# Model-Independent Price Jump Indicators

1. Extreme returns indicator: a price jump occurs at time t if the return at time t is above some threshold. The threshold value can be selected by two ways: it can be selected globally—one threshold value for the entire sample, for example, when the threshold is a given centile of the distribution of returns over the entire data set. Or, it can be selected locally, and consequently, some sub-samples may have different threshold values. A global definition of the threshold allows to compare the behavior of returns over the entire sample, however, the distribution of returns can vary, e. g., the width of the distribution can change due to changes in market conditions, and thus the global definition of the threshold is not suitable to directly compare price jumps over periods with different market conditions. This group is represented by the works of Ait-Sahalia (2004), Ait-Sahalia et al. (2009) and Ait-Sahalia and Jacod (2009a, b). The indicators have well-defined analytic properties; but they do not identify price jumps one by one but rather measure the jumpiness of the given period. These methods are more suitable to assess the jumpiness of ultra-high-frequency data.

2. Temperature. Kleinert (2009) shows that high-frequency returns at a 1-min frequency for the S&P 500 and the NASDAQ 100 indices have the property that they have purely an exponential behavior for both the positive as well as negative sides. The distribution can fit the Boltzmann distribution:

B(r) = 2T exp f (1)

where T is the parameter of the distribution conventionally known as the temperature, and r stands for returns. The parameter T governs the width of the distribution; the higher the temperature of the market, the higher the volatility. Kleinert and Chen (2007) and Kleinert (2009) document that this parameter varies slowly, and its variation is connected to the situation on the market.

3.

p-dependent Realized Volatility. The general definition of the p-dependent realized volatility can be written as:

where the sample over which the volatility is calculated is represented by a moving window of length T (Dacorogna 2001). The interesting property of this definition is that the higher the p is, the more weight the outliers have. Since price jumps are simply extreme price movements, the property of realized volatility can be translated into the following statement: the higher the p is, the more price jumps are stressed. The ratio of two realized volatilities with different p can be thus used as an estimator of price jumps.

4. Price Jump Index. The price jump index jT, t at time t (as employed by Joulin et al. 2010) is defined as

Гг

T-1

тУІ rt-i

і =0

where T is the market history employed. Gopikrishnan et al. (1999), Eryigit et al. (2009) and Joulin et al. (2010) take normalized price time series—the normalization differs across these papers—and define the scaling properties of the tails of the distributions. This technique has its roots in Econophysics, it is based on the scaling properties of time series known in physics, see e. g. Stanley and Mantegna (2000).

5. Wavelet Filter. The Maximum Overlap Discrete Wavelet Transform (MODWT) filter represents a technique that is used to filter out effects at different scales. In the time series case, the scale is equivalent to the frequency, thus, the MODWT can be used to filter out high frequency components of time series. This can be also described as the decomposition of the entire time series into high – and low-frequency component effects (Gencay et al. 2002). The MODWT technique projects the original time series into a set of other time series, where each of the time series captures effects at a certain frequency scale.

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