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When Frictions are Fractional: Rough Noise in High-Frequency Data

Carsten H. Chong, Thomas Delerue, Guoying Li

Abstract

The analysis of high-frequency financial data is often impeded by the presence of noise. This article is motivated by intraday return data in which market microstructure noise appears to be rough, that is, best captured by a continuous-time stochastic process that locally behaves as fractional Brownian motion. Assuming that the underlying efficient price process follows a continuous Itô semimartingale, we derive consistent estimators and asymptotic confidence intervals for the roughness parameter of the noise and the integrated price and noise volatilities, in all cases where these quantities are identifiable. In addition to desirable features such as serial dependence of increments, compatibility between different sampling frequencies and diurnal effects, the rough noise model can further explain divergence rates in volatility signature plots that vary considerably over time and between assets.

When Frictions are Fractional: Rough Noise in High-Frequency Data

Abstract

The analysis of high-frequency financial data is often impeded by the presence of noise. This article is motivated by intraday return data in which market microstructure noise appears to be rough, that is, best captured by a continuous-time stochastic process that locally behaves as fractional Brownian motion. Assuming that the underlying efficient price process follows a continuous Itô semimartingale, we derive consistent estimators and asymptotic confidence intervals for the roughness parameter of the noise and the integrated price and noise volatilities, in all cases where these quantities are identifiable. In addition to desirable features such as serial dependence of increments, compatibility between different sampling frequencies and diurnal effects, the rough noise model can further explain divergence rates in volatility signature plots that vary considerably over time and between assets.

Paper Structure

This paper contains 19 sections, 15 theorems, 129 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Central limit theorem for variation functionals
  4. Estimating the roughness parameter and integrated price and noise volatilities
  5. Asymptotically mixed normal estimators
  6. Feasible implementation
  7. Finite-sample considerations
  8. Monte Carlo simulation
  9. Empirical analysis
  10. Conclusion and future directions
  11. Does microstructure noise exist in continuous time?
  12. A multivariate central limit theorem for variation functionals
  13. Proof of Theorem \ref{['thm:CLT:mixedSM']}
  14. Size estimates
  15. Estimates for fractional kernels
  16. ...and 4 more sections

Key Result

Proposition 2.3

Assume that $Y$ is an mfBM, that is, $Y=X+Z$ where $X={\sigma} B$ and $Z= \rho B^H$ for some $\rho,{\sigma}\in(0,\infty)$, $B$ is a Brownian motion and $B^H$ is an independent fBM with Hurst parameter $H\in(0,\frac{1}{2})$. For any $T>0$, the laws of $(Y_t)_{t\in[0,T]}$ and $(Z_t)_{t\in[0,T]}$ are m

Figures (9)

  • Figure 1: (a) Volatility signature plot and (b) variance plot for 2019 SPY transaction data (top). The same plots on a log--log scale (middle) reveal a divergence rate of $-0.04$ for RV and a shrinkage rate of increments of $0.96$ for the whole year. (The divergence rate of RV is $\alpha$ if $\mathrm{RV}\sim C_1\Delta_n^{{\alpha}}$ for some $C_1>0$; the shrinkage rate of increments is $\beta$ if $\mathop{\mathrm{\mathrm{Var}}}\nolimits(\Delta^n_i Y)\sim C_2\Delta_n^{\beta}$ for some $C_2>0$.) The histograms (bottom) show the daily divergence rates in volatility signature plots and the daily shrinkage rates of price increments in 2019 SPY transaction data. Each data point corresponds to one trading day.
  • Figure 2: (a) Estimators of $\mathop{\mathrm{\mathrm{Var}}}\nolimits({\varepsilon})$ and (b) estimators of $r(1)$ including 95%-confidence intervals. The analysis is based on 2019 SPY transaction data sampled at $\Delta_n=1\,\mathrm{sec}$.
  • Figure 3: Sample quantiles of $(H^n-H)/\sqrt{\mathbb{V}^H_n}$ against standard normal quantiles.
  • Figure 4: Bias, SE and RMSE of $H^{n}$, $\widetilde{H}^{n}_{\text{DMS}}$, $\widetilde{H}^n_{\text{VS}}$ and $\widetilde{H}^{n}_{\text{acf}}$ in absolute numbers.
  • Figure 5: Histogram of daily estimates of $H$ (a) and of the NSR (b) based on 1 second SPY transaction data over a period of ten years, with the mean indicated by a red line.
  • ...and 4 more figures

Theorems & Definitions (33)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Corollary 4.1
  • Theorem 4.2
  • ...and 23 more