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24
Asymptotic error distributions for the Euler method for stochastic differential equations
 THE ANNALS OF PROBABILITY
, 1998
"... We are interested in the rate of convergence of the Euler scheme approximation of the solution to a stochastic differential equation driven by a general (possibly discontinuous) semimartingale, and by the asymptotic behavior of the associated normalized error. It is well known that for Itô’s equatio ..."
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Cited by 103 (8 self)
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We are interested in the rate of convergence of the Euler scheme approximation of the solution to a stochastic differential equation driven by a general (possibly discontinuous) semimartingale, and by the asymptotic behavior of the associated normalized error. It is well known that for Itô’s equations the rate is 1 / √ n; we provide a necessary and sufficient condition for this rate to be 1 / √ n when the driving semimartingale is a continuous martingale, or a continuous semimartingale under a mild additional assumption; we also prove that in these cases the normalized error processes converge in law. The rate can also differ from 1 / √ n: this is the case for instance if the driving process is deterministic, or if it is a Lévy process without a Brownian component. It is again 1 / √ n when the driving process is Lévy with a nonvanishing Brownian component, but then the normalized error processes converge in law in the finitedimensional sense only, while the discretized normalized error processes converge in law in the Skorohod
How often to sample a continuoustime process in the presence of market microstructure noise
 Review of Financial Studies
, 2005
"... 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 closedform expression. But even with optimal sampling, usi ..."
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Cited by 86 (13 self)
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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 closedform expression. But even with optimal sampling, using say 5min 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 logreturns 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 We are grateful for comments and suggestions from the editor, Maureen O’Hara, and two anonymous
Ultra high frequency volatility estimation with dependent microstructure noise
"... We analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration. We show that combining two time scales for tha ..."
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Cited by 56 (10 self)
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We analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration. We show that combining two time scales for that purpose will work even when the noise exhibits time series dependence, analyze in that context a refinement of this approach based on multiple time scales, and compare empirically our different estimators to the standard realized volatility.
MICROSTRUCTURE NOISE, REALIZED VARIANCE, AND OPTIMAL SAMPLING
, 2005
"... Observed asset prices are known to deviate from their efficient values due to market microstructure frictions. This paper studies the effects of market microstructure noise on nonparametric estimates of the efficient price integrated variance. Specifically, we consider both asymptotic and finite sam ..."
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Cited by 49 (5 self)
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Observed asset prices are known to deviate from their efficient values due to market microstructure frictions. This paper studies the effects of market microstructure noise on nonparametric estimates of the efficient price integrated variance. Specifically, we consider both asymptotic and finite sample effects of general market microstructure noise on realized variance estimates. The finite sample results culminate in a variance/bias tradeoff that serves as a basis for an optimal sampling theory. Our theory also considers the effects of prefiltering the data and proposes a novel biascorrection. We show that this theory is easily implementable in practise requiring only the calculation of sample moments of the observable highfrequency return data.
Variation, jumps, market frictions and high frequency data in financial econometrics
, 2005
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Estimating covariation: Epps effect and microstructure noise
 Journal of Econometrics, forthcoming
, 2009
"... This paper is about how to estimate the integrated covariance 〈X, Y 〉T of two assets over a fixed time horizon [0, T], when the observations of X and Y are “contaminated ” and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previoustick covariance e ..."
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Cited by 26 (3 self)
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This paper is about how to estimate the integrated covariance 〈X, Y 〉T of two assets over a fixed time horizon [0, T], when the observations of X and Y are “contaminated ” and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previoustick covariance estimator is biased, and the size of the bias is more pronounced for less liquid assets. This is an analytic characterization of the Epps effect. We also provide optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms, including nonsynchronous trading, microstructure noise, and time discretization. Finally, a twoscales covariance estimator is provided which simultaneously cancels (to first order) the Epps effect and the effect of microstructure noise. The gain is demonstrated in data.
Edgeworth expansions for realized volatility and related estimators
, 2005
"... This paper shows that the asymptotic normal approximation is often insufficiently accurate for volatility estimators based on high frequency data. To remedy this, we derive Edgeworth expansions for such estimators. The expansions are developed in the framework of smallnoise asymptotics. The results ..."
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Cited by 15 (4 self)
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This paper shows that the asymptotic normal approximation is often insufficiently accurate for volatility estimators based on high frequency data. To remedy this, we derive Edgeworth expansions for such estimators. The expansions are developed in the framework of smallnoise asymptotics. The results have application to CornishFisher inversion and help setting intervals more accurately than those relying on normal distribution.
Are Volatility Estimators Robust with Respect to Modeling Assumptions
 Bernoulli
, 2007
"... We consider microstructure as an arbitrary contamination of the underlying latent securities price, through a Markov kernel Q. Special cases include additive error, rounding and combinations thereof. Our main result is that, subject to smoothness conditions, the two scales realized volatility is rob ..."
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Cited by 12 (5 self)
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We consider microstructure as an arbitrary contamination of the underlying latent securities price, through a Markov kernel Q. Special cases include additive error, rounding and combinations thereof. Our main result is that, subject to smoothness conditions, the two scales realized volatility is robust to the form of contamination Q. To push the limits of our result, we show what happens for some models that involve rounding (which is not, of course, smooth) and see in this situation how the robustness deteriorates with decreasing smoothness. Our conclusion is that under reasonable smoothness, one does not need to consider too closely how the microstructure is formed, while if severe nonsmoothness is suspected, one needs to pay attention to the precise structure and also the use to which the estimator of volatility will be put.