### Non-parametric Daily Tests

The non-parametric daily jump tests developed with the methodology of Barndorff-Nielson and Shephard (2006) that describes the asymptotic behavior of the Bipower Variation, as a measure for the daily integrated volatility, which is robust to jumps. In case a significant difference between the Realized Variance and this measure is determined with a certain level of confidence, then we can conclude that at least one significant change in price was realized in the respective day.

The test can provide an answer regarding the existence of sudden price changes in a respective time frame, once all the prices are realized.

### Non-parametric Intradaily Tests – Lee-Mykland and Lee-Hannig Tests

The importance of sudden changes in the dynamics of stock prices was acknowledged in the framework of continuous-time jump-diffusion modeling by Merton (1976) and their identification was always considered as an important econometric issue that necessitated sophisticated numerical estimation techniques, highly computationally intensive. The field developed in the direction of parametric estimation of jumps with, among others, the work of Jorion (1988), Maheu and McCurdy (2003), Andersen, Benzoni and Lund (2002), Bates (2000) and Chernov, Gallant, Gysels and Tauchen (2003) on a stream of research aiming at calibrating processes that allow for time-varying intensities of jumps with different model specifications that allow for diffusions with stochastic volatilities and jumps. This stream of research allowed for the semi-martingale setting to become the standard framework for the modeling of stock returns. However, the parametric method of jump identification requires a heavy modeling aparatus with sophisticated numerical methods for its estimation, which makes them less useful especially at the intra-day level.

The new stream of non-parametric methods has gained momentum. The work of Barndorff-Nielsen and Shephard (2006) was essential for a new stage in the jump-detection process. Their main contribution to the literature consists in the use of Bipower Variation as a nonparametric (model-free) robust-to-jumps volatility estimator. Their research provided one of the mostly used framework for detection of daily jumps, which relies on the fact that the difference between the Realized Variance (as a measure of Integrated Volatility for a trading day) and the Bipower Variation (a measure for the same statistic – the Integrated Volatility – that is robust to jumps) is a variable that has stable distribution asymptotically and allows for the identification of jumps in case of significance.

The seminal paper of Lee and Mykland (2008) opened the door to nonparametric identification of jumps at the intraday frequency based on the local estimation of volatility, while the series of academic work cosigned by Ait-Sahalia developed a framework for the decomposure of continuous dynamics, small jumps and large jumps, from which we mostly acknowledge the Ait-Sahalia and Jacod (2009) that rooted, among others, the important work of the computation of Truncated Variation developed by Lee and Hannig (2010) for the differentiation between big jumps and small jumps at the high frequency level, and culminated with Ait-Salahia and Jacod (2012) that represents an important breakthrough in the setting of a background for jump identification tests across power variations, truncation rates and frequencies.

We provide here an example of the use of the Lee-Mykland test and the Lee-Hannig test on the EURUSD for the 5-minute log-returns on a time series starting on July 20th 2013. The specification of the Lee-Mykland test is the following:

where is the jump test, t is the time-frame used for the computation of our analysis, i.e. the time sample (usually it has the size of a day), while i counts the moments in this time-frame (for ex. 288 5-minute time samples for a continuously traded asset). The is the standard deviation computed for this time sample. In the case of the Lee-Mykland test this standard deviation is actually replaced by the estimated standard deviation:

where M is the number of observations i in the sample t and BVt is the Bipower Variation computed for the M returns in the sample t. As opposed to this framework, the Lee-Hannig test is using the Truncated Variation instead of the Bipower Variation.

where g>0 and are used for the computation of the thresholds needed to eliminate the large returns from the series used in the computation of the volatility.

The following graphs show the difference in the two tests. We can notice that the Lee-Mykland test provides more jumps than the Lee-Hannig.

One of the drawbacks of intra-day measures is the fact that the high-frequency returns are known to be influenced by many factors that generate periodic dynamics, i.e. have time-dependent structure. In search for a suited measure of volatility at the intra-day frequency, Andersen and Bollerslev (1997) show that the dynamics of returns and volatilities for different financial assets present strong periodicity, which is likely to mislead the statistical analyses aiming at characterizing the intra-day variation of financial assets’ returns. This conjecture generated the adjustment of the jump tests using various both parametric and non-parametric measures of periodicity introduced by Boudt, Croux and Laurent (2011), as Median Absolute Deviation, the Shortest Half scale estimator on one hand, and the Truncated Maximum Likelihood estimator on the other hand.

References:

*Ait-Sahalia, Y., Jacod, J., (2009) Estimating the Degree of Activity of Jumps in High Frequency Data, The Annals of Statistics*
*Lee, S. S., Hannig, J., (2010) Detecting Jumps from Levi Jump Diffusion Processes, Journal of Financial Economics;*

*Ait-Sahalia, Y., Jacod, J., (2012) Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data, Journal of Economic Literature;*

*Andresen, T. G., Bollerslev, T., (1997) Intraday Periodicity and Volatility Persistence in Financial Markets, Journal of Empirical Finance;*

*Andersen, T.G., Benzoni, L., Lund, J., (2002) An Empirical Investigation of Continuous-Time Equity Return Models, The Journal of Finance;*

*Barndorff-Nielsen, O. E., Shephard, N., (2006) Econometrics of Testing for Jumps in Financial Economics using Bipower Variation, Journal of Financial Econometrics;*

*Bates, D.S., (2000), Post-87′ Crash Fears in the S&P 500 Futures Option Market, Journal of Econometrics;*

*Boudt, K., Croux, C., Laurent, S., (2011) Robust Estimation of Intraweek Periodicity in Volatility and Jump Detection, Journal of Empirical Finance;*

*Chernov, M., Gallant, A.R., Gysels, E., Tauchen, G., (2003) Alternative Models for Stock Price Dynamics, Journal of Econometrics;*

*Jorion, P., (1988) On Jump Processes in the Foreign Exchange and Stock Markets, The Review of Financial Studies;*

*Cont, R., (2001) Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues, Quantitative Finance;*

*Lee, S. S., Mykland, P. A., (2008) Jumps in Financial Markets: A New Nonparametric Test and Jump Dynamics, The Review of Financial Studies;*

*Maheu, J.M., McCurdy, T.H., (2004) News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns, The Journal of Finance;*

*Merton, R. C., (1976) Option Pricing when Underlying Returns are Discontinuous, Journal of Financial Economics.*