Microstructure-based order placement in a continuous double auction agent based model

2.1Order placement problem

The trading process is a trade-off between execution cost and delay risk, and comprises two types of trading decisions: order scheduling (break-up large orders or trade all-at-once) and order submission (type and placement). In this paper we will tackle only the latter, which is actually the foundation for designing a trading-driven ABM. Proper execution of individual orders involves a set of micro-trading decisions, i.e. the order can be articulated into a market order, a limit order or can be split between a market order and a limit order. In other words, the trader faces a trade-off between execution certainty and a more favorable transaction price. At one extreme, a market order does not carry any execution risk, but has a higher transaction cost consisting in market impact. On the other side, preferring a limit order saves the cost of immediacy associated with the market order alternative and can further improve trading costs with the relative limit distance. The drawback is that limit orders encounter the risk of remaining unexecuted, as they are conditioned on the uncertain future event of being matched by a counter party (for a broader introduction into transaction costs, market impact and timing risk, see Kissell and Glantz (2003)).

In the microstructure literature, a large range of factors driving traders to act more aggressively or passively have been identified, and which can be classified into liquidity-, price- and time-based factors (for a comprehensive review see Johnson (2010)). Firstly, the choice in favor of a market order is found to be highly dependent on the instant liquidity reflected by the market tightness, i.e. the bid-ask spread, and by the order book height, i.e. the potential price impact of a market order walking up the book. Aggressive orders are more probable when the cost of immediacy is low, while a high liquidity cost due to higher spreads encourages liquidity suppliers to place more limit orders. Beber and Caglio (2005) found also evidence for a non-linear relation, showing that particularly wide spreads favor in-spread limit orders rather than market or far away off-spread orders. Pascual and Veredas (2009) concluded that wide spreads discourage especially small market orders, increasing the frequency of larger market orders. With respect to order size in general, Cont and Kukanov (2012) state that market orders are usually larger than limit orders.

Another liquidity measure, i.e. order book depth, influences the general order aggressiveness in two ways. An overall supply-demand imbalance drives traders to price their orders more aggressively when their side of the book is crowded in order to increase their order execution probability (competition effect). Conversely, traders become less aggressive when the opposite side is thicker, forecasting a favorable short-term order flow (strategic effect). If only the thickness of the opposite side of the market is taken into consideration, both Beber and Caglio (2005) and Pascual and Veredas (2009) identify an asymmetricbehavior – sellers are more impatient to trade than buyers and thus are more willing to take advantage of the available liquidity by issuing larger aggressive orders; contrary, buyers show more patience and place less aggressive orders.

Zovko and Farmer (2002) found that short-term volatility at least partially drives the relative limit prices and also suggest that such a feedback loop may contribute to volatility clustering. On one side, the probability of executing further placed limit orders increases while, on the other side, the picking-off risk due to adverse selection is also higher. Another price-based factor is the momentum indicator proposed by Beber and Caglio (2005), defined as the ratio between the current price and its exponential moving average. The direction of the short-term market trend asymmetrically affects the execution probability of limit orders and ultimately leads to a change in order pricing aggressiveness. Moreover, higher previously traded volumes, acting as a proxy for market information, lead to an even bigger increase in aggressiveness in the direction of the market trend.

Several papers have studied various order placement strategies for limit order book markets in a stochastic process based framework. Lillo (2007) defines an optimization problem within the framework of expected utility maximization, where the probability for a limit order of getting executed is given by the first passage time distribution of a random walk, i.e. the probability that the stochastic price reaches the limit price Δ by a certain time – thus, the hitting time probability is a function of Δ, time horizon and volatility. Kovaleva and Iori (2012) develop a model which discriminates between placing a market order and a limit order. In their model, the total time-to-fill is not only given by the first passage time distribution which measures the probability of reaching the beginning of the queue, but also by a random delay – sampled from an exponential distribution with constant intensity – which stands for the order’s effective execution. By assuming stochastic log-normal processes for the trajectories of the bid and ask prices, an analytical solution is identified by maximizing a mean-variance utility function. Cont and Kukanov (2012) formulate a convex optimization problem for the decision of splitting between a market and a limit order placed at the best bid or ask. The optimization function penalizes the execution price with an execution risk which takes into consideration factors such as order size, the existing queue size at the front of the book and the cumulative distribution function of the queue outflow. If a functional form for the outflow CDF is assumed, a parametric numerical solution can be computed or, alternatively, a non-parametric solution when the empirical distribution is based on past order fills.

2.2Order placement model

The main difference from the previous stochastic models is that our contributed model does not make any assumptions about the general distributions of the underlying price or order flow processes, but takes into consideration only the current market microstructure factors and intrinsic agent characteristics – the former can be easily observed within the limit order book trading mechanism. Therefore, the proposed submission strategy, applied as an ABM component, reflects the different facets of the current market state making agents more reactive and the entire system more endogenous, in the spirit of ABM where reflexivity is a key concept. The drawback is that compact analytical solutions are not derivable anymore and, instead, algorithmic decision rules have to be determined.

By the way we formalize the optimization problem, with the goal of minimizing the risk adjusted execution cost, our model is more similar to Cont and Kukanov (2012). One distinction is that the decision is restricted to choosing between a market and a limit order for the entire quantity, i.e. no market-limit order split.⁴ On the other side, the limit order is allowed to be placed further away from the best quotes and thus a supplementary decision about the optimal limit price has to be made.

Specifically, the objective function f ( _M , Δ) describes the trade-off between execution cost and non-execution risk, balanced by agent’s sense of urgency λ _u . The sense of urgency can reflect a mix of risk aversion, degree of informativeness, strategy time-frame or just time pressure, i.e. waiting time of the trading process. The decision variables are the binary _M , discriminating between a market and a limit order, and the relative limit distance Δ for the limit order case ( _M  = 0).

(1)

(2)

The cost function cost ( _M , Δ) captures what is known as the implementation shortfall, i.e. the difference between a given benchmark BM _is and the effective order execution price as defined in Equation (3).⁵ The market impact function mk . imp (V) is influenced by the current order book state and can be computed as the percentage change in price where the entire order size is executed. Actually, all measures involved (Δ, BM _is , mk . imp) are scaled as percentage returns relative to a base price, which is the best bid for sell orders or the best ask for buy orders, respectively.⁶ An example of how these measures are related for different types of buy orders is depicted in Fig. 3. Finally, the order volume V is usually expressed as a percentage of the average daily volume (ADV).

(3)

The cost function in Equation (4) wraps around the implementation shortfall by adjusting it with a volatility-threshold downside price change penalty in order to discourage execution prices which are too far away beyond σ _is , i.e. a multiple of the short-term volatility σ, corresponding to highly unfavorable executions.⁷ The single parameter β > 1 controls for the weight of this penalty. The choices for this particular functional form, as well as for the following relations between decision variables and microstructure factors, represent sensible approximations of the reported empirical behaviors. The actual functional forms are unknown and alternative choices could be justified only through extensive calibration or if representative microdata would exist.

(4)

On the risk side, the execution probability of a given limit order depends on (i) order flow proxied by the order book imbalance flow (OBI), (ii) short-term market volatility dyn (Δ),⁸ and (iii) order queue in front of the limit order queue (Δ). Moreover, (iv) an opportunity cost size (V) as a penalty function of order size is also included. The aggregate non-execution risk function risk (Δ) is given by (5). The parameters α ₀, α ₁, α ₂ tune the general preference for market and limit orders, as well as the distribution of relative limit distances. Eventually, each individual decision will also depend on the agent’s urgency factor λ _u and on the specific state of the market.

A key model component is the expected order-flow flow (OBI), which drives the short-term price returns and affects the execution probability of outstanding limit orders. In this implementation, order flow is more liquidity- rather than price-based and relies on the Order Book Imbalance indicator (OBI). OBI quantifies the difference between the cumulated volumes up to a certain depth level N on each side of the order book. By its definition in (6), OBI takes values between -1 and 1, the extremes corresponding to the cases when one book-side is empty. When the order book is unfavorable leaned, OBI is positive, flow (OBI) is larger and the ultimate non-execution risk increases leading to a bigger incentive of placing more aggressive limit orders.⁹ Overall, flow (OBI) takes values between 1/μ and μ – asymmetric around zero – meaning that it also acts as a complementary penalty factor when OBI is unfavorable. The constant μ in Equation (7) weights the relative importance of the order flow effect in assessing the aggregate non-execution risk and in setting the optimal relative limit distance Δ, e.g., OBI is neutral when μ = 1.

(6)

The opportunity component size (V) is an increasing function of order size, always bigger than one because of the exponential.¹⁰ The intuition behind this penalty is that an outstanding limit order is associated with a “signaling” risk, as well as a “jump-over” effect – the bigger the order, the more likely other limit orders get placed in front. Furthermore, a non-executed limit order is expected to be transformed into a market order at a worse transaction price than the initial one, because the market is assumed to have moved in an unfavorable direction, i.e. adverse selection.

The functional form of the market dynamics effect dyn (Δ) describes a sub-linear increasing risk of non-execution for limit orders inside the volatility bands, defined by a central benchmark BM _dyn and σ _dyn , expressed as a multiple of the short-term volatility. When the relative limit distance is outside this interval, the risk increases faster, penalizing far away orders.¹¹ Potential candidates for BM _dyn can be for example the bid-ask midpoint, last trade price or previous close price.

The effective order queue effect queue (Δ) reflects the cumulative size of the book queue BQ _Δ situated in front of the client limit order – placed at the relative distance Δ. The order queue can be immediately computed within an observable order book and is expressed as a percentage of ADV, without assuming any functional form.

2.3Order placement strategy

There is one issue in trying to analytically deal with the optimization problem defined in Subsection 2.2 – unless a functional form for the order queue effect queue (Δ) is assumed, an analytical solution cannot be derived. However, if a functional form for queue (Δ) based, for example, on an average order book shape would be assumed, any connection to the temporal specific structure of the book would be lost. Therefore, we implement an iterative numerical procedure for identifying the optimal relative distance Δ ^* of a potential limit order.¹²

By replacing imp . sh ( _M , Δ) in (4) with its definition in (3), the conditions of the multi-part cost function can be rewritten and the optimization problem can be forked accordingly. The selection decision between a market or a limit order placed at relative distance Δ ^* becomes equivalent to choosing the minimum of the following three branches: f ( _M ), f (Δ ^* ≥ BM _is  - σ _is |not ( _M )) and f (Δ ^* < BM _is  - σ _is |not ( _M )).

I. When _M  = 1 (market order, negative price change relative to BM _is ), one can discriminate between two cases – inside or outside the volatility bands – depending on the market impact size:

(11)

II. When _M  = 0 (limit order) and Δ ≥ BM _is  - σ _is (positive price change or negative price change smaller than the volatility threshold, i.e. inside the bands – identified in Fig. 4 (top) with Δ ^II+ and Δ ^II-, respectively). Let A = λ _u  flow (OBI) size (V). It follows that:

(12)

III. When _M  = 0 (limit order) and Δ < BM _is  - σ _is (negative price change larger than the volatility threshold, i.e. outside the bands – Δ ^III in bottom Fig. 4). As Δ > 0 (non-crossing limit orders), the precondition BM _is  - σ _is  > 0 must apply.

(13)

In a simulation framework, the non-execution risk function can be based on the effective order queue component, which takes into account the actual state of the order book, assuring the endogeneity of the model. This queue function increases in steps at random values because of the expected book gaps and stochastic depth sizes at various book levels. Thus, the queue function is not derivable and a numerical procedure has to be implemented. We propose an iterative procedure, where a trajectory of potential solutions Δ _i starting at 0 – corresponding to a spread limit order – is evaluated step by step with respect to the objective of minimizing f (Δ _i |not ( _M )) and the best candidate Δ ^* is stored. Finally, the fitness of the best candidate for a limit order f (Δ ^*|not ( _M )) can be compared to the fitness of a market order f ( _M ) and the appropriate order type can be chosen.

If the queue effect is temporarily ignored, i.e.α ₂ = 0, the functional forms of the cost and risk functions can be exploited with the goal of identifying a stopping point for the numerical procedure, reached where the derivative of the fitness function g (Δ) = f (Δ|not ( _M ) , α ₂ = 0) is zero.¹³ Identifying this critical point also allows for considering a sparser search space, by jumping from one book level to the next – since the only inflexion point is at the end of the search interval, all intermediary potential Δ _i  ≤ Δ ^S situated within the order-book gaps can be ignored. Thus, the stopping point corresponding to the second branch, i.e. Equation (12), is given by:

If BM _is  - σ _is  > 0, the solution corresponding to the third branch, i.e. Equation (13), is:

4.1Market price impact

The market impact function reflects the relationship between market order size and price impact, captured by the shift between the pre-trade and post-trade market equilibrium. Lillo et al. (2002, 2003) provide a method for computing the average market impact. Firstly, trades with the same time-stamp are aggregated and treated as a single transaction, as these are assumed to be part of a single market order which is matched against several outstanding limit orders. The price impact associated with each transaction is reflected by the difference in the logarithmic mid-quote price. Originally, the transaction size was measured in dollars, but we adopt the approach from Cui and Brabazon (2012) where the size of the market order is relative to the total daily trading volume. Finally, the data is divided into ten bins based on order size and the average price impact of each bin is computed. We also remove the upper outliers with respect to order size using a modified interquartile range rule,¹⁸ otherwise the number of observations in the higher bins would be very low if not even zero.

A functional form of the market price impact is provided in the literature – Lillo et al. (2002, 2003) and Plerou et al. (2002) consider the empirical market impact function for trade by trade data to be a power function of order size η ν ^γ , with exponent γ taking values between 0.2 and 0.6 for stocks traded at New York Stock Exchange. Especially for London Stock Exchange data, Farmer et al. (2004) have computed an estimate γ = 0.26 for a selection of three highly capitalized stocks – Lloyds (LLOY), Shell (SHEL) and Vodafone (VOD).¹⁹ It must be noted that this functional form is only an average property of the entire market. Since the order timing process is not observable and also not completely random – one can assume at least some intelligent trading taking into consideration available market liquidity – we are confronted with an endogeneity issue which does not allow for the identification of the “true” relationship between order size and market impact only by analyzing historical transaction data. Moreover, if the unconditional impact function would be concave, there would be no incentive to split a large order, as the total market impact of the smaller trades would be larger than the initial impact. Actually, Weber and Rosenow (2005) found that a virtual price impact function – computed by inverting the available order book depth as a function of return – is convex and is increasing faster than the concave average price impact function associated with effective market orders. Still, the overall average market impact represents a stylized fact which should emerge in a market with rational tradingagents.

The average market impact computed on binneddata – as in the standard procedure described in the beginning of the current section – is depicted in the left plots of Fig. 6. In order to identify the relation with respect to the normalized order size, we also estimate the coefficients of the power function β ₁ v ^{β ₂} using nonlinear least squares. Before discussing the results, it is worth mentioning that these are quite robust to small variations of the considered parameters. The ad-hoc calibration could not influence significantly – neither improve, nor deteriorate – the results in any qualitative and only very slightly in a quantitative way. The robustness check procedure and results are detailed in Subsection 4.3.

The results presented in Table 2 show a convex market impact function for the CB model, which is in line with the instantaneous price impact of Weber and Rosenow (2005) expected for randomly timed trading. On the other side, the Micro model associated function is concave with an exponent β ₂ = 0.67. Both estimates are statistically significant different from zero and one – the restriction β ₂ = 1 has been rejected by the likelihood-ratio test. Obviously, the market impact associated with the Micro model is much more realistic than in the random approach, even if the exponent is larger than the empirical estimates. The difference can be explained by the lack of a thoroughly automated calibration, on one side, and by the major simplifications with respect to the investment process, on another side.

Since the sample size for the binned data is very small, we repeat the estimations on all available data, still the results are very similar.²⁰ Both the individual as well as the average market impact are smaller in the Micro model, which can be explained by the selective trading strategy pursued by the agents aware of their market impact. This result is also in accordance with Weber and Rosenow (2005) who found that the virtual price impact is more than four times stronger than the actual one. The right panels of Fig. 6 present scatter plots (hexagonal binned) of the market impact for all transactions, as well as the fitted market impact function. In the CB model case, we observe that the data on the market impact axis is slightly bimodal, which raises the question of the validity of the mean estimate especially for the superior bins. The mode close to 0.001 is caused by the market maker’s intervention to prevent the order book from getting empty – even during executing of transactions – and resetting the bid-ask spread to its default value.²¹

4.2Relative price distribution of the off-spread limit orders

Zovko and Farmer (2002) define the relative limit price of a limit order δ as the difference between the limit price and the best quote on the same side of the market, i.e. the best bid for a buy order and the best ask for a sell order. Furthermore, the standard procedure introduced in Zovko and Farmer (2002) takes into consideration only off-spread limit orders, i.e. only limit orders with positive relative prices (δ > 0), while the rest – crossing, in-spread and spread – are discarded. The distribution of δ was found to decay asymptotically as a power-law, meaning that even if most of the limit orders are concentrated close to the best quotes, there are enough orders which are priced much less aggressively such that the distribution exhibits long-tails. Different values for the characteristic exponent α associated with the power law probability distribution function p (δ) ∼ δ ^-α have been computed in the literature: Zovko and Farmer (2002) found α ∼ 1.49 for data from London Stock Exchange, while Bouchaud et al. (2002) and Potters and Bouchaud (2003) found α ∼ 1.6 for Euronext and NASDAQ.

Fig. 7 presents the relative price distributions of the off-spread limit orders generated by the two compared models, both as regular histograms and as Zipf plots.²² In the Micro model case, the power-law tail of the relative limit price distribution can graphically be assessed by the approximately linear shape of the Zipf’s plot on a log-log scale. On the other side, the CB model does not reproduce the same stylized fact due to the design of the off-spread limit order random number generator. However, other intraday ABMs which explicitly model the economics of investment decision, such as Chiarella et al. (2009), could be able to replicate this stylized fact.

We have also estimated the distribution exponent αˆ=1.66 for the Micro model, when xmin is set to 0.5, using the R package ‘poweRlaw’ which was developed based on the Santa Fe Institute recommendations, although the estimation is highly sensitive to the chosen value of the xmin cut-off parameter. A key role, in replicating this stylized fact, is played by the bimodal distribution of urgency coefficients λ _u , which allows for a number of passive limit orders with no chance of immediate execution to be sent by agents with very low urgency. Otherwise, if the investment process would be considered, these passive orders could be the result of different agents’ evaluations or strategies, e.g., fundamentalists might place orders away from the current trading price.

4.3Robustness checks

Since the results of the simulation presented in the previous two subsections are conditioned on the specific set of model parameters, it is important to analyze whether these qualitative findings are robust to the variation of the default parametrization. Similar to Harting (2015), we have run 3,000 new Monte Carlo simulations over 100 different configurations – 30 runs per individual configuration, where each of the 14 Micro model specific parameters in Table 1 are determined by independent random draws from uniform distributions with a 60% range centered around the default model settings. The results with respect to the estimated exponent of the market impact function (β ₂), both for the binned and individual data, are captured in the top and middle plots of Fig. 8 and show that the concavity property is not lost. Also, in the case of the power-law function exponent (α), the distribution of the estimator and the 95% confidence bands presented in the bottom plot of Fig. 8 exhibit values close to the empirically estimated coefficient. Even if the selected parametrization in not totally arbitrary, the qualitative results – consisting in the reproduction of two stylized facts – are apparently not affected by the variation of the initial parametrization, underlining the robustness of the results.