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

Classical statistics suggest that for inference purposes one should always use as much data as is available. We study how the presence of market microstructure noise in high-frequency financial data can change that result. We show that the optimal sampling frequency at which to estimate the parameters of a discretely sampled continuous-time model can be finite when the observations are contaminated by market microstructure effects. We then address the question of what to do about the presence of the noise. We show that modelling the noise term explicitly restores the first order statistical effect that sampling as often as possible is optimal. But, more surprisingly, we also demonstrate that this is true even if one misspecifies the assumed distribution of the noise term. Not only is it still optimal to sample as often as possible, but the estimator has the same variance as if the noise distribution had been correctly specified, implying that attempts to incorporate the noise into the analysis cannot do more harm than good. Finally, we study the same questions when the observations are sampled at random time intervals, which are an essential feature of transaction-level data.

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 based on multiple time scales, and compare empirically our different estimators to the standard realized volatility. --Market microstructure,Serial dependence,High frequency data,Realized volatility,Subsampling,Two Scales Realized Volatility

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 based on multiple time scales, and compare empirically our different estimators to the standard realized volatility.

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 compute Edgeworth expansions for such estimators. Unlike the usual expansions, we have found that in order to obtain meaningful terms, one needs to let the size of the noise to go zero asymptotically. The results have application to Cornish-Fisher inversion and bootstrapping.

Luxury Goods and the Equity Premium

This paper evaluates the equity premium using novel data on the consumption of luxury goods. Specifying household utility as a nonhomothetic function of the consumption of both a luxury good and a basic good, we derive pricing equations and evaluate the risk of holding equity. Household survey and national accounts consumption data overstate the risk aversion necessary to match the observed equity premium because they mostly reflect basic consumption. The risk aversion implied by equity returns and the consumption of luxury goods is more than an order of magnitude less than that implied by national accounts data. For the very rich, the equity premium is much less of a puzzle.

A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High Frequency Data

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.

Viability problem with perturbation in Hilbert space

This paper deals with the existence result of viable solutions of the differential inclusion $$\dot{x}(t) \in f(t,x(t)) + F(x(t))$$ $$x(t) \in K \quad \text{on } [0,T],$$ where $K$ is a locally compact subset in separable Hilbert space $H,$ $(f(s,\cdot))_s$ is an equicontinuous family of measurable functions with respect to $s$ and $F$ is an upper semi-continuous set-valued mapping with compact values contained in the Clarke subdifferential $\partial_{c} V(x)$ of an uniformly regular function $V.