Jump detection in high-frequency order prices

TL;DR

This work develops a jump-detection framework for high-frequency price dynamics under a one-sided, lower-bounded microstructure-noise model (LOMN). By leveraging local minima of best-ask quotes and a global maximum-based statistic, the authors construct both local and global tests for jumps, establish uniform spot-volatility estimation, and derive stable and extreme-value asymptotics that enable online jump detection. The main theoretical contributions include a $n^{-1/3}$-rate for local alternatives (faster than the MMN benchmark) and a Gumbel limit for the global test, along with consistent localization of jump times and sizes. Empirical analysis on JPM data and simulations show that LOMN-based methods detect more jumps, offer speed advantages for online detection, and complement traditional mid-quote MMN analyses, with practical implications for high-frequency trading and macroeconomic inference.

Abstract

We propose methods to infer jumps of a semi-martingale, which describes long-term price dynamics, based on discrete, noisy, high-frequency observations. Different to the classical model of additive, centered market microstructure noise, we consider one-sided microstructure noise for order prices in a limit order book. We develop methods to estimate, locate and test for jumps using local minima of best ask quotes. We provide a local jump test and show that we can consistently estimate jump sizes and jump times. One main contribution is a global test for jumps. We establish the asymptotic properties and optimality of this test. We derive the asymptotic distribution of a maximum statistic under the null hypothesis of no jumps based on extreme value theory. We prove consistency under the alternative hypothesis. The rate of convergence for local alternatives is determined and shown to be much faster than optimal rates for the standard market microstructure noise model. This allows the identification of smaller jumps. In the process, we establish uniform consistency for spot volatility estimation under one-sided noise. Online jump detection based on the new approach is shown to achieve a speed advantage compared to standard methods applied to mid quotes. A simulation study sheds light on the finite-sample implementation and properties of the new approach and draws a comparison to a popular method for market microstructure noise. We showcase how our new approach helps to improve jump detection in an empirical analysis of intra-daily limit order book data.

Figures (14)

• Figure 1: Lines interpolate best ask (blue) and best bid (red) prices of AAPL over 10 minutes. Above (below) prices of other active limit ask (bid) orders are plotted.
• Figure 2: Simulated efficient price with one jump and local averages of MMN observations (left) and local minima of LOMN observations (right).
• Figure 3: Left panel: Kernel density estimates of the standardized version of the test statistic $T^{BHR}$ under the null (black solid line) and under the alternative (dashed-dotted line) for $\vert \text{jump size}\vert = 0.3\%$ and noise level $q = 0.1\%$. The gray solid line depicts the density function of the standard Gumbel distribution. The black and gray vertical lines (almost indistinguishable) are the $95\%$-quantile of the standardized version of the test statistic $T^{BHR}$ under the null and the $95\%$-quantile of the standard Gumbel distribution. Right panel: Jump times vs. inferred jump times for $\vert \text{jump size}\vert = 0.3\%$ and noise level $q = 0.1\%$.
• Figure 4: Left panel: Kernel density estimates of the test statistic $T^{BHR}$ in Scenario 2 under the null (black solid line) and under the alternative (dashed-dotted line) for $\vert \text{jump size}\vert = 0.125\%$, noise level $q = 0.05\%$ and $X_0 =\log 50$ for the optimal $nh_n=5$. Right panel: Kernel density estimates of the test statistic $T^{LM}$ in Scenario 2 under the null (black solid line) and under the alternative (dashed-dotted line) for $\vert \text{jump size}\vert = 0.125\%$, noise level $q = 0.05\%$ and $X_0 =\log 50$ for the optimal $nh_n=12$. The vertical lines refer to the $95\%$-quantile of the respective test statistic under the null.
• Figure 5: Log mid quotes over time provided in seconds after midnight. Areas highlighted in gray contain the detected jumps. Left: negative jump on 24th July 2007 (34800 is 9:40am) detected by all three tests. Right: positive jump on 30th Jan 2008 (51300 is 2:15pm) detected by all three tests.

Theorems & Definitions (21)

• Proposition 2.1
• Corollary 2.2
• Theorem 1
• Corollary 2.3
• Remark 1
• Theorem 2
• Proposition 2.4
• Corollary 2.5
• Lemma 1
• proof