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

2.1 Order 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.

2.2 Order 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.

risk ( Δ ) = flow ( OBI ) size ( V ) ( α 0 + α 1 dyn ( Δ ) + α 2 queue ( Δ ) )(5)

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)

f l o w ( O B I ) = μ O B I(7)

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.

S i z e ( V ) = exp ( V η )(8)

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.

d n y ( △ ) = σ d y n ( △ − B M d y n σ d y n ) 2(9)

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.

q u e u e ( △ ) = B Q △(10)

2.3 Order 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:

4 Experimental setup, results and discussions

Both the Micro model and the replication of the CB are implemented in the same software framework in order to keep differences to the minimum. Actually, the microtrading agent is an extension of the original CB trader, overwriting only the placement decision. Everything else – e.g., matching engine, agent pooling, order cancellation, random seeds, market making – is kept unchanged.

The parameters of the CB implementation have the same values as in Cui and Brabazon (2012), which were originally estimated from a dataset for Barclays Capital (BARC.L) traded at London Stock Exchange. 16 The Micro model maintains the same parametrization as in Cui and Brabazon (2012), where it applies: λ _o  = 0.9847, λ _m  + λ _l  = 0.008, λ _c  = 0.0073, μ _size  = 8.2166, σ _size  = 0.9545, xmin offspr = 0.05, β _offspr  = 1.7248, initial mid-quote price 300.00, initial spread 0.50, tick size 0.01. For convenience, all model parameters are summarized in Table 1.

The remaining parameters regarding the microtrading strategy are set in order to replicate the same order type frequencies as the ones generated by the CB model: 4% market orders, 10% in-spread and 86% off-spread limit orders. Also, we have tried to roughly reproduce the stylized facts discussed in the rest of this section, but no intensive or automated calibration has been pursued. The chosen set of parameters is: book depth levels N = 3, OBI base μ = 2, size penalty exponent η = 0.8, α ₀ = 0.1, α ₁ = 0.5, α ₂ = 0.25, β = 2.5. The parameters of the distribution mixture associated with the sense of urgency λ _u are: μ ₁ = 0.4, σ ₁ = 0.2, μ ₂ = 1.1, σ ₂ = 0.2, and the probability of drawing from the first normal is 40%. The two benchmarks BM _is  = BM _dyn are chosen as the exponential moving average of trading price p ¯ t = 0.95 p ¯ t - 1 + 0.05 p t, because this indicator is more stable than the mid-price. The index t corresponds to the trade price time series and thus p ¯ is updated every new trade. Similar, the two volatility bands σ is = 7 σ ¯ from (4) and σ dyn = 7.5 σ ¯ from (9) are multipliers of the estimated time-varying instant standard deviation computed as σ ¯ t = 0.95 σ ¯ t - 1 2 + 0.05 r t 2, where r is the percentage return. Additionally, the default average daily volume is initialized with ADV = 77m. 17

For exemplification and according to the adage “a picture is worth a thousand words”, single one-day realizations of the price time-series for each of the two models are depicted in Fig. 5. At the beginning of each trading day, the model is warmed up for 3,600,000 milliseconds (1 hour). Both models are run for 30 artificial days, each day with a different random seed and all data, except for the warm-up period, is aggregated in one dataset over which two stylized facts are investigated. “Stylized facts”, i.e. empirical regularities exhibited by a wide range of financial time series, are commonly used to validate ABM designs and parameterizations. A wide range of stylized facts, both for high-frequency and aggregated data, are described in the literature, e.g., Chen et al. (2012), Cont (2001, 2011), Daniel (2006), Pacurar (2008). The class of intraday stylized facts can be associated to transaction data, order book shape and order flow. In this paper, we have chosen one stylized fact related to limit orders which states that the distribution of the relative limit prices decays asymptotically as a power-law, and one associated with market orders which were found to generate a non-linear concave price impact function of trade size. No stylized facts related to return or order flow are selected since both models assume purely random investment decisions and inter-events durations.

5 Conclusion and outlook

We have modeled the agents’ order placement decision as an optimization problem which minimizes the risk adjusted execution cost, taking into consideration various microstructure factors – such as order book liquidity, order flow proxied by the order book imbalance and transient volatility, as well as intrinsic agents characteristics – such as the sense of urgency. We have derived an order submission strategy based on an iterative numerical procedure which allows for the efficient identification of the potential optimal limit price, taking into account the effective state of the order book. Next, we have integrated the order submission model into a zero-intelligence agent based model, providing a realistic test environment which can be used for assessing the collective behavior of high-frequency trading algorithms, its implications on financial market systemic risk, as well as for virtual testing of potential regulatory measures.

The results show that, when replacing random order placement decisions by a micro-trading strategy, the intraday market properties are more similar to those of real markets, at least with respect to two high-frequency stylized facts. The model comprising the microstructure-based order placement component has successfully reproduced the power-law tail of the relative price distribution of off-spread limit orders, even if there is no explicit power-law component assumed nor hard-coded into the agents’ design – thus it can be considered an emergent property. Regarding market orders, both the binned-average price impact as well as the individual price impact functions exhibit a realistic concave shape – a trace of rational selective trading. Moreover, the conducted robustness checks show that the results are valid not only for the specific parameter setting. On the opposite, in the absence of intelligent trading, the expected market price impact shape is convex – confirmed by the results obtained with the alternative zero-intelligence agent based model. It is to be noted that we did not intend, at this point, to explain what causes the results, but rather try to obtain a realistic high-frequency simulation environment. Extensive testing of current and alternative functional assumptions, which could also shed more light on the causal relations between model ingredients and emerging behaviors, requires the implementation of a proper calibration and model selection procedure. Given the complexity of the model, this is not a trivial task and will be addressed separately in a future paper.

However, both the order submission model, as well as the underlying agent based model, have some limitations which could be tackled in future implementations. The investment process is purely random and there is no relationship between agents’ type and/or wealth and the size of their orders – usually volumes are correlated with strategy time-frame – or with risk aversion and trading urgency, respectively. Moreover, no learning component is implemented – micro-trading strategies are constant during the short running time span of a single trading session. Even if this assumption seems reasonable, agents are not able to adapt and exploit current market conditions and also, as consequence, market conditions do not change over time. Basically, the agents are completely uninformed and could easily be taken advantage of by third party strategies. On the other side, the implementation of various missing model components could lead to significant improvement, as well as to relaxing some of the current assumptions – e.g., the bimodal distribution of urgency coefficients could be removed if heterogeneity is introduced through explicit modeling of the investment decision.

If the process of investment would be developed, order execution could be enriched by adding algorithmic trading techniques, which consist in splitting large orders’ submission over time. The actual order placement model can also be extended by taking into consideration other factors, e.g., intraday market trend, time of day, available trading time, prior order aggressiveness based on trading events clustering. Several other stylized facts related to order book shape, e.g., order book gaps, could be analyzed. Also, even under the current conditions where the investment process is stochastic, some stylized facts related to price returns, e.g., volatility clustering, or to order flow might emerge. Even more interesting would be the analysis of possible feedback loops around price volatility and order book imbalance. Moreover, a systematic model calibration encompassing both the estimation of parameters, as well as the selection of individual components, with the objective of fitting various quantitative behavioral aspects, could be pursued. Also, besides a highly liquid stock, the model should also be analyzed under various other market conditions, such those provided by a small-cap, and the potential differences assessed.

Acknowledgments

The author would like to thank Deutscher Akademischer Austauschdienst (DAAD) for the awarded PhD scholarship. Also, the author is grateful to Prof. Peter Winker, two anonymous referees and participants of the GENED meeting at TU Darmstadt (29th–30th September 2014) and of the CFE-ERCIM 2014 Pisa conference for their helpful comments and suggestions.

6 Appendix

The formalization of the optimization problem for the market-limit order split

In the market-limit order split case, the binary decision variable _M is replaced with continuous 0 ≤ m ≤ 1, representing the share of the original order executed as a market order, the complementary fraction (1 - m) being placed as a limit order. The optimization problem and the objective function in (1) become: 23

min m , Δ f ( m , Δ )(18) f ( m , Δ ) = c o s t ( m , Δ ) + λ u ( 1 − m ) r i s k ( m , Δ ) ︸ adjusted risk for limit orders(19)

The implementation shortfall is now the result of the mixed execution, while the wrapping cost function can be easily rewritten:

imp . sh ( m , Δ ) = m ( BM is + mk . imp ( m V ) ) ︸ market order part(20) + ( 1 - m ) ( BM is - Δ ) ︸ limit order part = BM is + m mk . imp ( m V ) - ( 1 - m ) Δ(21) c o s t ( m , Δ ) = { i m p . s h ( m , Δ ) for i m p . s h ( · ) ≤ σ i s β σ i s ( i m p . s h ( m , Δ ) / σ i s ) 2 for i m p . s h ( · ) > σ i s(22)

The non-execution risk remains unchanged, but the opportunity component in (8) is now a function of the limit order size (1 - m) V.

risk ( m , Δ ) = flow ( OBI ) size ( ( 1 - m ) V ) ( α 0 + α 1 dyn ( Δ ) + α 2 queue ( Δ ) )(23)