Optimal Execution in Presence of Short-Term Trading

Hidden Order. Criscuolo, A. M. and Waelbroeck, H.

Optimal Execution in Presence of Short-Term Trading Adriana M. Criscuolo1 and Henri Waelbroeck Portware, LLC, 233 Broadway, 24th Floor, New York, NY, 10279, USA (February 2014) Abstract

Starting from basic hypotheses on how footprints from hidden orders are interpreted by short-term traders, we derive a fair price model that predicts market impact for non-uniform participation rate schedules. We use this model to derive an optimal execution schedule for a risk-averse trader. The optimal schedule delays front-loading to avoid the information shock of an abrupt start. We also consider optimal strategies with respect to the Volume-Weighted Average Price (VWAP) benchmark. We show that the VWAP-optimized schedule for a large order is similar to the risk-averse one. In an example, we compute the cost of front-loading, and the additional cost of the information shock that results from an aggressive trade start.

Key Words: Hidden order, information leakage, market impact, trading cost, optimal trade execution.

1

Correspondence Address: Research Group, Portware, LLC, 233 Broadway, 24 th Floor, New York, NY, 10279, USA. Email: [email protected]. Fax number: 1-212-5714634.

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Introduction In portfolio management, implementation shortfall is the portfolio loss resulting from the difference between the realized price of a trade and the market price at the time of the decision. The aggregate cost society pays for active investing has been estimated to represent annually for the US stock market (French, 2008); at a 2011 market valuation this would represent . Part of this cost is realized through the implementation shortfall of institutional trades. However, fund performance could be ameliorated by the design of execution strategies that minimize implementation shortfall. In this paper, we consider the optimal execution of a single large portfolio transaction, realized through a hidden order that must be split into smaller slices and executed incrementally over time. We consider the case where the execution proceeds without interruption, at a sufficient rate and for a long enough time to cause observable effects on the order flow. A participation rate schedule consists in a sequence of participation rates, each of them applied in a particular interval. The optimization problem is to find the participation rate schedule that minimizes the risk-adjusted cost while completing the trade in a given maximum amount of time. One must first specify a model for market impact to optimize the risk-adjusted cost. Different impact models lead to different optimal execution solutions (Kyle, 1985), (Bertismas and Lo, 1998), (Almgren and Chriss, 2000), (Almgren et al., 2005), (Obizhaeva and Wang, 2013). From the hypotheses established in our theory, we will obtain a market impact function that, in each transactional interval, depends non-linearly on both the total number of shares traded from the start of the trade until the observed time, and the participation rate at that interval. Additionally, impact at each step will be non-stationary because of its dependency on the time elapsed from the start of the trade. In comparison, Almgren and Chriss assumed that the market prices are driven by an arithmetic random walk overlaid with a stationary market impact process. Impact is proposed to be the linear sum of permanent and temporary components, where the permanent impact depends linearly on the number of traded shares and the temporary impact is a linear function of the trading velocity. They find that, with risk-aversion, it is optimal to trade fastest at the beginning and slow down as the trade progresses (front-loaded execution) according to a hyperbolic sine function. This type of front-loaded participation rate profile is widely used by industry participants to reduce risk, yet it is also recognized that front-loading executions increases impact early in the trade resulting in higher trading costs. A related concern is that liquidity exhaustion Page 2 of 47


or increased signaling risk could also lead to a higher variance in trade results (Hora, 2006), defeating the main purpose of frontloading. The non-linearity of temporary impact in the trading velocity has been addressed previously in (Almgren, 2003), (Almgren et al., 2005); the optimization method has also been adjusted for non-linear phenomenological models of temporary impact (Loeb, 1983; Lillo et al., 2003). However, these studies share the assumption that the effect of trading on price is stationary for a uniform participation schedule; i.e., temporary impact remains constant for the duration of the trade (explicitly independent of time). Instead, practitioners find that total impact is a concave function of the amount of time that a uniform participation algorithm has been engaged. In previous work (Farmer et al., 2013) (FGLW) proposed a nonstationary model for market impact for constant participation rate . In this theory, the basic assumption is that traders are able to detect the existence of a hidden order by observing an imbalance in the order flow, caused by the execution algorithm. Temporary impact represents expectations of further activity from this algorithm. Since a hidden order that has been observed for a longer period is more likely to continue, temporary impact grows. The continuation probability depends on the distribution of hidden order sizes. If one assumes that the hidden order size has a Pareto distribution with a tail exponent of 1.5, the model predicts that temporary and permanent impact of a constant-participation algorithm both grow as a square root of time. Integrating over time to compute the average price, it follows that total impact is also a square root of the trade size. This result is in agreement with phenomenological models including the Barra model (Torre, 1997). See also, (Chan and Lakonishok, 1993), (Chan and Lakonishok, 1995), (Almgren et al., 2005), (Bouchaud et al., 2008), (Moro et al., 2009). In the FGLW approach and in this paper, the impact of an individual transaction depends on the prior activity of the hidden order. Other models have considered impact to be the sum of independent transactional impacts with a decay kernel; for example: (Bouchaud et al., 2004), (Alfonsi et al., 2008), (Gatheral, 2010), (Obizhaeva and Wang, 2013). In (Huberman and Stanzl, 2004), it is argued that permanent impact must be linear in order to prevent the existence of systematic profits from price manipulation. However, this argument assumes that the permanent impact functions are the same for the buy and sell periods of a round-trip trade. Permanent impact is the expected information content of a trade; our framework deals with trades that are large enough to be detectable so market observers, who see a large buy followed by a large sell, cannot be expected to believe that both trades are equally informed. This article is organized as follows. In the first section, we extend FGLW’s theory to allow for a variable participation rate. The Page 3 of 47


generalized theory has additional hypotheses and is self-contained. In the second and third sections, we use numerical methods to find optimal solutions that minimize the risk-adjusted cost and optimize the performance to the VWAP benchmark. Section fourth discusses the implications our model has for institutional trading desks. An Appendix contains detailed derivations of some formulae.

1. Short-Term Trading and Hidden Orders Our paper addresses the situation of a large institutional trade of size that is executed over time through a sequence of smaller transactions. For simplicity, we will consider a single institutional trade executing in a market where prices are driven by an arithmetic random walk. The trade is executed according to an execution schedule with participation rates , which represents the probability that a market transaction belongs to the institutional trade. For large enough trades and high enough participation rates, it seems reasonable to assume that market observers will be able to detect the presence of the institutional trade through statistical means, with a degree of confidence that increases over time. For example, one might observe a significant imbalance in order flow, or short-term changes in the security’s price relative to some index. A class of short-term trading strategies arises with the aim to profit from information about hidden orders. If short-term traders believe that a hidden order will continue, they will expect further impact and take a position on the same side. On the other hand, if they believe the hidden order will stop, they will expect reversion and take an opposite position. The competition between short-term traders sets the fair value of market impact. Our next step is to capture order flow inputs and other concepts as hypotheses for a theory of impact. In Section 1.2, we will explicitly obtain the mathematical function that incorporates all the hypotheses and models impact. The equation of price dynamics that incorporates impact will be exposed in Section 1.3.

1.1 Hypotheses 

Hidden Order Detection.

A hidden order with participation rate can be detected at or after the end of an interval of market transactions. Justification: We will consider an anonymous marketplace where order arrivals can be classified as either buyer-initiated (+sign) or seller-initiated (- sign). Page 4 of 47


We assume that in absence of a hidden order, both signs are equally likely. A hidden order originates only buy or only sell orders. We will call the difference between buyer and seller-initiated transactions in an observation interval, the order flow imbalance. We suppose that all market transactions are for the same number of shares . Market participants, who detect an order flow imbalance, wish to determine whether it originates from a hidden order executing with a participation rate where or as a random occurrence from equiprobable buy and sell transactions. Given such a hidden order, the expected imbalance will be for a market observation period of transactions. If there were no hidden order, the expected imbalance would be null with a standard deviation of (for ). Therefore, the hidden order becomes detectable with when . Otherwise, if , the imbalance may be interpreted as market noise in a situation when no hidden order is transacting. For simplicity, we will discretize the space of execution schedules by considering piecewise flat schedules that maintain a constant participation rate in the minimal interval of detection In practice, there is no particular reason why an institutional trader should choose to change participation rates after precisely market transactions. However, we will show in Section 4.2 that the qualitative features of these optimal solutions remain the same if one chooses coarser discretizations. We will call intervals satisfying the detection criterion “detectable intervals” and the index will represent the detectable interval. We consider as an execution schedule a finite set of detectable intervals, where the institution maintains a constant participation rate within each interval. The choice of a trading trajectory by the institution affects how an order becomes detectable; a higher participation rate will cause detection to occur sooner. If an institution is executing a hidden order with a participation rate of , it should expect to become detectable after market transactions. During this , the institution expects to execute a total of partial fills. For example, a hidden order executing with a uniform participation rate of is detectable after a segment of 100 market transactions. In this interval, it will have accumulated 10 partial fills. The average trade size in the US, at the time of this writing, is approximately shares; therefore, such a hidden order becomes detectable after 2500 filled shares. The order will be traded in a finite amount of time; then, a finite number of detectable steps will take place. This number can be represented by an integer that we will call Page 5 of 47


where . In other words, represents the last detectable interval. A hidden order will generally not stop precisely at the end of a detectable interval . A remaining time of trading could exist that is not sufficient to complete another detectable interval of market transactions. We set , where is the most recently observed rate. Therefore, an execution schedule will extend over a total number of intervals , which need not be an integer. The number will be determined by the trade size and the transactional time horizon in a constrained optimization problem that we will explain in Section 2. In this paper, we will not concern ourselves with the relation between the transactional time interval and actual clock time. Each interval has a different length as the rate varies and it is dimensionless. For the purposes of this paper, the number of market transactions is the natural scale to measure time. We define the non-random variables of the process of executing a hidden order to be That means the trade will be executed by the institution with the participation rates schedule and size The number of variables will be reduced once we set the constraints of the optimization problem for the total cost of the trade. Figure 1 (see page 40) illustrates the scenario we present in this paper. The execution schedule is divided in intervals, each with a uniform participation rate. Time flows from left to right. The number of market transactions and shares filled are shown together with the changes of participation rate. The process of price dynamics will be modeled as a stochastic process; therefore, we will need to define a distribution function for market prices at any time or the type of process itself. 

Process of Price Dynamics.

Before introducing the second hypothesis that describes the evolution of prices, we will explain some concepts and definitions. H.2.1 Prices and Impacts As explained before, our model is designed for the case where a market is driven, in part, by an institution that executes a hidden order. Observers can expect two different situations at interval : the institution either stops or continues the execution of the order. As a result, two kinds of market prices and two kinds of impacts arise.

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By reversion price, we will mean the future price per share conditioned on observers determining that the hidden order ended after completing the interval . The reversion price accounts for the permanent impact from prior intervals as:

We will denote by the execution price per share in the interval. It is the market price conditioned on that the hidden order continues through interval The execution price contains the sum of the temporary impact and the reversion price at the same interval. However, relative to an earlier interval, it accounts for both permanent and temporary impacts what we call simply Impact. That is:

The reversion price is hypothetical if the institution does not stop at the end of the interval. Figure 2 shows these concepts graphically. Figure 2. Definitions of Prices and Impacts. The expected market price at is if the institution continues after ; if it stops, the market price is the reverted price Temporary Impact at is the difference and Permanent Impact at is the difference Interval Market Impact at is and it is the sum of Permanent and Temporary Impacts at

Expected Market Price

If Continues

Interval Temporary Impact

Impact

Temporary Impact

Market

Permanent Impact If Stops

Interval

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We could construct the dynamics of prices from any of the expressions given above. However, we are interested in whether a hidden order that has completed intervals will continue or stop. Following the definitions of prices, if the institution stops, the price will reach the reversion price ; if it continues, it will reach the market execution price The difference will comprise two conceptually different contributions: 1) The predictable one known as market impact, or simply Impact as explained in , caused by the participation and detection of the institution at Market Impact at has, at the same time, a temporary and a permanent contribution. Temporary Impact at will vanish once the institution stops at the end of with the price falling to the reversion . Permanent Impact at is the remaining part of the market impact after the institution stops, relative to the reversion price . 2) The unpredictable noise term, as we will describe next. H.2.2 Variables and functions of the dynamic process Following with the ideas exposed above, we will construct the prices of the asset at each step as functions depending on the variables of the execution schedule, , and on the independent random 2 variables of Gaussian white noise, We will define as the interval market impact function at , depending solely on the variables of the execution schedule. This term represents the change of price derived by the law of supply and demand in the interval , principally driven by the institution and the observers who are able to detect it and try to make profits. We will assume in Hypotheses (H.3) and (H.5) that profits are zero on average, and show that this defines, in part, the shape of the impact function. The expression for will be qualitatively proposed in Hypothesis (H.4) and explicitly derived in Section 1.2. On the other hand, the stochastic aspect of prices will be modeled by the noise term, which will consequently be proportional to the volatility of the asset and depending on the random variables of noise. In this paper, we will consider a short-term trading in nonhighly volatile markets; therefore, the volatility can be considered 2

In trading, noise means all factors that change the price in an unpredictable manner and, consequently, should be studied statistically through their correlations and cumulants related to a distribution function. Gaussian white noise is defined by a Gaussian process with zero mean and delta correlation. It has been chosen in this paper for its mathematical simplicity and is widely used in models of trading.

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as a constant independent of the prices. Mathematically, the noise term in the interval will be represented by a small change in a variable which follows a Weiner process of pure Brownian motion or diffusion in the short-term trading. Explicitly, this means . Additionally, if the function is defined as the Gaussian distribution on the independent random variables of white noise, it corresponds to a Gaussian mean Markov correlations and standard deviation

.

Until here, we have collected all the elements necessary for the introduction of our next hypothesis. H.2.3 Evolution of prices Inspired by the literature in optimal trading execution, we will consider the process of price dynamics as a discrete-time Langevin process3. The intervals between steps and will be determined by market transactions. We will suppose the drift term and the volatility independent of the prices. This kind of discrete-time process is usually called arithmetic random walk. Summarizing:

The process of trading drives market prices as:

with the definitions given in H.2.1 and H.2.2

In hypothesis we are considering the variables nonrandom and independent of the variables or the same: . That can be valid in non-highly volatile markets for algorithms that are designed to track a specific execution schedule. Then, when evaluating the expected value on equation the impact term will remain the same and the noise term will vanish. The expected change in price during the interval will depend on the trading schedule through the particular shape of the impact function as:

3

See for example (Cobb, 1998).

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The expectation is over the Gaussian function variables

, with fixed

Comments: It follows from

4

that:

On short-term trading in non-highly volatile markets, the distribution function of the market prices is a Gaussian distribution of a Weiner process with constant drift term. Therefore, market prices are determined uniquely by their expectation values and standard deviations (first and second cumulants). The expectation value is related to the total impact of the trade, while the standard deviation is proportional to the square root of the elapsed transactional time.5 

Breakeven.

The expected reversion price following a trade that completed after intervals is equal to its weighted average execution price:

4

Because a Weiner process postulates the independency of all its Gaussian variables then their sum is also Gaussian. In Section 1.3, we will derive equation that gives the expression for the market execution price at an interval as the sum of the variables of noise and the expectation value of the execution price, which is the total impact accumulated since the start of the trade. Therefore, we conclude that the market price is a Gaussian random variable. In this paper, we are interested in the expression of the expected price in terms of the market impact and the execution schedule and we will never use the explicit form of the Gaussian distribution on the price random variable. However, we note that the arithmetic process makes in comparison with a geometric process ln is the transactional time.

ln

, where

5

We know that this result is controversial in general cases, where stock prices could be strenuously fluctuating and follow other process as complicated as fractal ones. In those cases, cumulants of order greater than 2 are required. See for example (Peters, 1994), (Bouchaud and Sornette, 1994), (Ghashghaie et al, 1996), (Mandelbrot, 1997). However, for non-highly volatile markets we are no so far from reality if we work with cumulants to order two and in a diffusive regime (or pure Brownian motion) for the short term trading, where the variance is proportional to the time.

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Here, and

is the number of institutional transactions to step is the number of institutional transactions in each step

Justification: The breakeven hypothesis states that the value of the shares bought (or sold), after reversion, is the same as the average price paid (or received) in the execution. For example, suppose a portfolio manager places a buy order when the price is $20. As the hidden order executes, the price rises to a peak of $21, with an average price of $20.7. After the trade ends, price reverts part way and closes at $20.7, leaving no profit after reversion. In this example, we assumed the noise was zero for clarity. Actually, some trades will be profitable after reversion and others unprofitable, Hypothesis states that breakeven will hold when averaged over the Gaussian noise distribution. We regard the breakeven hypothesis primarily as an empirical observation, see (Bershova and Rakhlin, 2013) and below. However, it raises an interesting question: if the security price were to revert to $20.7 in the above example, why would the manager continue to buy as it rises above this level? The main reason is that noise term in equation creates uncertainty in the execution price, which represents risk to the portfolio manager proportional to the elapsed transactional time. To manage this risk, portfolio managers expect trades to be completed in a fixed amount of time. Should one try to rush the trade to complete the order before price reaches $20.7, market impact will increase faster making the realized execution price even higher. Given the constraints on the total size of the order and the total amount of time available to trade, reversion is an inevitable consequence of the removal of order flow expectations after a hidden order has stopped. The breakeven hypothesis contrasts with Kyle’s informed trader model. In Kyle, the insider knows the final price and, therefore, would not logically buy above this level; impact is linear, and there is no reversion. As a result, Market Makers in Kyle’s model systematically lose money to hidden orders and generate counterbalancing profits at the expense of the uninformed traders. Kyle’s model predicts a net transfer of wealth from uninformed traders to informed traders. We instead theorize with the breakeven hypothesis that hidden order expected profits are null; therefore, our model does not require noise traders to lose money systematically. Both models are idealizations of situations that arise in the real world6.

6

Reality lies somewhere in between these two idealizations. For example, portfolio managers need at times to execute liquidity trades in spite of an expected loss. Vice-verse, it may be possible in some circumstances to identify informed traders where the portfolio manager makes a market-to-market profit using techniques from data mining.

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We test this hypothesis using datasets provided by Alpha Vision clients. The data contains 86253 trades larger than 1% of daily volume from 94 portfolio managers in European and US cash equities, over various time windows from May 2008 through November 2012. To confirm that the equality is fulfilled, we first need to adapt this equation to a real market situation. Next, we describe how we proceed. 1. It is common for various portfolio managers at a given asset manager to place orders in the same security, in which case the desk usually aggregates orders into one and executes the merged trade as a unit. Therefore, we will merge overlapping orders from a given firm. In addition, we will also merge orders that are separated by no more than 60 market minutes, where we count the time from the market close to the next-day open as zero market minutes. We will bin merged trades by their size , which is defined as the total number of shares traded. We will suppose that for each interval , there exists a hidden order that completes its entire order size in intervals. Therefore, and just for the purpose of the verification of breakeven, we will use order sizes in place of intervals. We will call the right side of equation and the left “the reversion price”

“the averaged capture” ,

2. Averaged Capture and Shortfall: We calculate the average execution price to fill the entire merged trade or average capture as a volume-weighted average fill price of all partial fills. For each merged trade, we consider the midpoint price at the start of the trade that we call start price or We take

at the starting point of the trade. The relative difference between the average capture and the start price is the shortfall. In basis points, this is:

where

of the trade is

for a buy and

for a sell.

3. Reversion Price and Return:

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The reversion price is the price relaxation after a trade has ended. It is difficult to measure because portfolio managers often add positions in subsequent days. We remove the effect of these followon trades by subtracting their estimated impact, using an empirical impact model that we explain below in “First- Interval Impact” We will call the adjusted amount the free price because it represents the value of the security free of further activity. The reversion price is the free price when the market determines that the hidden order has stopped. It is difficult to establish when an order has ended. One of the reasons is that the market does not know whether the institution is making a pause or stopping. The data show that very large trades that require several days to execute can take two to five market days to complete reversion. Generally, a full day is sufficient to observe complete reversion. Here we will consider the following options for the reversion price: 1) the price at the market close on the day the trade was finished or close price “T” 2) the price on the close of the next day “T ” and 3) the price on the close after two days “T ” Our purpose is to compare the shortfall to the return,

Below we show Figure 3 that compares the shortfall with the return at the close, at the close after one day and at the close after two days, as a function of the trade size. We are using a logarithmic scale for a better view.

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Return vs Arrival [bps]

Breakeven vs Trade Size

100

Shortfall

Close Close+1 Day Close+2 Days

10

1

10

100 1000 10000 Trade size (X) [x 250 shares]

100000

Figure 3. Breakeven is verified by comparing the implementation shortfall (thick line) and the return, with trades binned by trade size . On the close, we observe some unrealized profits (reversion is not complete) in the economically important range . By Close+1 Day, the difference is insignificant with t-stat , which is the same as in Hypothesis . Therefore, the expression for market impact in a model with finite detection probability remains the same, as do the optimal execution solutions. In relation to the detection of hidden orders, we used a discretization of the time line into “detectable” intervals comprising transactions. This choice was based on the argument that the imbalance in order flow should violate the hypothesis of a null bias with t-stat=1, which only provides a minimal level of confidence in detecting a hidden order. In practice, one would expect competing market makers to devise their own detection model. They will be looking not only at order flow, but also at sector-relative returns, signs of repeated order placements, available liquidity in dark pools, etc. The advantage goes to the one who is first able to detect the hidden order, which is why we chose the low value t-stat=1. However, while differences in the hidden order detection mechanism will lead to different discretizations of the execution process, the structure of the model remains the same as long as the detection interval remains proportional to . The shape of the optimal execution solutions described in this paper remains qualitatively similar if one chooses to define a detectable interval with a higher level of confidence. In Figure 13, we show the optimal participation rate profile for the non-risk-averse case and t-stat=2. While not identical to Figure 7, the main qualitative feature of a back loaded execution schedule remains intact.

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L=0 Optimal Trajectory with t-stat=2 0.16

Participation Rate

0.14 0.12 0.1

0.08 0.06 0.04 0.02 0 0

200

400

600

800

1000

Transactional Time

Figure 13. Optimal participation rate profile without risk aversion and with a detection model based on a level of confidence t-stat=2. The optimal solution executes the first 600 transactions with a participation rate of approximately 7%, similar to the solution with t-stat=1 found in Figure 7. The execution back-loading and peak participation rate of 15% is also similar to the previous case. Although, the solutions are different in detail, they are qualitatively similar. The optimal execution framework is robust to minor variations in the assumptions regarding hidden order detection.

Acknowledgements The authors are in debt to F. Lillo for conversations on the proposed generalization of the FGLW theory to variable speed trading. We are also grateful to the referees for their insightful comments and to N. Bershova and D. Rakhlin for their review. One of us (HW) wishes to thank J. D. Farmer and the Santa Fe Institute for their hospitality in numerous productive visits. The phenomenological results regarding the first-interval impact model and breakeven conditions are part of ongoing research in collaboration with Carla Gomes, which will be published separately.

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Figure 1. A detailed representation of the institutional activity inside the market transactional time. The grey cells show an example when the step

consists of

market transactions, while the institution participates twice with a rate

⟶

.

↔ Institutional Rates Steps

⟶

⟶

⟶

↔ Stop

Market transactional intervals Market transactional time Institutional transactions filled at each interval Total filled Institutional transactions at

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Appendix A. Derivation of Market Impact for Constant Participation Rate Here we will derive the interval market impact for a constant participation rate schedule. This expression was first obtained in (Farmer et al, 2011) but we will reproduce the derivation for completeness following our particular notations and definitions. Market participants, who are able to detect the presence of a hidden order at a particular interval, will formulate expectations about the remaining size after . To quantify those expectations, we will define the probability that the hidden order stops at (or to be . We also define the probability that the hidden order continues at given that it was detected at the interval. Then, by the definition of conditional probability, we write:

The second equality comes from the definition of probability of the union of disjoint sets, and represents the maximum size of any trade in step units. It is natural to take lim which means , when and is very large. By definition, we write:

Hypothesis

and

imply that

Alternatively the same, By using the breakeven Hypothesis

in

, we write:

Here, we denoted Equation

By applying

is also:

in

, we get:

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and from

Moreover, we could set: where is a constant, and , to avoid the contradictions and with . In addition, equations are reduced by recursion to:

Introducing in the sequence using we get

Matching (A.7) with

and

, we obtain

From we arrive at the equation that establishes the relation between the market impact and the reversion (or permanent impact): This, using

and (A.6), can be written as

Additionally, we get:

Finally, from impact for

Here,

and changing

and

per

, we obtain the interval

, as:

.

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The empirical data show (Gopikrishnan et al., 2000), (Gabaix et al., 2006) that the hidden order sizes have a Pareto distribution with parameter :

In this case, Hurwitz Zeta function.

where

is the

Formula should be consistent with , therefore it should be proportional to powers of the total transactions , and the participation rate , which in this case are . We know that and setting and , we obtain:

Formula enables us to make the connection with the impact at for a variable participation schedule, when , or the same This is:

We re-obtained equation

when

and

Appendix B. Derivation of Market Prices as Functions of the Participation Rate Schedule The series of equations (7) may be proven by complete (or strong) induction. We will show that equation (7) is valid for and then is valid for First, note that we may write:

where

Here, is the notation for the permanent impact in each interval, and is the sum of the permanent and temporary impacts at the interval as:

Demonstration of equations

:

We write, following definitions H.2.1 and the equation hypothesis , that:

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Proof of equations For For

:

equation follows from and formula we have from , and that:

The breakeven hypothesis is valid for 13 extend it to taking so Then, after applying becomes:

, but we may .

. Finally, we apply the same reasoning to

and we get:

that after reordering terms becomes: . Therefore, formula

is valid for all

by strong induction.

Comments:  Comparing formulas with we get the expressions for temporary and permanent impacts in the approximation as:

 If instead of approximation we take formulas will become: , 13

In the first interval, the hidden order has not yet been detected therefore expectations of continuing order flow imbalances are small. This makes the difference and which means that market impact in the first interval consists predominantly of permanent impact.

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. Moreover, for formula

, the correction , with

and is only a function on function on and

to the cost will be:

Because

are constants

the correction term is only a

Appendix C. Derivation of the Cost of the Trade and its Variance We will arrive to equations and

and

, by using the notations . From definition

, we have:

By using formulas

with

, this may be written as:

containing the the total impact, as:

The second summand in switched as:

contributions and

consists of two sums that may be

then:

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Summarizing,

as:

On the other hand, by using definition write:

and formula

, we can

where

The second summand in the noise term contains a double sum that can be switched as:

Then, it results:

After that, it is simple to calculate the variance of the cost . Explicitly, it is:

By using

and Markov correlations

then

and

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