How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise

In theory, the sum of squares of log returns sampled at highfrequency estimates their variance. When market microstructurenoise is present but unaccounted for, however, we show thatthe optimal sampling frequency is finite and derives its closed-formexpression. But even with optimal sampling, using say 5-minreturns when transactions are recorded every second, a vastamount of data is discarded, in contradiction to basic statisticalprinciples. We demonstrate that modeling the noise and usingall the data is a better solution, even if one misspecifiesthe noise distribution. So the answer is: sample as often aspossible.     

On the jump activity index for semimartingales

Empirical evidence of asset price discontinuities or “jumps” in financial markets has been well documented in the literature. Recently, Aït-Sahalia and Jacod (2009b) defined a general “jump activity index” to describe the degree of jump activities for asset price semimartingales, and provided a consistent estimator when the underlying process contains both a continuous and a jump component. However, only large increments were used in their estimator so that the effective sample size is very small even for large sample sizes. In this paper, we explore ways to improve the Aït-Sahalia and Jacod estimator by making use of all increments, large and small. The improvement is verified through simulations. A real example is also given.     

EVALUATING HEDGING ERRORS: AN ASYMPTOTIC APPROACH

We propose a methodology for evaluating the hedging errors of derivative securities due to the discreteness of trading times or the observation times of market prices, or both. Utilizing a weak convergence approach, we derive the asymptotic distributions of the hedging errors as the discreteness disappears in several situations. First, we examine the hedging error due to discrete-time trading when the true strategy is known, which generalizes the result of Bertsimas, Kogan, and Lo (2000) to continuous Itô processes. Then we consider a data-driven strategy, when the true strategy is unknown. This strategy is free of parametric model assumptions, therefore it is expected to serve as a benchmark for the evaluation of parametric strategies. Finally, we consider a case study of the Black-Scholes delta-hedging strategy when the volatility is unknown in the proposed framework. The results obtained give us a prospect for further developments of the framework under which various parametric strategies could be compared in a unified manner.     

Edgeworth expansions for realized volatility and related estimators

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 small-noise asymptotics. The results have application to Cornish–Fisher inversion and help setting intervals more accurately than those relying on normal distribution.     

Jumps in equilibrium prices and market microstructure noise

Asset prices observed in financial markets combine equilibrium prices and market microstructure noise. In this paper, we study how to tell apart large shifts in equilibrium prices from noise using high frequency data. We propose a new nonparametric test which allows us to asymptotically remove the noise from observable price data and to discover jumps in fundamental asset values. We provide its asymptotic distribution to decide when such jumps occur. In finite samples, our test offers reasonable power for distinguishing between noise and jumps. Empirical evidence indicates that it is necessary to incorporate the presence of jumps in equilibrium prices.     

A Gaussian calculus for inference from high frequency data

In the econometric literature of high frequency data, it is often assumed that one can carry out inference conditionally on the underlying volatility processes. In other words, conditionally Gaussian systems are considered. This is often referred to as the assumption of “no leverage effect”. This is often a reasonable thing to do, as general estimators and results can often be conjectured from considering the conditionally Gaussian case. The purpose of this paper is to try to give some more structure to the things one can do with the Gaussian assumption. We shall argue in the following that there is a whole treasure chest of tools that can be brought to bear on high frequency data problems in this case. We shall in particular consider approximations involving locally constant volatility processes, and develop a general theory for this approximation. As applications of the theory, we develop an ANOVA for processes with multiple regressors, and give an estimator for error bars on the Hayashi–Yoshida estimator of quadratic covariation. Other applications are considered in other papers.     

An analysis of Hansen–Scheinkman moment estimators for discretely and randomly sampled diffusions

We derive closed-form expansions for the asymptotic distribution of Hansen and Scheinkman [1995. Back to the future: generating moment implications for continuous-time Markov processes. Econometrica 63, 767–804] moment estimators for discretely, and possibly randomly, sampled diffusions. This result makes it possible to select optimal moment conditions as well as to assess the efficiency of the resulting parameter estimators relative to likelihood-based estimators, or to an alternative type of moment conditions.     

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 that purpose will work even when the noise exhibits time series dependence, analyze in that context a refinement of this approach is based on multiple time scales, and compare empirically our different estimators to the standard realized volatility.