ModelDependent Price Jump Indicators
1. The Difference Between Bipower Variance and Standard Deviation
The method is based on two distinct measures of overall volatility, where the first one takes into account the entire price time movement while the second one ignores the contribution of the modeldependent price jump component. BarndorffNielsen and Shephard (2004a) discuss the role of the standard variance in the models where the underlying process follows Eq. (1): the standard variance captures the contribution from both the noise and the price jump process unlike the realized variance, which does not take into account the term with price jumps. It is called the realized bipower variance. The difference between the standard and the bipower variance can be used to define indicators that assess the jumpiness of the market. Generally, there are two ways to employ bipower variance: the differential approach and the integral approach.
ot2 = t T 1 
t1 T1 r.T r,i)2 
1.1 The Differential Approach. The standard variance is defined as
The bipower variance is defined according to BarndorffNielsen and Shephard (2004b) as


The higher the ratio Of /Ot, the more jumps are contained in the past T time steps back.
1.2 The Integral Approach. The integral approach is motivated by the work of Pirino (2009). The integral approach employs the two cumulative estimators for the total volatility over a given period. The first one is the cumulative realized volatility estimator defined as
RVday = (rz)2 (6)
day
The second estimator is the bipower cumulative volatility estimator defined in an analogous way:
Ж
BPVday = 2 I rz I I rz— 1  (7)
day
Analogously the ratio of the two cumulative estimators RVday/BPVday can be considered as a measure of the relative contribution of price jumps to the overall volatility over the particular period.
2. Bipower Test Statistics. The bipower variance can be used to define the proper statistics for the identification of price jumps one by one. This means testing every time step for the presence of a price jump as defined in Eq. (1). These statistics were developed by Andersen et al. (2007) and Lee and Mykland (2008) and are defined as





Following Lee and Mykland, the variable £ is defined as





n
Where An is the tested region with n observations and the parameters are defined as
ln ж + ln (ln n) 1
2c(2 ln n)2


The variable £ has in the presence of no price jumps the cumulative distribution function P(£ < x) = exp(ex). The knowledge of the underlying distribution can be used to determine the critical value £cv at a given significance level. Whenever £ is higher than the critical value £cv, the hypothesis of no price jump is rejected, and such a price movement is identified as a price jump. In contrast, when £ is below the critical value, we cannot reject the null hypothesis of no price jump. Such a price movement is then treated as a noisy price movement. These statistics can be used to construct a counting operator for the number of price jumps in a given sample. However, the main disadvantage of bipower variationbased methods lie in the sensitivity of the intraday volatility patterns, which leads to a high rate of jump misidentification.
Jiang and Oomen (2008) modified this approach. They suggest to calculate Swap Variance as:
n
SwV = 2J2 Ri – r
i =2
where
Pi – Pii
Pi
Pi = exp Oi)
ri = Pi – Pi 1
Jiang and Oomen claim that employing swap variance further amplifies the contribution coming from price jumps and thus makes the estimator less sensitive to intraday variation. The JiangOomen statistics is defined as
ю _ nBV f1 RV ^
= PoSwV SwV
JO is asymptotically equal to z ~ N (0,1) and tests the null hypothesis that a given window does not contain any price jump. The indicator for a price jump is defined as those price movements for which JOt — 1 < Ф — 1(a) and JOt > Ф — 1(a). The authors claim that their test is better than the one based on bipower variation since it amplifies the discontinuities to a larger extent, as they show with a comparative analysis using Monte Carlo simulation. The amplification of discontinuities tends to suppress the effects of intraday volatility patterns.
To define extreme events on tick scale Nanex methodology can be applied. This methodology defines down (up) shock if stock had to tick down (up) at least 10 times before ticking up (down)—all within 2 s and the price change had to exceed 0.8 %. Tick means a price change caused by trade(s).
Hanousek (2011) performed an extensive simulation study to compare the relative performance of many pricejump indicators with respect to false positive and false negative probabilities. The results suggest large differences in terms of performance among the indicators: in the case of false positive probability, the bestperforming pricejump indicator is based on thresholding with respect to centiles, in the case of false negative probability, the best indicator is based on bipower variation. The differences in indicators is very often significant at the highest significance level, which further supports the initial suspicion that the results obtained using different pricejump indicators are not comparable.
Another problem specific for any statistical filter is spurious detection. The problem is that performing the tests for many days simultaneously results in conducting multiple testing, which by nature leads to making a proportion of spurious detections equal to the significance level of the individual tests (Bajgrowicz and Scaillet 2011). Bajgrowicz and Scaillet (2010) treat the problem of the spurious identification of price jumps by adaptive thresholds in the testing statistics. The problem with most of the pricejump indicators lies in what model they are built upon and there is the need for robustness of each filter when dealing with price jumps. Bajgrowicz and Scaillet (2011) developed a method to eliminate spurious detections that can be applied very easily on top of most existing jump detection tests, a Monte Carlo study shows that this technique behaves well infinite sample. Applying this method on highfrequency data for the 30 Dow Jones stocks over the 3year period between 2006 and 2008, authors found that up to 50 % of days selected initially as containing a jump were spurious detections. Abramovich et al. (2006) introduce the data adaptive thresholding scheme based on the control of the false discovery rate (FDR). FDR control is a recent innovation in simultaneous testing, which ensures that at most a certain expected fraction of the rejected null hypothesis correspond to spurious detections. Bajgrowicz and Scaillet (2010) use the FDR to account for data snooping while selecting technical trading rules. The choice of which threshold to use: universal or FDR, depends on the application. If the main purpose of research is the probability of a jump conditional on a news release, the FDR threshold is more appropriate as it reduces the likelihood of missing true jumps. If the goal is to study what kind of news cause jumps, it is better to apply the universal threshold in order to avoid looking vainly for a news when in fact the detection is spurious.
Documenting the presence of cojumps and understanding their economic determinants and dynamics are crucial for a risk measurement and management perspective.
Bajgrowicz and Scaillet (2011) did not detect cojumps affecting all stocks simultaneously for the sample including highfrequency data for the 30 Dow Jones stocks over the 3year period between 2006 and 2008, which supports the assumption in Merton (1976) that jump risk is diversiflable and thus does not require a risk premium.
Other empirical studies of cojumps include Bollerslev et al. (2008) who examine the relationship between jumps in a sample of 40 largecap U. S. stocks and the corresponding aggregate market index. To more effectively detect cojumps authors developed a new cross product statistic, termed the cpstatistic, that directly uses the crosscovariation structure of the highfrequency returns and examines cross comovements among the individual stocks. Employing this statistic allows to detect many modestsized cojumps. Cross product statistic defined by the normalized sum of the individual highfrequency returns for each withinday period:
where j = 1, …, M, M—total number of observations in a day, n—number of stocks. The cpstatistic provides a direct measure of how closely the stocks move together.
Lahaye et al. (2009) investigated cojumps between stock index futures, bond futures, and exchange rates in the relation with news announcements, they found that exchange rates experience frequent but relatively small jumps because they are subject to news from two countries and because they probably experience more idiosyncratic liquidity shocks during slow trading in the 24h markets. Forex jumps tend to be smaller than bond or equity jumps because national macro shocks produce much smaller changes in expected relative fundamentals between currencies. Equity and bond market cojumps are much more strongly associated with news releases than foreign exchange cojumps. But also authors admit that most of news does not cause jumps. A generic announcement only produces an exchange rate jump about 12 % of the time and a bond or equity jump only about 34 % of the time.
By investigating a set of 20 high cap stocks traded at the Italian Stock Exchange, Bormetti et al. (2013) found that there is a large number of multiple cojumps, i. e. minutes in which a sizable number of stocks displays a discontinuity of the price process. As mentioned above, they show that the dynamics of these jumps is not described neither by a multivariate Poisson nor by a multivariate Hawkes model, which are unable to capture simultaneously the time clustering of jumps and the high synchronization of jumps across assets. Authors introduce a one factor model approach where both the factor and the idiosyncratic jump components are described by a Hawkes process. They develop a robust calibration scheme which is able to distinguish systemic and idiosyncratic jumps and show that the model reproduces very well the empirical behaviour of the jumps of the Italian stocks.
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