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In empirical finance it is customary to sample prices sparsely on evenly spaced grids, such as one or five minutes. When doing so, prices are often flat, and many price returns are zero. For example, when sampling at one minute, the percentage of zero returns in stocks belonging to the S&P 500 index in more than 17 years is 36.06%; at five minutes, it is 14.78%; and at the relatively low frequency of ten minutes, it is still 8.97%. Hence, flatness is a non-negligible feature of the data. Nevertheless, the vast majority of models used in finance postulate the mere impossibility of zero returns in the price dynamics.
In this paper, while being agnostic about the sources of zero returns, we focus on the distortions induced by zeros on popular statistics used in finance. More specifically, we study the distortions of power and multipower estimators, which are used to estimate volatility, detect jumps and assess the dynamical features of the price process. All these estimates are basic ingredients in many financial applications, especially in derivative pricing and risk management, the most popular estimator in the power/multipower class being realized volatility. This paper studies the impact of a general and mathematically tractable form of flatness on all multipower estimators, including realized volatility as a special case.
We show that flatness is heavily detrimental for reliable jump inference. First, traditional jump tests are strongly distorted toward rejection of the null, inducing a large number of false positives which are actually due to flat prices. Second, traditional measures of jump activity on data which contain zero returns are strongly negatively biased. Thus, due to the price flatness, the presence of the Brownian motion component in price dymamics can be spuriously rejected in favour of a pure-jump process.
We provide limit theorems for multipower variation under flat trading which allow to quantify the bias, and propose a simple recipe for its correction. We use the flatness-robust multipowers to reappraise the statistical features of jumps in empirical finance. We first show that jumps appear to be much less frequent and much less contributing to price variation that what found by the empirical literature so far. We then show that estimates of the activity index of a panel of individual stocks, of the stock index exchange-traded fund, and of the VIX index are not significantly different from the value implied by the Brownian motion.
