Intertrade Durations Research Articles

A multifactor regime-switching model for inter-trade durations in the high-frequency limit order market

Modeling behaviors of inter-trade durations has been an important step to understand the market microstructure. Given that most existing models focused on intraday transaction data, we analyze limit order book (LOB) data in a high-frequency (HF) market and find that stocks’ inter-trade durations follow bimodal distributions peaking at levels of second and millisecond. To characterize this empirical property, we develop a multifactor regime-switching duration (MF-RSD) model, in which the modes of inter-trade durations are characterized as regimes and the transitions of regimes are determined by LOB-factor-dependent transition probability matrices. Using the model, we analyze the impact of LOB factors on inter-trade duration dynamics and find that the bimodal phenomenon is caused by the switchings of traders’ trading strategies between market making and price speculation. Our empirical studies show that the MF-RSD model not only produces good in-sample fit but also outperforms some benchmark models in out-of-sample predictions of inter-trade durations. • Inter-trade durations in high-frequency LOB exhibit a bimodal distribution. • A multifactor regime-switching model is built for inter-trade durations in HF. • The regimes of inter-trade durations are associated with the trader’s behaviors. • Our model has better out-of-sample performance compared with some benchmarks. • The summarized effects of LOB factors reveal principle of market microstructure.

A Multifactor Regime-Switching Model for Inter-Trade Durations in the Limit Order Market

This paper studies inter-trade durations in the NASDAQ limit order market and finds that inter-trade durations in ultra-high frequency have two modes. One mode is to the order of approximately 10^{-4} seconds, and the other is to the order of 10^0 seconds. This phenomenon and other empirical evidence suggest that there are two regimes associated with the dynamics of inter-trade durations, and the regime switchings are driven by the changes of high-frequency traders (HFTs) between providing and taking liquidity. To find how the two modes depend on information in the limit order book (LOB), we propose a two-state multi-factor regime-switching (MF-RSD) model for inter-trade durations, in which the probabilities transition matrices are time-varying and depend on some lagged LOB factors. The MF-RSD model has good in-sample fitness and the superior out-of-sample performance, compared with some benchmark duration models. Our findings of the effects of LOB factors on the inter-trade durations help to understand more about the high-frequency market micro-structure.

Modeling macroeconomic series with regime-switching models characterized by a high-dimensional state space

The Markov-switching multifractal process, and recent extensions such as the factorial hidden Markov volatility model, correspond to tightly parametrized hidden Markov models characterized by a high-dimensional state space. Because the central component in these models is a Markov chain restricted to have positive support, the applicability of such models has been so far limited to the modeling of positive processes such as volatilities, inter-trade durations and trading volumes. By adapting the factorial hidden Markov volatility model, we develop a new regime-switching process for capturing time variation in the conditional mean of a time series with support on the whole real line. We show its promising performance to fit 21 widely used macroeconomic data sets.

Drift in Transaction‐Level Asset Price Models

We study the effect of drift in pure‐jump transaction‐level models for asset prices in continuous time, driven by point processes. The drift is assumed to arise from a non‐zero mean in the efficient shock series. It follows that the drift is proportional to the driving point process itself, that is, the cumulative number of transactions. This link reveals a mechanism by which properties of intertrade durations (such as heavy tails and long memory) can have a strong impact on properties of average returns, thereby potentially making it extremely difficult to determine long‐term growth rates or to reliably detect an equity premium. We focus on a basic univariate model for log price, coupled with general assumptions on the point process that are satisfied by several existing flexible models, allowing for both long memory and heavy tails in durations. Under our pure‐jump model, we obtain the limiting distribution for the suitably normalized log price. This limiting distribution need not be Gaussian and may have either finite variance or infinite variance. We show that the drift can affect not only the limiting distribution for the normalized log price but also the rate in the corresponding normalization. Therefore, the drift (or equivalently, the properties of durations) affects the rate of convergence of estimators of the growth rate and can invalidate standard hypothesis tests for that growth rate. As a remedy to these problems, we propose a new ratio statistic that behaves more robustly and employ subsampling methods to carry out inference for the growth rate. Our analysis also sheds some new light on two long‐standing debates as to whether stock returns have long memory or infinite variance.

Large scale simulation of synthetic markets

High-frequency trading has been experiencing an increase of interest both for practical purposes within financial institutions and within academic research; recently, the UK Government Office for Science reviewed the state of the art and gave an outlook analysis. Therefore, models for tick-by-tick financial time series are becoming more and more important. Together with high-frequency trading comes the need for fast simulations of full synthetic markets for several purposes including scenario analyses for risk evaluation. These simulations are very suitable to be run on massively parallel architectures. Aside more traditional large-scale parallel computers, high-end personal computers equipped with several multi-core CPUs and general-purpose GPU programming are gaining importance as cheap and easily available alternatives. A further option are FPGAs. In all cases, development can be done in a unified framework with standard C or C++ code and calls to appropriate libraries like MPI (for CPUs) or CUDA for (GPGPUs). Here we present such a prototype simulation of a synthetic regulated equity market. The basic ingredients to build a synthetic share are two sequences of random variables, one for the inter-trade durations and one for the tick-by-tick logarithmic returns. Our extensive simulations are based on several distributional choices for the above random variables, including Mittag-Leffler distributed inter-trade durations and alpha-stable tick-by-tick logarithmic returns.

Unveiling correlations between financial variables and topological metrics of trading networks: Evidence from a stock and its warrant

Traders develop and adopt different trading strategies attempting to maximize their profits in financial markets. These trading strategies not only result in specific topological structures in trading networks, which connect the traders with the pairwise buy–sell relationships, but also have potential impacts on market dynamics. Here, we present a detailed analysis on how the market behaviors are correlated with the structures of traders in trading networks based on audit trail data for the Baosteel stock and its warrant at the transaction level from 22 August 2005 to 23 August 2006. In our investigation, we divide each trade day into 48 rolling time windows with a length of 5 min, construct a trading network within each window, and obtain a time series of over 11,600 trading networks. We find that there are strongly simultaneous correlations between the topological metrics (including network centralization, assortative index, and average path length) of trading networks that characterize the patterns of order execution and the financial variables (including return, volatility, intertrade duration, and trading volume) for the stock and its warrant. Our analysis may shed new lights on how the microscopic interactions between elements within complex system affect the system’s performance.

A Compound Multifractal Model for High-Frequency Asset Returns

This paper builds a model of high-frequency equity returns by separately modeling the dynamics of trade-time returns and trade arrivals. Our main contributions are threefold. First, we characterize the distributional behavior of high-frequency asset returns both in ordinary clock time and in trade time. We show that when controlling for pre-scheduled market news events, trade-time returns of the highly liquid near-month E-mini S&P 500 futures contract are well characterized by a Gaussian distribution at very fine time scales. Second, we develop a structured and parsimonious model of clock-time returns by subordinating a trade-time Gaussian distribution with a trade arrival process that is associated with a modified Markov-Switching Multifractal Duration (MSMD) model. This model provides an excellent characterization of high-frequency inter-trade durations. Over-dispersion in this distribution of inter-trade durations leads to leptokurtosis and volatility clustering in clock-time returns, even when trade-time returns are Gaussian. Finally, we use our model to extrapolate the empirical relationship between trade rate and volatility in an effort to understand conditions of market failure. Our model suggests that the 1,200 km physical separation of financial markets in Chicago and New York/New Jersey provides a natural ceiling on systemic volatility and may contribute to market stability during periods of extremely heavy trading.

An analysis of the financial crisis in the KOSPI market using Hurst exponents

Recently, the study of the financial crisis has progressed to include the concept of the complex system, thereby improving the understanding of this extreme event from a neoclassical economic perspective. To determine which variables are related to the financial event caused by the 2008 US subprime crisis using temporal correlations, we investigate the diverse variables that may explain the financial system. These variables include return, volatility, trading volume and inter-trade duration data sets within the TAQ data for 27 highly capitalized individual companies listed on the KOSPI stock market. During 2008 and 2009, the Hurst exponent for the return time series over the whole period was less than 0.5, and the Hurst exponents for other variables, such as the volatility, trading volume and inter-trade duration, were greater than 0.5. Additionally, we analyze the relationships between the variation of temporal correlation and market instability based on these Hurst exponents and the degree of multifractality. We find that for the data related to trading volume, the Hurst exponents do not allow us to detect changes in market status, such as changes from normal to abnormal status, whereas other variables, including the return, volatility and weekly inter-trade duration, indicate a significant change in market status after the Lehman Brothers’ bankruptcy. In addition, the multifractality and the measurement defined by subtracting the Hurst exponent of the return time series from that of the volatility time series decrease sharply after the US subprime event and recover approximately 50 days after the Lehman Brothers’ collapse. Our findings suggest that the temporal features of financial quantities in the TAQ data set and the market complexity perform very well at diagnosing financial market stability.

High frequency lead/lag relationships — Empirical facts

Lead/lag relationships are an important stylized fact at high frequency. Some assets follow the path of others with a small time lag. We provide indicators to measure this phenomenon using tick-by-tick data. Strongly asymmetric cross-correlation functions are empirically observed, especially in the future/stock case. We confirm the intuition that the most liquid assets (short intertrade duration, narrow bid/ask spread, small volatility, high turnover) tend to lead smaller stocks. However, the most correlated stocks are those with similar levels of liquidity. These lead/lag relationships become more and more pronounced as we zoom on significant events. We reach 60% of accuracy when forecasting the next midquote variation of the lagger using only the past information of the leader, which is significantly better than using the information of the lagger only. However, a naive strategy based on market orders cannot make any profit of this effect because of the bid/ask spread.

LIMIT LAWS IN TRANSACTION-LEVEL ASSET PRICE MODELS

We consider pure-jump transaction-level models for asset prices in continuous time, driven by point processes. In a bivariate model that admits cointegration, we allow for time deformations to account for such effects as intraday seasonal patterns in volatility and nontrading periods that may be different for the two assets. We also allow for asymmetries (leverage effects). We obtain the asymptotic distribution of the log-price process. For the weak fractional cointegration case, we obtain the asymptotic distribution of the ordinary least squares estimator of the cointegrating parameter based on data sampled from an equally spaced discretization of calendar time, and we justify a feasible method of hypothesis testing for the cointegrating parameter based on the correspondingt-statistic. In the strong fractional cointegration case, we obtain the limiting distribution of a continuously averaged tapered estimator as well as other estimators of the cointegrating parameter, and we find that the rate of convergence can be affected by properties of intertrade durations. In particular, the persistence of durations (hence of volatility) can affect the degree of cointegration. We also obtain the rate of convergence of several estimators of the cointegrating parameter in the standard cointegration case. Finally, we consider the properties of the ordinary least squares estimator of the regression parameter in a spurious regression, i.e., in the absence of cointegration.

A Markov-Switching Multi-Fractal Inter-Trade Duration Model, with Application to U.S. Equities

We propose and illustrate a Markov-switching multi-fractal duration (MSMD) model for analysis of inter-trade durations in financial markets. We establish several of its key properties with emphasis on high persistence (indeed long memory). Empirical exploration suggests MSMD's superiority relative to leading competitors.

A Markov-switching multifractal inter-trade duration model, with application to US equities

We propose and illustrate a Markov-switching multifractal duration (MSMD) model for analysis of inter-trade durations in financial markets. We establish several of its key properties with emphasis on high persistence and long memory. Empirical exploration suggests MSMD’s superiority relative to leading competitors.

A Markov-Switching Multifractal Inter-Trade Duration Model, with Application to US Equities

A Markov-Switching Multifractal Inter-Trade Duration Model, with Application to US Equities

NONEXTENSIVE STATISTICAL MECHANICS DISTRIBUTIONS AND DYNAMICS OF FINANCIAL OBSERVABLES FROM THE NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

We present nonlinear stochastic differential equations, generating processes with the q-exponential and q-Gaussian distributions of the observables, i.e. with the long-range power-law autocorrelations and 1/fβpower spectral density. Similarly, the Tsallis q-distributions may be obtained in the superstatistical framework as a superposition of different local dynamics at different time intervals. In such approach, the average of the stochastic variable is generated by the nonlinear stochastic process, while the local distribution of the signal is exponential or Gaussian one, conditioned by the slow average. Further we analyze relevance of the generalized and adapted equations for modeling the financial processes. We model the inter-trade durations, the trading activity and the normalized return using the superstatistical approaches with the exponential and normal distributions of the local signals driven by the nonlinear stochastic process.

Long-term correlations and multifractal nature in the intertrade durations of a liquid Chinese stock and its warrant

The intertrade duration of equities is an important financial measure, characterizing trading activities; it is defined as the waiting time between successive trades of an equity. Using the ultrahigh-frequency data of a liquid Chinese stock and its associated warrant, we perform a comparative investigation of the statistical properties of their intertrade duration time series. The distributions of the two equities can be better described by the shifted power-law form than the Weibull form, and their scaled distributions do not collapse onto a single curve. Although the intertrade durations of the two equities have very different magnitude, their intraday patterns exhibit very similar shapes. Both detrended fluctuation analysis (DFA) and detrending moving average analysis (DMA) show that the 1 min intertrade duration time series of the two equities are strongly correlated. In addition, both multifractal detrended fluctuation analysis (MFDFA) and multifractal detrending moving average analysis (MFDMA) unveil that the 1 min intertrade durations possess multifractal nature. However, the difference between the two singularity spectra of the two equities obtained from the MFDMA is much smaller than that from the MFDFA.

Long memory in intertrade durations, counts and realized volatility of NYSE stocks

We study the persistence of intertrade durations, counts (number of transactions in equally spaced intervals of clock time), squared returns and realized volatility in 10 stocks trading on the New York Stock Exchange. A semiparametric analysis reveals the presence of long memory in all of these series, with potentially the same memory parameter. We introduce a parametric latent-variable long-memory stochastic duration (LMSD) model which is shown to better fit the data than the autoregressive conditional duration model (ACD) in a variety of ways. The empirical evidence we present here is in agreement with theoretical results on the propagation of memory from durations to counts and realized volatility presented in Deo et al. (2009).

High Frequency Traders and Asset Prices

Do high frequency traders affect transaction prices? In this paper we derive distributions of transaction prices in limit order markets populated by low frequency traders (humans) before and after the entrance of a high frequency trader (machine). We find that the presence of a machine is likely to change the average transaction price, even in the absence of new information. We also find that in a market with a high frequency trader, the distribution of transaction prices has more mass around the center and thinner far tails. With a machine, mean intertrade duration decreases in proportion to the increase in the ratio of the human order arrival rates with and without the presence of the machine; trading volume goes up by the same rate. We show that the machine makes positive expected profits by sniping out human orders somewhat away from the front of the book. This explains the shape of the transaction price density. In fact, we show that in a special case, the faster humans submit and vary their orders, the more profits the machine makes.

The distribution of first-passage times and durations in FOREX and future markets

Possible distributions are discussed for intertrade durations and first-passage processes in financial markets. The view-point of renewal theory is assumed. In order to represent market data with relatively long durations, two types of distributions are used, namely a distribution derived from the Mittag–Leffler survival function and the Weibull distribution. For the Mittag–Leffler type distribution, the average waiting time (residual life time) is strongly dependent on the choice of a cut-off parameter t max , whereas the results based on the Weibull distribution do not depend on such a cut-off. Therefore, a Weibull distribution is more convenient than a Mittag–Leffler type if one wishes to evaluate relevant statistics such as average waiting time in financial markets with long durations. On the other hand, we find that the Gini index is rather independent of the cut-off parameter. Based on the above considerations, we propose a good candidate for describing the distribution of first-passage time in a market: The Weibull distribution with a power-law tail. This distribution compensates the gap between theoretical and empirical results more efficiently than a simple Weibull distribution. It should be stressed that a Weibull distribution with a power-law tail is more flexible than the Mittag–Leffler distribution, which itself can be approximated by a Weibull distribution and a power-law. Indeed, the key point is that in the former case there is freedom of choice for the exponent of the power-law attached to the Weibull distribution, which can exceed 1 in order to reproduce decays faster than possible with a Mittag–Leffler distribution. We also give a useful formula to determine an optimal crossover point minimizing the difference between the empirical average waiting time and the one predicted from renewal theory. Moreover, we discuss the limitation of our distributions by applying our distribution to the analysis of the BTP future and calculating the average waiting time. We find that our distribution is applicable as long as durations follow a Weibull law for short times and do not have too heavy a tail.

Trading Networks

We use network analysis to quantify the flow of information through financial markets. Using unique ultra high frequency data, we compute network and financial variables for transactions that occured during August 2008 in the nearby E-mini S&P 500 futures contract ‐ the cornerstone of price discovery for the S&P 500 Index. We find that network variables presage the information represented by financial variables. Most notably, we find that network variables strongly Granger-case intertrade duration and trading volume, suggesting that network metrics serve as primitive measures of information flow. Finally, we find that the dynamics of returns and volatility are rooted in the network mechanics of the information arrival process ‐ as evidenced both in our data and the results of an agent-based simulation model.

Detrended fluctuation analysis of intertrade durations

The intraday pattern, long memory, and multifractal nature of the intertrade durations, which are defined as the waiting times between two consecutive transactions, are investigated based upon the limit order book data and order flows of 23 liquid Chinese stocks listed on the Shenzhen Stock Exchange in 2003. An inverse U -shaped intraday pattern in the intertrade durations with an abrupt drop in the first minute of the afternoon trading is observed. Based on a detrended fluctuation analysis, we find a crossover of power-law scaling behaviors for small box sizes (trade numbers in boxes) and large box sizes and strong evidence in favor of long memory in both regimes. In addition, the multifractal nature of intertrade durations in both regimes is confirmed by a multifractal detrended fluctuation analysis for individual stocks with a few exceptions in the small-duration regime. The intraday pattern has little influence on the long memory and multifractality.