Investigating Intertrade Durations using Copulas: An Experiment with NASDAQ Data

Electronic limit order markets represent a market for immediacy where limit order submitters offer terms of trade to potential market orders. Therefore, limit order submitters must continuously monitor the market for changing conditions or risk being arbitraged by traders who arrive subsequent to them but closer to the trade.

Limit orders are a source of market liquidity (Biais et al. (2015), Pani (2021a)), and price discovery (Brogaard et al. (2019)). Technological advancement has aided this role of limit orders by lowering the costs to monitor the market and cancel or modify limit orders (Hasbrouck and Saar (2013)). The competitive conditions in fragmented markets has encouraged exchanges to offer innovative maker-taker pricing, providing further incentives to traders to place limit orders.

In limit order markets, determining who demands liquidity and who supplies it is a non-trivial issue. Market makers take positions on both sides of the market, adjusting their spread on either side depending on their inventory, adverse selection risk and other market insights.

Hasbrouck (1991) vector autoregressive framework remained the key method for a long period. The method involves investigating how the variable is correlated to its lagged versions. Usually, one variable is investigated at a time. The method adopted in our study looks at the contemporaneous cross-sectional relationship among variables.

Engle and Russell (1998) introduced the autoregressive conditional duration (ACD) model to present a duration analysis in a dynamic framework. Financial market events, such as trades and quote changes, occur in clusters and ACD and ARCH models attempt to study them. Bauwens and Giot (2000) introduced a logarithmic version of the ACD model, implying a nonlinear relation between the duration and its lags.

Dufour and Engle (2000) extend Hasbrouck (1991) vector autoregressive framework to account for time varying transaction intensity when measuring the information content of trades. They find that duration is informative, and higher trading intensity is associated with a higher price impact of trades, a faster price adjustment to new trade related information and a stronger autocorrelation of trades.

The majority of studies to date have predominantly used trades and mid quotes for the analysis. Empirical observations in our experiments and others in limit order books show that trades walk up the book with regularity. Hence, the earlier approaches would ignore a lot of information content of orders. Chakravarty and Pani (2021) introduces a new data handling paradigm to deal with data from limit order markets. This data paradigm can be used to handle data even beyond the best bid and ask to unravel the information content of the limit orders. The following are a few studies that use order book data.

Cenesizoglu and Grass (2018) investigate the differences between bid- and ask-side liquidity in the NYSE limit order book data. Their focus is to observe the changes around the 2008 financial crisis. Their dataset includes millisecond timestamped snapshots of the LOB, including prices and quantities at its first ten levels on the ask and bid sides for stocks traded on the NYSE. A new measure of LOB liquidity called the marginal cost of immediacy (MCI) is computed for both bid and ask side. This measure is defined as the volume-weighted transaction costs (relative to the mid price) of investors instantaneously accepting all ask (bid) offers in the first ten levels of the book, scaled by the total dollar volume they acquire (sells). The paper also uses this to examine asymmetries in order book liquidity. Although the study utilises level II data, its focus is more to explain events in level I, i.e., trades. It attempts to use liquidity measures (liquidity and imbalance) to predict daily returns and monthly returns. In contrast, the focus in the current paper is on the high resolution view.

Hautsch and Huang (2012) analyze order book data of 30 stocks traded at Euronext Amsterdam. The paper models quotes and depth simultaneously and also the trade and order arrival process. Their quote and depth model is estimated by Johansen’ full information maximum likelihood estimator. They show that limit orders have significant market impacts and cause a dynamic rebalancing of the book. The strength and direction of quote and spread responses depend on the aggressiveness (price) and size(quantity) of incoming orders and the state of the book. Cross-sectional variations in the magnitudes of price impacts are well explained by the underlying trading frequency and relative tick size. Passive order placement through limit orders reveals intention and hence incurs significant market impact even if the order is not been executed. Several exchanges offer hidden order types (such as iceberg orders or hidden orders) to participants.

Hall et al. (2003) model the simultaneous buy and sell trade arrival process in a limit order book market using data from Australian Stock Exchange. The variables selected to reflect the state of the order book are market depth, tightness, and the cumulated volume in the bid / ask queue. Changes of the order book induced by limit order arrivals are captured as time varying covariates. The paper finds that the state of the order book has an impact on the buy/sell intensity.

How private information is assimilated into the markets has been an important theme for early microstructure models. The idea may be viewed as a model of a semi-strong efficient market. For example, Copeland and Galai (1983), Kyle (1985), Glosten and Milgrom (1985), Diamond and Verrecchia (1987), Admati and Pfleiderer (1988), Easley and O’Hara (1987), Easley and O’Hara (1992).

BIDASK: The price innovation in a mark refreshes the limit order book and hence the spread (BIDASK). Before the computation of the spread the order book is sorted on the basis of price and then time. The lowest ask price and the highest bid give the spread.

BIDASK = ASK _lowest - BID _highest

Unlike quote driven markets where the bids and asks from the dealer arrive synchronously, in order driven markets the spreads may turn out to be negative at times. This issue is more real in the markets dominated by algorithmic traders.

BMU, AMU: The transaction rates (BMU, AMU) is the number of transactions (trades) per unit time. The innovation can be computed as a reciprocal of the duration. These we refer to as realised observations. A transaction is a cross between a resting bid (ask) order and an incoming market ask (bid). The resting order defines the bid or ask marker. If such a distinction is not available in the dataset, a transaction should update both bid side and ask side durations.

Assume there is transaction in which a resting bid order is crossed with an incoming market sell order. This is usually referred to as a bid side trade. Such a transaction will result in an update in the order book. Let, t _latest= timestamp of the trade transaction, t _{bidTrade.last}= timestamp of last bidside trade transaction. Define trade duration on bid side as the following:

t _{bidTrade.duration} = t _latest - t _{bidTrade.last}

BMU = 1/t _{bidTrade.duration}

In this case there is no update in the AMU. Similarly,

AMU = 1/t _{askTrade.duration}

BLAMBDA, ALAMBDA: The order rates (BLAMBDA, ALAMBDA) similarly is the number of orders (bids/asks) per unit time. These are represented by reciprocal of bid durations or ask durations. Let, t _latest= timestamp of the latest bid/ask incoming order, t _bid.last= timestamp of last bid. Define bid order duration as:

t _bid.duration = t _latest - t _bid.last

BLAMBDA = 1/t _bid.duration

In this case there is no update in the BLAMBDA. Similarly,

ALAMBDA = 1/t _ask.duration

BIDDEPTH, ASKDEPTH: Traditionally, orders resting in the order book are considered to be providing liquidity to the market. Orders rest as per the price (buy/sell) embedded. Each price is named a price level. There could be a large number of orders at every price. Transaction priority is given on the basis of time within each price level. Similar to Brogaard et al. (2019) and Pani (2021a) we have considered five price levels each on bid and ask side. While any large order can walk down the order book even deeper than five price levels, we note that in liquid markets orders are usually split into child orders. The liquidity residing in the five price levels is significant.

More important is the question, whether the orders resting at different price levels have equal opportunity to trade in an ongoing auction? The orders residing in the best bid and ask prices have the highest probability to trade in an auction. The chance to get traded decreases as we move away from the best bid and ask in either direction. We arbitrarily choose a linearly decreasing scale to enforce the contribution to liquidity of orders resting in any price level.

We define BIDDEPTH as the cumulative volume demanded in the five best price levels on the bid side, weighted by the distance from the best bid. Without the weighting, this is usually qualified as depth profile. Let n ₁to n ₅ be the number of orders in the five best price levels in bid side. Here, subscript 1 represents the best bid price level

Where, V _ij is the quote volume ordered in the jth bid in the ith price level. w ₁ = 1, w ₂ = 0.8, w ₃ = 0.6, w ₄ = 0.4, w ₅ = 0.2. We update the BIDDEPTH (liquidity on the bid side) in the book at each incoming event with quote innovations (Orders –bid, ask, or cancel, Trade).

Similarly, we calculate liquidity on the ask side and refer to it as ASKDEPTH.

Orders are considered as the unit of information (O’Hara (2015)). Orders arrive and depart within the intertrade durations. Our purpose is to investigate intertrade durations and events within them at the tick level. The vector of observations associated with these events are timestamped to at least a microsecond and upto nanosecond level resolution in most modern stock exchanges.

Bedford and Cooke (2002) introduced graphical models called Vines or Regular Vines to represent dependent random variables. A regular vine representation although not essential for a pair copula construction comes in handy when dealing with high dimensional distributions. As Bedford and Cooke (2002) explain vines generalize the Markov trees used in modelling high dimensional distributions but differ from both Markov trees and Bayesian belief nets. Vines allow for various forms of conditional dependence. Vines can be used to specify multivariate distributions by specifying various marginal distributions and the ways in which these marginals are to be coupled.

The joint density in (1) can be written as the product of the marginal density. Consider the vector, X = (X ₁, X ₂, X ₃, X ₄, X ₅, X ₆, X ₇) of random variables with the joint density function f (x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇). This density can be factorised as:

and this decomposition is unique upto a relabelling of the variables. Thus the insight is that a joint distribution function contains the marginal distribution of individual variables and the dependency structure of those variables. Aas et al. (2009) extend marks Bedford and Cooke (2002) to use copulae to describe the dependency structure with a method now known as ’pair copula construction’.

The following general formula can be used to decompose every term in equation (3) into a product of bivariate pair copula and conditional marginal density. v _j is an arbitrarily chosen component of the vector v and v _-j is the vector without the component j.

Thus a multivariate density can be factorised and represented as products of pair-copula and several different conditional probability distributions. It is possible to have several different re-parameterisations for any factorisation.

The pair copula construction method that applies the techniques has been implemented in various use cases. See Aas (2016) for a review of the application in finance. The applications include market risk, capital asset pricing, credit risk, operational risk, liquidity risk, systematic risk, portfolio optimisation and option pricing. Our paper adds to literature as it is an attempt to use pair copula construction in market microstructure models using ultra high frequency intraday market data.

We proceed with the following steps:

There are 3727 bivariate relationships (edges in a network with the variables representing the nodes) in our sample. They represent relationships with highest measure of correlation (Kendall’s Tau). The count (percentage) of these bivariate relationships and the fitted copulas are presented in this section. Table 2 presents the count of the relationships in the sample.

Bivariate relationships and the top 10 fitted copulas pairs contribute to 75.5% of all the relationships in the first tree. The significant pair is between the bid side and ask side trade duration (BMU-AMU). 20.1% of the bivariate pairs is from this pair and these are, to a large extent (19.5%) fitted with the t-copula, suggesting a tail relationship between the two variables. Later in this section we also evaluate the strength (relative strength in the first tree) of these variables. The strong dependence between trade duration is expected based on a large volume of market microstructure literature. However, the nature of the best fitted copula (t-copula) is a significant result. It comments on the nature of trading in the algorithmic world where buyers and sellers trade with strong stop losses and within narrow price bands. It also indicates that there exists thresholds of transaction success or failure at which the traders move between market orders and limit orders. The literature reported in section 2 suggests such changes to be intermediated by changes in spread or liquidity (depth).

Interestingly, bid side durations (BMU) is present in 3 among the top 10 pairs, while the ask side durations (AMU) is related only to BMU. The bid side durations mediate the information pathways between the ask side durations and other variables in many instances - the high BMUdeg (2.08), as given in Table 3, emphaises this point.

The following essential pair is the bid side and ask side order durations (BLAMBDA-ALAMBDA: 11.6%). Of this, tail relationships exist in 5.1% of the pairs fitted with bb7, bb8, clayton, and t copulas, while in 4.7% of pairs, explained by frank copulas there is no tail relationship. Thus, order durations are important. This finding is in line with the results in Brogaard et al (2019) and Pani (2021). In an algorithmic trading world, the order durations are more important than liquidity or spread. The fear of stale quotes being picked-off leads to a behaviour of trying to time the arrival. Availability of technology enables the execution. Van Kervel et al. (2020) finds that Institutional investors coordinate their trading in real-time.

When compared with the trade durations, the order durations carry lesser importance and a relatively weaker dependence relationship. This is expected when observed from a queue theoretic framework. Trades are signals with higher precision; they are expected to carry more information, leading to the algorithmic and non-algorithmic traders emphasising the cues from trades, thus reacting through the durations. Orders possess relatively lower precision and maybe some noise. The more sophisticated algorithmic traders use these signals. As discussed in section 5.2 the bid durations (BMU) is a stronger variable. One reason is that it interacts with both order durations (14%). Further the order durations interact with the volume or liquidity variables that may be considered as the relevant inventory (inventory supplied into the limit order book). This paper has considered orders from the five best price levels on the bid side as well as ask side. In market microstructure, the best bid and best ask prices are considered as the expectation of the trade prices. When order durations from the best bid - best ask are considered the current inference may not be obvious. We have not examined this in our current design.

The number of dependence relationships between the order durations and liquidity on the same side (BLAMBDA-BIDDEPTH, ALAMBDA-ASKDEPTH) and the opposite side (BLAMBDA-ASKDEPTH, ALAMBDA-BIDDEPTH) is similar. This block of pairs constitutes 16.9% of the total. The buyers and sellers attain such a balance between them, through coordinated trading responses of the algorithms. Algorithms achieve such optimisation at the trading system level through continuous monitoring, adjusted responses and feedback loops between buy and sell side.

The dependence relationships between the order durations (BLAMBDA,ALAMBDA) with the trade durations (BMU, AMU) and the liquidity or relevant inventory (BIDDEPTH, ASKDEPTH) is 31.4% of the pairs. The BLAMBDAdeg (1.47) and ALAMBDAdeg (1.52) as given in table 3, only emphasise this aspect. The degree measure for the order variables is consistent through out the trading day (other than the first period where is low). The fitted copulas is diverse. We infer that the role of order durations is important, but the nature and strength of the relationship is an aspect of the microstructure of the traded stock.

The liquidity or volume on bid and ask side (BIDDEPTH-ASKDEPTH pair) occupy 7.1% of the pairs. This relationship is described by several copulas, the more prominent in number (3.1%) being t-copula. The strength of the relationship is lower than the trade durations. In the top 10 pairs, the spread (BIDASK) appears only with the depth (BIDASK-ASKDEPTH-4.5% and BIDASK-BIDDEPTH: 4%). A change in spread can change the response of algorithmic traders from liquidity supply to liquidity demand or vice-versa. The spread continues to be an important variable with reasonable strength (discussed in section 5.2). However, we observe that it is not an active variable in all periods leading to lower occurence.

The share of the variable is computed using the normalised cumulative Kendall’s Tau attributed to the variable in the first tree of pair copula construction. Fig. 1 describes the distribution of the strength of the variables. The distributions in Fig. 1 illustrate remarkable properties of limit order markets. The mechanism or market design leads to well defined central tendencies for each variable. At the same time, the presence of outliers in the data illustrates differences arising most likely from market microstructure properties of the markets. The presence or absence of traders, market makers, venues where the stock is traded, tick size, lot size, value of each share, the bid-ask spread. The strength of BIDASK is relatively low along with the presence of outliers. We examine this later in this section.

Table 3 gives the degree, the number of variables with which each variable is paired, to describe dependence. Table 4 shows the time-varying nature (dynamic through intraday periods of trading) of the strength of these variables. Additionally, this data is illustrated in fig. 2. The BIDASK has good relative shares in the first half of the trading day and later it has low relative share in the dependence structure. All the other variables show a time-varying nature of relative share in dependence. There are periods where they go out of the dependence structure and periods where they enjoy the highest share in the dependence structure.