - PubMed

Affiliation: Department of Economics and Management, University of Florence, Florence, Italy.

ABSTRACT
The recent availability of high frequency data has permitted more efficient ways of computing volatility. However, estimation of volatility from asset price observations is challenging because observed high frequency data are generally affected by noise-microstructure effects. We address this issue by using the Fourier estimator of instantaneous volatility introduced in Malliavin and Mancino 2002. We prove a central limit theorem for this estimator with optimal rate and asymptotic variance. An extensive simulation study shows the accuracy of the spot volatility estimates obtained using the Fourier estimator and its robustness even in the presence of different microstructure noise specifications. An empirical analysis on high frequency data (U.S. S&P500 and FIB 30 indices) illustrates how the Fourier spot volatility estimates can be successfully used to study intraday variations of volatility and to predict intraday Value at Risk.

No MeSH data available.


Related in: MedlinePlus

Comparison of cumulative density functions.Cumulative density functions of standard normal sample (red solid line), of the standardized returns obtained using the true volatility (green dotted line) and of the standardized returns obtained using the Fourier spot volatility estimates (blue dash-dot line) when Δ t = 10 seconds (left upper panel), Δ t = 30 seconds (right upper panel), Δ t = 1 minute (left lower panel) and Δ t = 3 minutes (right lower panel).
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pone.0139041.g004: Comparison of cumulative density functions.Cumulative density functions of standard normal sample (red solid line), of the standardized returns obtained using the true volatility (green dotted line) and of the standardized returns obtained using the Fourier spot volatility estimates (blue dash-dot line) when Δ t = 10 seconds (left upper panel), Δ t = 30 seconds (right upper panel), Δ t = 1 minute (left lower panel) and Δ t = 3 minutes (right lower panel).

Mentions: We further investigate the accuracy of the Fourier spot volatility estimates using the standardized returns defined by:zt=rtσtΔt,(14)where rt = pt+Δ t − pt is the log-return. These standardized returns are random variables normally distributed with zero mean and variance equal to one when the sample interval is sufficiently small. Specifically, we compute the standardized returns and obtained using the true and the Fourier spot volatilities, respectively, and we compare their cumulative density functions with the theoretical one, (0,1). Fig 4 shows the results of this comparison when Δ t = 10 seconds (Fig 4 left upper panel), Δ t = 30 seconds (Fig 4 right upper panel), Δ t = 1 minute (Fig 4 left lower panel) and Δ t = 3 minutes (Fig 4 right lower panel). The cumulative density functions are obtained elaborating the data of one realization of the log-price variable observed in one day. In addition, we measure the performance of the Fourier estimator by generating 504 replications of the standardized return and for various values of the sampling interval. Then we use the Kolmogorov-Smirnov (KS) and the Jarque-Bera (JB) test at the 5% significance level to determine whether the 504 random samples have the hypothesized standard normal cumulative density function.


Fourier Spot Volatility Estimator: Asymptotic Normality and Efficiency with Liquid and Illiquid High-Frequency Data.

Mancino ME, Recchioni MC - PLoS ONE (2015)

Comparison of cumulative density functions.Cumulative density functions of standard normal sample (red solid line), of the standardized returns obtained using the true volatility (green dotted line) and of the standardized returns obtained using the Fourier spot volatility estimates (blue dash-dot line) when Δ t = 10 seconds (left upper panel), Δ t = 30 seconds (right upper panel), Δ t = 1 minute (left lower panel) and Δ t = 3 minutes (right lower panel).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4589382&req=5

pone.0139041.g004: Comparison of cumulative density functions.Cumulative density functions of standard normal sample (red solid line), of the standardized returns obtained using the true volatility (green dotted line) and of the standardized returns obtained using the Fourier spot volatility estimates (blue dash-dot line) when Δ t = 10 seconds (left upper panel), Δ t = 30 seconds (right upper panel), Δ t = 1 minute (left lower panel) and Δ t = 3 minutes (right lower panel).
Mentions: We further investigate the accuracy of the Fourier spot volatility estimates using the standardized returns defined by:zt=rtσtΔt,(14)where rt = pt+Δ t − pt is the log-return. These standardized returns are random variables normally distributed with zero mean and variance equal to one when the sample interval is sufficiently small. Specifically, we compute the standardized returns and obtained using the true and the Fourier spot volatilities, respectively, and we compare their cumulative density functions with the theoretical one, (0,1). Fig 4 shows the results of this comparison when Δ t = 10 seconds (Fig 4 left upper panel), Δ t = 30 seconds (Fig 4 right upper panel), Δ t = 1 minute (Fig 4 left lower panel) and Δ t = 3 minutes (Fig 4 right lower panel). The cumulative density functions are obtained elaborating the data of one realization of the log-price variable observed in one day. In addition, we measure the performance of the Fourier estimator by generating 504 replications of the standardized return and for various values of the sampling interval. Then we use the Kolmogorov-Smirnov (KS) and the Jarque-Bera (JB) test at the 5% significance level to determine whether the 504 random samples have the hypothesized standard normal cumulative density function.

Bottom Line: The recent availability of high frequency data has permitted more efficient ways of computing volatility.We address this issue by using the Fourier estimator of instantaneous volatility introduced in Malliavin and Mancino 2002.We prove a central limit theorem for this estimator with optimal rate and asymptotic variance.

View Article: PubMed Central - PubMed

Affiliation: Department of Economics and Management, University of Florence, Florence, Italy.

ABSTRACT
The recent availability of high frequency data has permitted more efficient ways of computing volatility. However, estimation of volatility from asset price observations is challenging because observed high frequency data are generally affected by noise-microstructure effects. We address this issue by using the Fourier estimator of instantaneous volatility introduced in Malliavin and Mancino 2002. We prove a central limit theorem for this estimator with optimal rate and asymptotic variance. An extensive simulation study shows the accuracy of the spot volatility estimates obtained using the Fourier estimator and its robustness even in the presence of different microstructure noise specifications. An empirical analysis on high frequency data (U.S. S&P500 and FIB 30 indices) illustrates how the Fourier spot volatility estimates can be successfully used to study intraday variations of volatility and to predict intraday Value at Risk.

No MeSH data available.


Related in: MedlinePlus