It is a common practice in finance to estimate volatility from the sum of frequently-sampled squared returns. However market microstructure poses challenges to this estimation approach, as evidenced by recent empirical studies in finance. This work attempts to lay out theoretical grounds that reconcile continuous-time modeling and discrete-time samples. We propose an estimation approach that takes advantage of the rich sources in tick-by-tick data while preserving the continuous-time assumption on the underlying returns. Under our framework, it becomes clear why and where the "usual" volatility estimator fails when the returns are sampled at the highest frequency.

This article introduces a new nonparametric test to detect jump arrival times and realized jump sizes in asset prices up to the intra-day level. We demonstrate that the likelihood of misclassification of jumps becomes negligible when we use high-frequency returns. Using our test, we examine jump dynamics and their distributions in the U.S. equity markets. The results show that individual stock jumps are associated with prescheduled earnings announcements and other company-specific news events. Additionally, S&P 500 Index jumps are associated with general market news announcements. This suggests different pricing models for individual equity options versus index options. (JEL G12, G22, G14) Financial markets sometimes generate significant discontinuities, so-called jumps, in financial variables. A number of recent empirical and theoretical studies proved the existence of jumps and their substantial impact on financial management, from portfolio and risk management to option and bond pricing and hedging (see Merton, 1976;Bakshi et al., 1997Bakshi et al., , 2000Bates, 1996;Liu et al., 2003;Naik and Lee, 1990;Duffie et al., 2000, andJohannes, 2004). Despite advances in asset pricing models and their inference techniques, the studies have found that jumps are empirically difficult to identify, because only discrete data are available from continuous-time models, in which most of the aforementioned applications were studied. Our goal in this article is first to propose a new jump detection technique to resolve such identification problems. Further, we show that our technique provides a model-free tool for characterizing jump dynamics in individual equity and S&P 500 Index returns, which allows us to investigate different model structures for their option pricing.

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In theory, the sum of squares of log returns sampled at high frequency estimates their variance. When market microstructure noise is present but unaccounted for, however, we show that the optimal sampling frequency is finite and derives its closed-form expression. But even with optimal sampling, using say 5-min returns when transactions are recorded every second, a vast amount of data is discarded, in contradiction to basic statistical principles. We demonstrate that modeling the noise and using all the data is a better solution, even if one misspecifies the noise distribution. So the answer is: sample as often as possible.Over the past few years, price data sampled at very high frequency have become increasingly available in the form of the Olsen dataset of currency exchange rates or the TAQ database of NYSE stocks. If such data were not affected by market microstructure noise, the realized volatility of the process (i.e., the average sum of squares of log-returns sampled at high frequency) would estimate the returns' variance, as is well known. In fact, sampling as often as possible would theoretically produce in the limit a perfect estimate of that variance.We start by asking whether it remains optimal to sample the price process at very high frequency in the presence of market microstructure noise, consistently with the basic statistical principle that, ceteris paribus, more data are preferred to less. We first show that, if noise is present but unaccounted for, then the optimal sampling frequency is finite, and we

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