# Model-Dependent Price Jump Indicators

1. The Difference Between Bi-power 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 model-dependent price jump component. Barndorff-Nielsen 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 bi-power variance. The difference between the standard and the bi-power variance can be used to define indicators that assess the jumpiness of the market. Generally, there are two ways to employ bi-power variance: the differential approach and the integral approach.

 ot2 = t T 1

 t-1 T-1 r.-T r,-i)2

1.1 The Differential Approach. The standard variance is defined as

The bi-power variance is defined according to Barndorff-Nielsen and Shephard (2004b) as

 (5)

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 bi-power 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. Bi-power Test Statistics. The bi-power 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

 Гг_ 2

 Lt

 (8)

Following Lee and Mykland, the variable £ is defined as

 maxz C An 1 Lz 1 Cn

 £

 (9)

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

 (11) (12)

The variable £ has in the presence of no price jumps the cumulative distribution function P(£ < x) = exp(e-x). 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 bi-power variation-based 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 – Pi-i
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 Jiang-Oomen 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 bi-power 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 price-jump 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 best-performing price-jump 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 price-jump 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 price-jump 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 high-frequency data for the 30 Dow Jones stocks over the 3-year 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.

1 Cojumps

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 high-frequency data for the 30 Dow Jones stocks over the 3-year 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 large-cap U. S. stocks and the corresponding aggregate market index. To more effectively detect cojumps authors developed a new cross product statistic, termed the cp-statistic, that directly uses the cross-covariation structure of the high-frequency returns and examines cross co­movements among the individual stocks. Employing this statistic allows to detect many modest-sized cojumps. Cross product statistic defined by the normalized sum of the individual high-frequency returns for each within-day period:

where j = 1, …, M, M—total number of observations in a day, n—number of stocks. The cp-statistic 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 24-h 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 1-2 % of the time and a bond or equity jump only about 3-4 % 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.