Emilio Said

Ahmed Bel

Hadj Ayed

Alexandre Husson

Frédéric Abergel

Frederic Abergel

Market Impact: A Systematic Study of Limit Orders

Keywords: Market microstructure, statistical finance, market impact, fair pricing, automated trading, limit orders

come   L'archive ouverte pluridisciplinaire

Introduction

It is a commonly acknowledged fact that market prices move during the execution of a tradethey go up for a large buy order and decrease for a large sell order. This is, loosely stated, the phenomenon known as market impact.

The market impact of large trading orders that are split into pieces -better known as metaorders -and executed incrementally through a sequence of orders of smaller sizes is crucial in describing and controlling the behaviour of modern financial markets. Being able to quantify this impact is clearly a question of great relevance when studying the price formation process, and it has also become a major practical issue for optimal trading. Indeed, in order to know whether a trade will be profitable, it is essential to monitor transaction costs, which are directly linked to market impact. Measuring and modelling market impact has therefore become a central question of interest, both for researchers and practitioners, in the field of market microstructure.

Given the importance of the subject, there exists but a few research articles pertaining to the empirical estimation of market impact, mostly due to the scarcity of data. In fact, trades and quotes, or even order book databases are not sufficient to perform the analysis: what is required is a clear identification of metaorders. Metaorders have started being recorded as such in a systematic way only recently, and mostly in proprietary databases that are not readily accessible to academic researchers in the field of market microstructure. The analyses presented in [START_REF] Almgren | Direct estimation of equity market impact[END_REF]] [Moro et al., 2009] [Gatheral, 2010] [Toth et al., 2011] [ [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF]] [Bacry et al., 2015] [Zarinelli et al., 2015] [Gomes and Waelbroeck, 2015] essentially cover all that is published about the market impact of large orders.

Although difficult to measure in practice, market impact has been studied from a theoretical point of view. In an economic theory perspective, the information held by investors, which governs their decisions, should have some predictive power over future prices. This point was thoroughly investigated by [START_REF] Farmer | How efficiency shapes market impact[END_REF], a paper which we find enlightening and use as reference for the theoretical measurements of market impact. In a related study, [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF] uses their own proprietary database to perform an empirical analysis of a set of large institutional orders, and validates some predictions of the [START_REF] Farmer | How efficiency shapes market impact[END_REF] model. Such a comparison will also be performed in the present work.

Our paper is a contribution to this strand of research, with a specific focus on the market impact of limit orders. As a matter of fact, market orders are generally not used by institutional investors because of the lack of control they imply. On the contrary, limit orders, whether they are aggressive -crossing the spread -or passive, form the vast majority of orders that are actually sent to the market during the execution of a large trade. As such, they should be the main subject of interest in a study of market impact. To the best of our knowledge, ours is the first academic study of market impact with an emphasis on limit orders. The statistical results presented in this paper are obtained from a proprietary database consisting of appropriately selected limit orders executed on the European equity market between January 2016 and December 2017. The originality of our approach lies in the fact that the study relies on an algorithmic-based identification and reconstruction of metaorders from the database of all orders.

The paper is organized as follows: Section 2 is a short review of the literature on market impact. Section 3 introduces our main definitions of metaorders and market impact measures. Section 4 presents our empirical results: They confirm that limit orders behave in agreement with some stylized facts already established in the literature, and shed a new light on the influence of passive orders in an execution strategy. Section 5 considers the so-called Square Root Law regarding to our metaorders. Section 6 deals with the fair pricing condition concerning our metaorders. Section 7 is a discussion of our results and their implications, including some comparisons with the literature. Appendix A recalls for reference the framework and main results of the theoretical, agent-based market impact model developed in [START_REF] Farmer | How efficiency shapes market impact[END_REF].

A short review of the market impact literature

The strategic reasons underlying the incremental execution of metaorders were originally analyzed by [Kyle, 1985], where a model considering an insider trader with monopolistic information about future prices is introduced. It is shown that the optimal strategy for such a trader consists of breaking its metaorder into pieces and execute them incrementally at a uniform rate, gradually incorporating its information into the price. In Kyle's theory the total impact is a linear function of size, and the price increases linearly with time as the auctions take place. The prediction of linearity also appeared in the work of [START_REF] Huberman | Price manipulation and quasi-arbitrage[END_REF]. They showed that, in the case of constant liquidity providing, and in order to prevent arbitrage, the permanent impact must be linear, i.e. incremental impact per share remains constant during the metaorder life. It is however the case that real data contradict these predictions: metaorders do not exhibit linear market impact. In fact, most empirical studies consistently highlight a concave impact, in sharp contrast with the theoretical linear shape. The first relevant empirical study of market impact is [START_REF] Almgren | Direct estimation of equity market impact[END_REF], which directly measures the market impact of large metaorders in the US equity market and provides empirical evidence of a concave temporary impact. Later studies [START_REF] Moro | Market impact and trading profile of hidden orders in stock markets[END_REF]] [Toth et al., 2011] [Bacry et al., 2015] also find a concave market impact, in rough agreement with a square root formula. Following these experimental findings, some theoretical efforts have been made to reconcile Kyle's model with a concave dependence on size, essentially by adding the hypothesis that larger metaorders contain less information per share than smaller ones. [START_REF] Farmer | How efficiency shapes market impact[END_REF] presents a model enriching Kyle's approach with this concave dependence on size. Whereas Kyle considers a single, monopolistic informed trader, [START_REF] Farmer | How efficiency shapes market impact[END_REF] introduces several competitive traders receiving a common information signal and then choosing independently the size of the order they submit to an algorithmic execution service. This set-up is close in spirit to the real-life organization of the major players in the market, which operate by setting up an internal market with a stakeholder recovering all the orders before executing them on the external market.

To be even more specific, market impact can be studied from two different perspectives. The first one, introduced in the previous paragraph, addresses the effect of a metaorder being executed on the price formation process. This effect is commonly termed the temporary market impact. It is clearly an important explanatory variable of the price discovery and is studied as such in several papers [Kyle, 1985] [Hautsch and Huang, 2012] [Farmer et al., 2004]. Temporary market impact is obviously the main source of trading costs, and models based on empirical measurements can be used in optimal trading schemes [START_REF] Almgren | Direct estimation of equity market impact[END_REF]] [Gatheral, 2010] [ [START_REF] Lehalle | Rigorous post-trade market impact measurement and the price formation process[END_REF]] [Almgren and Chriss, 2001] [Gatheral and Schied, 2013], or used by an investment firm in order to understand its trading costs [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF]] [Brokmann et al., 2015] [Mastromatteo et al., 2014]. One common conclusion to the studies is that the temporary market impact is mainly characterized by three components. The first, obvious one is the size of the metaorder, suitably rescaled by a quantity reflecting the traded volume of the security under scrutiny. The daily participation or trading rates capture most of the dynamics of this component. Note that some empirical studies such as [START_REF] Brokmann | Slow decay of impact in equity markets[END_REF] prefer to consider as a scaling factor the participation rate during the metaorder life. This approach presents the advantage to include a duration effect not captured by the daily participation rate, and it will be used in the present study. One must also allow for some dependency on the price uncertainty while the metaorder is executed, and the volatility or the bid-ask spread are typical measures of this uncertainty. Last but not least, it seems necessary to capture the information leakage generated by the metaorder, the number of orders executed during the metaorder or its duration being good proxies.

There is a second, more controversial type of market impact, pertaining to the persistence of a shift in the price after the metaorder is fully executed. This effect is commonly called the permanent market impact. Research papers dealing with permanent impact can be separated in two categories. The first one considers the permanent impact as the consequence of a mechanical process. The second ones consider the permanent impact as the trace of new information in the price. Among those who share the mechanical vision of the permanent market impact, there are two approaches. The first picture of [START_REF] Farmer | How efficiency shapes market impact[END_REF] and [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF] says that the permanent impact is important and roughly equals to 2/3 of the temporary impact. This is derived from a fair-pricing hypothesis. In the second picture [Bouchaud, 2010], there is actually no such thing as a permanent impact, the slow decay of the market impact being the result of the long memory of the order flow. These two approaches are incompatible. In the present paper, the studies performed on the price relaxation seem to advocate in favour of a permanent impact at the approximate two-thirds level, in agreement with the existing empirical literature as well as the fair pricing condition of [START_REF] Farmer | How efficiency shapes market impact[END_REF].

3 Definitions, Algorithm and Market Impact measures

Basic Definitions

Some basic concepts, and the algorithmic definition of a metaorder, are introduced here. Definition 1. A limit order is an order that sets the maximum or minimum price at which an agent is willing to buy or sell a given quantity of a particular stock. Definition 2. An aggressive limit order is one that instantaneously removes liquidity from the order book by triggering a transaction. An aggressive order crosses the Bid-Ask spread. In other words an aggressive buy order will be placed on the ask, and an aggressive sell order will be placed on the bid.

A limit order that is not aggressive is termed passive. Passive orders sit in the order book until they are executed or cancelled.

Loosely speaking a metaorder is a large trading order that is split into small pieces and executed incrementally. In order to perform rigorous statistical analyses, a more specific and precise definition of a metaorder is required, and given in Definition 3 below: Definition 3. A metaorder is a series of orders sequentially executed during the same day and having those same attributes:

• agent i.e. a participant on the market (an algorithm, a trader...);

• product id i.e. a financial instrument (a share, an option...);

• direction (buy or sell);

The advantage of adopting such a definition is that it is no longer necessary to work directly on raw metaorder data. Indeed, a series of orders executed by the same actor on the same product on the financial market will behave like a metaorder and therefore can be considered as such (Figure 1). Let us mention that the current study is not restricted to metaorders executed on a single market, due to the fact that an instrument can be simultaneously traded on several markets. Also note that orders executed during the same second are aggregated in order to avoid time stamping issues: the quantities are summed up, the local VWAP is set as the execution price, and the TIMESTAMP of the last order is retained for all orders during the same second. One can see below (see Figure 3) how those simplifications reduce the complexity of Figure 2:

Algorithmic procedure

To carry out the study of market impact, the algorithm is trained and run on a proprietary database. Figure 1 in the previous section represents the database in its initial state, as the input of the algorithm. Figure 2 shows an intermediate state of the data extracted from the initial data, during the metaorder reconstruction phase. Figure 3 displays the data in their final state, directly exploitable for statistical studies. Note that the recovery and cleaning of market data are done simultaneously during the metaorder reconstruction.

Market Impact definitions

The same framework as that in [START_REF] Bacry | Market impacts and the life cycle of investors orders[END_REF]] is adopted. Let Ω be the set of metaorders under scrutiny, that is, metaorders that are fully executed during a single market session, and pick ω ∈ Ω executed on an instrument S and during a given day d. Its execution starts at some time t 0 (ω) and ends the same day at time t 0 (ω) + T (ω). Thus T (ω) represents the duration of the metaorder. Denote by Q(ω) and N(ω) respectively the number of shares and the number of orders that have been executed during the life cycle of the metaorder ω. Hence Q(ω) is the size and N (ω) the length of ω. Let V (ω) be the volume traded the same day d on the instrument S (summed over all European trading venues) between time t 0 and t 0 + T , i.e. during the life cycle of ω. The sign of ω will be denoted by ϵ(ω) with ϵ = 1 for a buy order and ϵ = -1 for a sell order. Clearly, all the quantities introduced in this section depend on ω. For the sake of simplicity, we chose to omit this dependence whenever there is no ambiguity and will often write

T , N , Q, V , ϵ instead of T (ω), N (ω), Q(ω), V (ω), ϵ(ω).
The market impact curve of a metaorder ω measures the magnitude of the relative price variation between the starting time of the metaorder t 0 and the current time t > t 0 . Let I t (ω) be a proxy for the realized price variation between time t 0 and time t 0 +t. In line with many authors [START_REF] Almgren | Direct estimation of equity market impact[END_REF]] [Bershova and Rakhlin, 2013] [Bacry et al., 2015], we use the return proxy defined by

I t = P t -P t 0 P t 0 , (1)
where P t represents either the execution price of the financial instrument S during the execution part of the metaorder, or the mid-price during its relaxation part starting when the metaorder has been fully executed. This estimation relies on the assumption that the exogenous market moves W t will cancel out once averaged, i.e. as a random variable, W t should have finite variance and basically satisfy E(ϵ(ω)W t (ω)) = 0. One can thus write

ϵ(ω)I t (ω) = η t (ω) + ϵ(ω)W t (ω), (2)
where η t (ω) represents the market impact curve and W t (ω), the exogenous variation of the price corresponding to the relative price move that would have occurred if the metaorder had not been sent to the market.

4 Empirical study

Aggressive Limit Orders

The subject of interest of this section is the metaorders generated by aggressive limit orders, that is, limit orders that actually cross the spread in order to trigger an immediate transaction. Such orders are sometimes rather loosely considered as market orders in the modelling literature on limit order books, but it is clear that they behave differently, as their execution price is always equal to that of the best available limit and can never trigger transactions at higher (buy) or lower (sell) price. One can observe that, in agreement with the intuition, metaorders with shorter durations are more frequent (Figure 4). As already observed for the duration, shorter metaorders (in length) are more represented. This observation is obviously not a surprise, since the quantities N and T are expected to be highly positively correlated. A log-log scale (Figure 6) gives a more precise idea of the distribution of N . The apparently linear relation suggests that N follows a discrete Pareto distribution. P(N = n) stands for the natural frequency estimator of the probability P(N = n). As regards the parameter β introduced in [START_REF] Farmer | How efficiency shapes market impact[END_REF] -see Section A.5 -one obtains the estimate β ≈ 1.4. Therefore, N is distributed as a power law, in agreement with [Vaglica et al., 2008] who reconstructed metaorders on the Spanish stock exchange using data with brokerage codes and found that N is distributed as power law for large N with β ≈ 1.7. Similar to the observations for duration and length (see Figures 4 and5), metaorders with smaller participation rates are more represented. Again, and as expected, the quantities N and Q V are highly positively correlated (Figure 7).

Length distribution

Participation rate distribution

Market Impact curves

The main results of Section 4.2 are now given, namely, the market impact curves for aggressive metaorders. In order to plot the market impact dynamics, a bucketing method is used: Consider for example that one wants to plot y as a function of x, x, y being two arrays of data. One starts by ordering the couple of values (x i , y i ) according to the values of x and then divides the sorted (by x) distribution (x, y) sorted into N bucket . This procedure yields N bucket subsets of the distribution (x, y) sorted , (x i , y i ) i∈I 1 , (x i , y i ) i∈I 2 , ..., (x i , y i ) i∈I N bucket , and for each bucket I k the mean values (x k , y k ) is computed. The last step of this bucketing method is to plot the points (x 1 , y 1 ), (x 2 , y 2 ), ..., (x N bucket , y N bucket ).

Market Impact Dynamics

To study the dynamics of the market impact, one plots (ϵ(ω)I t (ω)) ω∈Ω,t 0 (ω)≤ t ≤t 0 (ω)+2T (ω) . The first sub-interval t 0 (ω) ≤ t ≤ t 0 (ω) + T (ω) corresponds to the execution of the metaorder, whereas the second t 0 (ω) + T (ω) ≤ t ≤ t 0 (ω) + 2T (ω) corresponds to the relaxation. The study of relaxation presents a degree of arbitrariness, since a choice has to be made as to the elapsed time after the metaorder is completed. For the sake of homogeneity, the relaxation is measured over the same duration as the execution. This choice seems to be a good compromise to cope with two antagonistic requirements, one being to minimize this elapsed time because of the diffusive nature of prices, the other being to maximize it so as to make sure that the relaxation is achieved. Further studies performed on our database actually show very little dependence of the permanent impact level on this time parameter, a result which we found quite comforting, and in line with some previous results in the literature, see e.g. [START_REF] Gomes | Is market impact a measure of the information value of trades? market response to liquidity vs. informed metaorders[END_REF].

In order to perform an extensive statistical analysis involving metaorders of varying lengths in physical and volume time, a rescaling in time is necessary, see e.g. [START_REF] Bacry | Market impacts and the life cycle of investors orders[END_REF]. With this convention, all orders are executed on the time interval [0, 1] and price relaxation occurs in the time interval [1,2]. For each metaorder ω, one considers

[0, 1] instead of [t 0 (ω), t 0 (ω) + T (ω)] [0, 1] = [t 0 (ω), t 0 (ω) + T (ω)] -t 0 (ω) T (ω)
for the execution part of ω and [1, 2] instead of [t 0 (ω) + T (ω), t 0 (ω) + 2T (ω)] for the relaxation part of ω, and then averages using the bucketing method previously described on the time-rescaled volume quantities.

The time variable t ∈ [0, 1] in Figure 8 is actually the volume time, i.e., the ratio between the part of the metaorder already executed at the time of the observation and the total size of the metaorder -of course, at the end of the execution part this quantity is always equal to 1. The analysis clearly yields an increasing, concave market impact curve. The decay observed in the last points of the curve is an artifact that can be explained by the larger number of metaorders of smaller lengths and with lower impact, as shown in Figure 5. Also note that on the three figures 8, 9 and 10, the larger the metaorders, the higher the impacts: 0.53, 0.60 and then 0.71 for the temporary market impact. However, the last curve 10 indicates that, for large metaorders, the impact reaches a plateau, as if the market has adjusted to the information it received in such a way that the execution of the metaorder no longer affects it. The blue points correspond to execution prices and the red points correspond to mid-prices observed at identical times starting from the end of the metaorder. The isolated points observed in Figures 11 and12 are due to the fact that we have considered all metaorders, so metaorders with a small length, especially those with length N = 2 are over-represented in volume time = 1.0, therefore inducing a bias towards the end of the curve. One can make this artifact vanish when considering only metaorders with larger sizes, see e.g. 13.

Figures 11, 12 and 13 clearly exhibit the concave shape of market impact during the execution part, followed by a convex and decreasing relaxation. Simply by eyeballing Figure 11, one can safely assume that relaxation is complete and stable at a level around 0.35. However, on Figures 12 and13, relaxation does not seem to be quite complete. This behaviour for larger metaorders is quite probably due to the fact that it is not always possible to reach the final time t 0 + 2T during the intraday observation period. Hence, the relaxation of larger metaorders may be hindered by the closing of the market. A conclusion to this section is that the concave shape of the temporary impact and the convex relaxation curve are in line with the empirical results in [START_REF] Bacry | Market impacts and the life cycle of investors orders[END_REF] and [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF]. Also, and more interestingly, the market impact and relaxation curves confirm the theoretical findings of [START_REF] Farmer | How efficiency shapes market impact[END_REF] that the impact should be concave and increasing, and that the final impact after the execution is performed should relax to about two-thirds of the peak impact.

Execution strategies

The purpose of this section is to analyse metaorders executed via passive limit orders, and compare the resulting market impact curves with those previously obtained. Note that the scales on the graphs concerning the aggressive and passive metaorders are exactly the same, thereby making comparisons possible.

At this stage, some comments are in order: by nature, passive limit orders are not always executed and therefore, the notion of market impact for such orders has to be taken with a grain of salt. When an agent places a new passive limit order, especially within the bulk of the order book, it is not clear whether this should indicate that the market is moving in any direction at all, and an impact should not always be expected. In fact, a strategy purely based on limit orders can be even considered to have only negative market impact, since the order is executed only after the price has moved in favour of the agent. And of course, an execution strategy relying only on limit orders will often fail to achieve its target, thereby facing an implementation shortfall that will have to be dealt at some point in the future.

In order to cope with this inherent difficulty and also to present results that are consistent with the task at hand, namely the study of market impact during the execution of a metaorder, the analyses in this section are performed on a database of execution strategies. Metaorders in this database are executed using (on average) ∼ 65% of passive limit orders and ∼ 30% of aggressive limit orders, the remaining 5% consisting of orders of various types (market, market on close...).

Data

• Study period : 1st Jan 2016 -31st Dec 2017

• Markets : European Equity Markets • Order types : All Orders 1

• Filters : metaorders ω ∈ Ω

• Number of metaorders : 74 552 1 ∼ 65% passive limit orders, ∼ 30% aggressive limit orders and 5% others As expected, and previously shown for the aggressive metaorders, see Figure 4, one observes again (Figure 14) that, the shorter the metaorders are, the more they are represented. Different from the case of the aggressive metaorders, the distribution is seen to have a fatter tail, an indication of the "intelligence" of the execution strategies. The shape of the distribution in Figure 15 is very similar to that obtained for the aggressive metaorders in Figure 5. Again, a linear relation in log-log scale seems rather clear (Figure 16), although it is noisier, with β ≈ 1.8 in this case. This tends to confirm the relevance of the discrete Pareto distribution independently of the nature of the orders. Once again the quantities N , T and Q V are highly positively correlated (Figure 17). Be-sides, and not surprisingly, the execution metaorders have larger participation rates than those observed in the aggressive case (Figure 7).

Length distribution

Participation rate distribution

Market Impact Dynamics

The market impact study proceeds along the same lines as those presented in 4.2.2.1. Hence, to plot (ϵ(ω)I t (ω)) ω∈Ω,t 0 (ω)≤ t ≤t 0 (ω)+T (ω) , one has to consider its rescaled time version (ϵ(ω)I s (ω)) ω∈Ω,0≤ s ≤1 , i.e. for each metaorder ω we consider [0, 1] instead of [t 0 (ω), t 0 (ω) + T (ω)] such as [0, 1] = [t 0 (ω), t 0 (ω) + T (ω)] -t 0 (ω) T (ω) .

Figure 18: Market impact dynamics during the execution part in the case of the execution metaorders (set: Ω, 74 552 metaorders, fit: y = 0.37 × x 0.60 ).

One observes a positive, increasing and concave market impact curve. The impacts appear to be smaller than in the aggressive case (Figure 8) although participation rates are higher and metaorders last longer. Moreover, the three curves Figures 18, 19 and 20 show that the impact is insensitive to the size of the metaorders. These two features are a clear indication that it is beneficial to use passive limit orders. An execution metaorder has the main constraint of executing on the market a known and predefined size in advance, whatever the chosen strategy is. The choice of an execution strategy less brutal than one relying solely on aggressive metaorders allows to limit and control its impact. At first glance, see Figures 21,22 and 23, the dynamic seems fairly similar to that obtained in the aggressive case, with a slightly rougher curve due to a much smaller data set for the execution metaorders: There is an increasing, concave temporary impact curve and a decreasing, convex relaxation curve that finishes at about two-thirds of the peak impact. However, the monotonicity and convexity of the relaxation curve are less pronounced. Two possible explanations come to mind. The first is that the relaxation is faster because the temporary impacts are less important and therefore, a return to equilibrium is easier. In this hypothesis we can consider the end of the relaxation curve as an other metaorder starting in the market. The second hypothesis is that this is a noisy artifact due to a smaller data set.

Figure 22: Market impact dynamics with relaxation in the case of the execution metaorders (set: Ω 10 , 47 423 metaorders, temporary impact: 0.36, permanent impact: 0.23).

Figure 23: Market impact dynamics with relaxation in the case of the execution metaorders (set: Ω 30 , 27 710 metaorders, temporary impact: 0.37, permanent impact: 0.24).

Square-Root Law

In this section we are interested in what is commonly called the Square-Root Law. The Square-Root Law is the fact that the impact curve should not depend on the duration of the metaorder. Indeed, almost all studies now agree on the fact that the impact is more or less close to be proportional to the square root of the volume executed. However, the so-called Square-Root Law states much more than that. It basically claims that the market impact does not depend on the metaorder duration. This last point remains a controversial matter: does the market impact depends on the metaorder duration or not.

At first glance, it does not seem clear that the market impact of a metaorder should depend solely on its size. In the industry, several models are based, at the first order, on a theoretical curve of the form σ Q V with σ a volatility factor and Q V a participation rate factor over the metaorder time scale, see e.g. [START_REF] Brokmann | Slow decay of impact in equity markets[END_REF]. Considering such a local participation rate already introduces a duration effect. Nevertheless, a scatter plot of the impacts in terms of the local participation rates (cf. Figures 24 and25)) does not highlight any additional dependency on the duration, apart from that already built in the participation rate. What one can see is that, for the same participation rate, the dispersion of market impact is higher for longer metaorders. This is simply due to the diffusive nature of the prices that creates a noisier impact.

Figure 24: Impact against participation rate, the duration of the order is increasing from dark blue to dark red in the case of the aggressive metaorders.

Figure 25: Impact against participation rate, the duration of the order is increasing from dark blue to dark red in the case of the execution metaorders.

Fair Pricing

In this final, short section, the fair pricing condition for metaorders is examined. In the interest of homogeneity, we consider normalized prices, so that for every stock and at every time, the value of the price is 1 whenever the execution of a metaorder begins. First of all, the VWAP of a metaorder ω is defined by

P V W AP (ω) = N -1 i=0 Q i (ω)P t i (ω) (ω) Q(ω) ,
where t 0 (ω), ..., t N -1 (ω) represent the transaction times of the metaorder ω and Q

(ω) = N -1 i=0 Q i (ω).
Hence, we want to compare 1 + R We conclude from Figures 26 and 27 that the fair pricing condition can reasonably be assumed to hold. It also appears that, the greater the absolute price variations, the more one moves away from the perfect fair pricing condition. This last point was predictable: high variations are generally associated to longer and larger metaorders that are therefore more affected by the diffusive nature of the prices. Note that, in the model of [START_REF] Farmer | How efficiency shapes market impact[END_REF], the discrete Pareto distribution of N of parameter β +1 = 2.5 associated with martingale and fair pricing conditions (see Appendix A) leads to a relaxation at 2/3 of the peak impact, a level that is experimentally verified in our study as well as in the already cited papers [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF] and [Zarinelli et al., 2015].

+ R V W AP = 1 + R t 0 +2T .

Conclusion

The work presented here is an empirical study of a large set of metaorders executed using limit orders in the European equity markets. A new algorithmic definition of a metaorder has been proposed. The statistical results show a good agreement with some predictions of the market impact model in [START_REF] Farmer | How efficiency shapes market impact[END_REF] in the case of limit orders.

Our study contains two distinct subgroups of orders: a set of aggressive limit orders, and a database of execution strategies predominantly composed of passive limit orders. In both cases, the analysis shows that the temporary market impact is increasing and concave, and it also confirms that the length distribution of metaorders follows a Pareto distribution of parameter β+1 with β ≈ 1.4 in the case of the aggressive metaorders and β ≈ 1.8 in the case of the execution metaorders. As for the relaxation phase, a convex, decreasing functional form is obtained, as mentioned in [START_REF] Bacry | Market impacts and the life cycle of investors orders[END_REF] and other related studies. More precisely, the price reversion after the completion of a trade yields a permanent impact such that its ratio to the maximum impact observed at the last fill is roughly 2/3 as predicted in the article of [START_REF] Farmer | How efficiency shapes market impact[END_REF] and highlighted empirically [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF].

Finally, we have shown that the square-root law and also the fair pricing condition seem to be empirically verified: the VWAP of a meatorder is equal to the final price of the security after the relaxation phase is over.

[ Toth et al., 2011] Toth, B., Lemperiere, Y., Deremble, C., De Lataillade, J., Kockelkoren, J., andBouchaud, J.-P. (2011). Anomalous price impact and the critical nature of liquidity in financial markets. Physical Review X, 1(2):021006.

[ Vaglica et al., 2008] Vaglica, G., Lillo, F., Moro, E., and Mantegna, R. N. (2008). Scaling laws of strategic behavior and size heterogeneity in agent dynamics. Physical Review E, 77(3):036110.

[ Zarinelli et al., 2015] Zarinelli, E., Treccani, M., Farmer, J. D., and Lillo, F. (2015)

A.4 Fair pricing condition

The martingale condition derived in the previous section only sets the value of the ratio R + t /R - t at each auction t. Another condition is required to derive the values of R + t and of R - t and therefore, to obtain the expression for the immediate and the permanent impact.

• The fair pricing condition states that for any N ,

π N = 1 N N t=1 S t -X N = 0.
This implies that we have

I N = 1 N N t=1 I t .
Assuming that the martingale condition holds for t = 1, ..., M and the fair pricing condition holds for t = 2, ..., M -1 leads to a system of 2M -2 homogeneous equations with 2M -1 unknowns, so we choose R + 1 as an undetermined constant. Proposition 2. The system of martingale conditions and fair pricing conditions has the solution

R + t =
I t = S t -X 0 = R + 0 + R + 1 1 + t-1 k=2 1 k 1 -P k P k 1 P 1 ...P k-1 , t ≥ 2.
Since X N -X 0 = X N -S N + S N -X 0 = I N -R - N , we have Corollary 2. The permanent impact is

I N = R + 0 + R + 1 1 + N -1 k=2 1 k 1 -P k P k 1 P 1 ...P k-1 - 1 N 1 P 1 ...P N -1
, N ≥ 2.

A.5 Dependence on the metaorder size distribution

According to [START_REF] Farmer | How efficiency shapes market impact[END_REF] which cite a host of other relevant studies, there is a considerable evidence that -in the large size limit and for most major equity markets -the metaorder size V is distributed as P(V > v) ∼ v -β , with β ≈ 1.5.

• P(V > v -1) -P(V > v) ∼ v -β 1 - 1 v -β
-1 ≈ β v β+1 : an exact zeta distribution for all n ≥ 1 is considered,

p n = P(N = n|m = 1) = 1 ζ(β + 1) 1 n β+1
so that

P t = ζ(1 + β, t + 1) ζ(1 + β, t) ≈ +∞ t+1 dx x β+1 +∞ t dx x β+1 = 1 + 1 t -β ∼ 1 - β t ;
• As a consequence,

R + t = 1 t 2+β ζ(1 + β) ζ(1 + β, t)ζ(1 + β, t + 1) R + 1 ∼ 1 t 2-β ;
• Thus, the immediate impact I t behaves asymptotically for large t as I t ∼ t β-1 f or β ̸ = 1 log(t + 1) f or β = 1 ;

• Recalling the fact that the fair pricing gives I N = 1 N N t=1 I t , there holds

I N ∼ 1 N N 0 x β-1 dx = 1 β N β-1 .

Finally

Proposition 3.

I N I N = 1 β .
For further use, one can observe that for a value of β = 1.5, we have

I N I N = 2 3 .

Figure 1 :

 1 Figure 1: Sample of the history of orders before the identification process of the algorithm. Each color corresponds to a different metaorder according to the methodology introduced in Definition 3.

Figure 2 :

 2 Figure 2: Sample of the history of orders after the identification process of the algorithm. Each color corresponds to a different metaorder according to the methodology introduced in Definition 3.

Figure 3 :

 3 Figure 3: Sample of the history of the orders executed corresponding to the previous figure after merging the orders. The yellow metaorder which was previously a metaorder of length N = 4 is now a metaorder of length N = 2.

•

  Figure 4: Duration distribution of the aggressive metaorders.

Figure 5 :

 5 Figure 5: Length distribution of the aggressive metaorders (mean: 8, median: 5).

Figure 6 :

 6 Figure 6: Fitting Length distribution of the aggressive metaorders.

  Figure 7: Q V distribution of the aggressive metaorders.

Figure 8 :

 8 Figure 8: Market impact dynamics during the execution part in the case of the aggressive metaorders (set: Ω, 1 561 505 metaorders, power law fit: y = 0.54 × x 0.45 ).

Figure 9 :

 9 Figure 9: Market impact dynamics during the execution part in the case of the aggressive metaorders (set: Ω 10 , 275 969 metaorders, power law fit: y = 0.63 × x 0.51 ).

Figure 10 :

 10 Figure 10: Market impact dynamics during the execution part in the case of the aggressive metaorders (set: Ω 30 , 65 683 metaorders, power law fit: y = 0.80 × x 0.53 ).

Figure 11 :

 11 Figure 11: Market impact dynamics with relaxation in the case of the aggressive metaorders (set: Ω, 1 561 505 metaorders, temporary impact: 0.53, permanent impact: 0.35).

Figure 12 :

 12 Figure 12: Market impact dynamics with relaxation in the case of the aggressive metaorders (set: Ω 10 , 275 969 metaorders, temporary impact: 0.60, permanent impact: 0.40).

Figure 13 :

 13 Figure 13: Market impact dynamics with relaxation in the case of the aggressive metaorders (set: Ω 30 , 65 683 metaorders, temporary impact: 0.71, permanent impact: 0.48).

Figure 14 :

 14 Figure 14: Duration distribution in seconds of the execution metaorders.

Figure 15 :

 15 Figure 15: Length distribution of the execution metaorders (mean: 50, median: 17).

Figure 16 :

 16 Figure 16: Length distribution of the execution metaorders with a log-log scale.

  Figure 17: Q V distribution of the execution metaorders.

Figure 19 :

 19 Figure 19: Market impact dynamics during the execution part in the case of the execution metaorders (set: Ω 10 , 47 243 metaorders, power law fit: y = 0.37 × x 0.61 ).

Figure 20 :

 20 Figure 20: Market impact dynamics during the execution part in the case of the execution metaorders (set: Ω 30 , 27 710 metaorders, power law fit: y = 0.40 × x 0.62 ).

Figure 21 :

 21 Figure 21: Market impact dynamics with relaxation in the case of the execution metaorders (set: Ω, 74 552 metaorders, temporary impact: 0.34, permanent impact: 0.22).

Figure 26 :

 26 Figure 26: Fair pricing condition in the case of the aggressive metaorders

Table 1 :

 1 Notations and definitionsRemark 1. As we only consider metaorders that have at least two executed transactions, Ω = Ω 2 .

	4.1 Notations
	Notation	Definition
	ω	a metaorder
	S(ω)	financial instrument of the metaorder ω
	t 0 (ω)	start time of the metaorder ω
	T (ω)	duration of the metaorder ω
	N (ω)	length of the metaorder ω
	Q(ω)	size of the metaorder ω
	V (ω)	volume traded on the day d(ω) on the instrument S(ω) during [t 0 (ω), t 0 (ω) + T (ω)]
	ϵ(ω)	sign of the metaorder ω
	P (ω)	price of S(ω)
	Ω	set of all the metaorders identified by the algorithm
	Ω n

* ⊂ Ω subset of the metaorders with N ≥ n *

  . Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate. MarketMicrostructure and Liquidity, 1(02):1550004.with:⋆ E[ S t+1 |F t , m = 1, N > t] = S t + R + t ⋆ E[ S t+1 |F t , m = 1, N = t] = S t -R - t ⋆ E[ S t+1 |F t , m = 0] = S t ⋆ P(N > t|F t , m = 1) = P(N ≥ t + 1|N ≥ t, m = 1) = P t .thus:S t = µ ′ t (P t ( S t + R + t ) + (1 -P t )( S t -R - t )) + (1 -µ ′ t ) S t P t R + t -(1 -P t )R - t = 0 Since ( S t) is a martingale, there holds:

	Proposition 1.
	R + t R -t	=	1 -P t P t	, t ≥ 2.

A The Market Impact Model of [START_REF] Farmer | How efficiency shapes market impact[END_REF] The main results of [START_REF] Farmer | How efficiency shapes market impact[END_REF] are recalled. The central goal of the model is to understand the way order splitting affects market impact.

A.1 Model description

• A filtered probability space (Ω, F, (F t ), P) is given.

• At t = 0, before the opening of the market, the K long-term traders have a common information signal α and each trader k = 1, ..., K formulate an order and submit it to the algorithmic trading firm that bundles them together into a metaorder which will be executed in lots of equal size.

• There are N auctions following each other t = 1, ..., N , N representing the number of orders necessary to execute fully the metaorder is bounded, N ≤ M .

• At t = N + 1, corresponding to the relaxation, the last instant in the game, i.e. the instant after the metaorder is fully executed, is announced with the final price X N = G(X 0 , α, η 1 , ..., η N ), N > 0, where X 0 is the initial price and G a function whose form is not important. We will use the tilde notation to refer to a quantity depending on (η t ), where (η t ) is a zero mean i.i.d random process modeling market noise.

A.2 Notations

• As regards statistical averages, S t denotes a specific realization of transaction prices, whereas S t stands for an average price over the signal η i . Likewise, X N is a specific realization of the final price at the end of the metaorder whereas X N is an average price over the signal η i ;

• The final price averaging over η t is denoted by

.., M and X M = S M thus R - M = 0, M corresponding to the end of the market session;

, where R + t = S t+1 -S t and R - t = S t -X t are the corresponding incremental average impacts;

• I t = S t -X 0 is the average immediate impact at t and I N = X N -X 0 denotes the average permanent impact;

• Let m ∈ {0, 1} an indicator variable where m = 1 if the metaorder is present and m = 0 if it is absent • We will consider p t = P(t ≤ N < t + 1 | m = 1) at each period t and P t the probability that the metaorder will continue given that it is still active at timestep t, i.e.

The possible price paths for a buy are summarized in the graph below: (extracted from [START_REF] Farmer | How efficiency shapes market impact[END_REF])

In Figure 28, the metaorder is supposed to be present and only expected price paths, averaged over the day trader's information (which is why the notation does not include tildes), are shown. The price is initially X 0 , after the first lot is executed it is S 1 = X 0 + R + 0 . If N = 1 it is finished and the price reverts to X 1 = S 1 -R - 1 , but if N > 1 another lot is executed and it rises to S 2 = S 1 + R + 1 . This proceeds similarly until the execution of the metaorder is completed. At any given point the probability that the metaorder has size N > t, i.e. that the order continues, is P t . A typical price path (rather than the expected price paths shown here) subject to a large day trader's noisy information signal would look more like a random walk with a time-varying drift caused by the impact of the metaorder.

A.3 Martingale condition

In this section, ( S t ) is supposed to be a martingale: E[ S t+1 |F t ] = S t . Therefore, the following properties hold true:

0], and:

t is the market maker's best estimate of the probability that the metaorder is present;

• For the price in the next period, it is necessary to average over three possibilities :


Keywords: Market microstructure, market impact, statistical finance, fair pricing, automated trading, limit orders, options market, implied volatility, high frequency

, we propose an algorithmic approach to identify metaorders, based on some implied volatility parameters, the at the money forward volatility and at the money forward skew. In both cases, we obtain results similar to the now well understood equity market: Square-Root Law, Fair Pricing Condition and Market Impact Dynamics.

Emilio Said *,*** , Ahmed Bel Hadj Ayed * , Damien Thillou * , Jean-Jacques Rabeyrin * , and Frédéric Abergel **,*** 1 Introduction

In recent years market impact has become a topic of interest for most market participants. The advent of algorithmic trading has significantly increased the traded volumes and the number of transactions. The whole point of electronic markets is to directly match participants that are willing to sell an asset with participants that are willing to buy it. This is mainly done via two types of orders: market orders and limit orders. Market orders are sent by participants that are willing to either buy or sell the asset immediately. Limit orders, however, do not share this urgency: these orders show the interest of the participant to buy or sell the asset at a pre-assigned price. Market orders are generally not used by institutional investors because of the lack of control they imply. On the contrary, limit orders, whether they are aggressivecrossing the spread -or passive, form the vast majority of orders actually sent to the market during the execution of an algorithmic trading strategy, for example a market making strategy or an optimal execution strategy. Most strategies referred to as algorithmic trading fall into the cost-reduction category. The basic idea is to break down a large order -a metaorder -into small orders and send them to the market over time. The reasons for the incremental execution of metaorders are originally due to what we have called the liquidity paradox : The volume of buy or sell limit orders typically available in the order book at a given instant of time is quite small and represents only the order of 1% of the traded daily volume, i.e. 10 -4 -10 -5 of the market cap for stocks [START_REF] Bouchaud | How markets slowly digest changes in supply and demand[END_REF]. Hence the fact that the outstanding liquidity is so small has an immediate consequence: trades must be fragmented.

Recently, market impact questions related to options market have appeared in option pricing issues. However, the constraints of non-arbitrage often impose some limitations such as linear market impact [START_REF] Bouchard | Hedging of covered options with linear market impact and gamma constraint[END_REF] [Loeper, 2018]. In these papers the authors examine how linear market impact on the spot market can affect option prices. The point of view adopted here is slightly different and seeks to highlight how dealing with options can affect directly volatility and therefore option prices.

Many studies have been conducted to understand the influences of metaorders on the price formation process. Most of them are concerned with the equity markets, see e.g. [START_REF] Almgren | Direct estimation of equity market impact[END_REF], [START_REF] Moro | Market impact and trading profile of hidden orders in stock markets[END_REF], [START_REF] Toth | Anomalous price impact and the critical nature of liquidity in financial markets[END_REF], [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF], [START_REF] Mastromatteo | Agentbased models for latent liquidity and concave price impact[END_REF], [START_REF] Bacry | Market impacts and the life cycle of investors orders[END_REF], [START_REF] Gomes | Is market impact a measure of the information value of trades? market response to liquidity vs. informed metaorders[END_REF], [START_REF] Brokmann | Slow decay of impact in equity markets[END_REF] and [START_REF] Said | Market impact: A systematic study of limit orders[END_REF]. All these studies have shown a common behavior during the execution of a metaorder, namely, a concave and temporary impact followed by a convex and decreasing relaxation. More recently, [START_REF] Donier | A million metaorder analysis of market impact on the bitcoin[END_REF] have observed similar effects in the bitcoin market, and a short note [START_REF] Tóth | The square-root impact law also holds for option markets[END_REF] indicated that the Square root law seems to be hold for the options market.

This paper is intended as an analysis of market impact in the options market. To the best of our knowledge, this is the first detailed, in-depth academic study of this phenomenon.

The paper is organized as follow: Section 2 recalls our algorithmic definition of an equity metaorder, introduces that of an options meatorder and presents some market impact measures. Section 3 introduces the data set of the study and presents our approach for options metaorders. Sections 4, 5 and 6 present our findings and empirical results: They confirm that the market impact law observed in the equity markets also hold true in the options markets. Section 7 is a discussion of our results and their implications.

2 Definitions, Algorithm and Market Impact measures

Basic Definitions

Some basic concepts, and the algorithmic definition of an options metaorder, are introduced here.

Definition 1. A limit order is an order that sets the maximum or minimum price at which an agent is willing to buy or sell a given quantity of a particular stock. Definition 2. An aggressive limit order is one that instantaneously removes liquidity from the order book by triggering a transaction. An aggressive order crosses the Bid-Ask spread. In other words an aggressive buy order will be placed on the ask, and an aggressive sell order will be placed on the bid.

A limit order that is not aggressive is termed passive. Passive orders sit in the order book until they are executed or cancelled.

Loosely speaking, a metaorder is a large trading order that is split into small pieces and executed incrementally. In order to perform rigorous statistical analyses, a more specific and precise definition of a metaorder is required, and given in Definition 3 below: Definition 3. A metaorder is a series of orders sequentially executed during the same day and having those same attributes:

• agent i.e. a participant on the market (an algorithm, a trader...);

• product id i.e. a financial instrument (a share, an option...);

• direction (buy or sell);

Clearly, Definition 3 must be adapted to fit the options market. Options are a bit more complex than equities. Traders buy and sell volatility and deal directly with the implied volatility surface, and therefore, with their implicit volatility parameters. As such, an options metaorder can naturally be defined as a sequence of transactions that generate some specific deformations of the volatility surface.

Definition 4. An options metaorder with respect to an implied volatility parameter θ is a series of orders sequentially executed during the same day and having those same attributes:

• agent i.e. a participant on the market (an algorithm, a trader...);

• underlying product id i.e. the underlying financial instrument;

• direction regarding the sign of

S θ := Q × ∂O ∂θ
where Q is the algebraic quantity (positive for a buy order and negative for a sell order) and O the price of the option traded;

Note that in Definition 4, the product id condition introduced in Definition 3 is dropped. As a matter of fact, trading an option with a given strike K and maturity T also affects those with nearby strikes and maturities, so that trades on options with different strikes and maturity can very well belong to the same metaorder.

This approach is in line with what is presented in [START_REF] Said | Market impact: A systematic study of limit orders[END_REF], and leads to a systematic study of options market impact.

In what follows we will use the term θ-metaorder to refer to an options metaorder with respect to the implied volatility parameter θ as defined in Definition 4.

Market Impact definitions

The framework is similar to that introduced in [START_REF] Said | Market impact: A systematic study of limit orders[END_REF]. Let Ω θ be the set of θmetaorders under scrutiny, that is, θ-metaorders that are fully executed during a single market session, and pick ω ∈ Ω θ executed on (possibly) several options with the same underlying and during a given day d. Its execution starts at some time t 0 (ω) and ends the same day at time t 0 (ω) + T (ω). Thus T (ω) represents the duration of the metaorder. Denote by N(ω) the number of orders that have been executed during the life cycle of the metaorder ω: N (ω) is the length of ω. Let t 0 (ω), t 1 (ω), ..., t N (ω)-1 (ω) be the transaction times of the metaorder ω, we define ω) as the sensitivity of the metaorder ω regarding to the parameter θ. Let

V θ (ω) := N (ω)-1 i=0 S θ t i (
V θ := t∈T market S θ t be the sensitivity traded the same day d on all the options of the universe, which means all the options traded by the algorithms summed over all market transactions T market occurred in the day d. Hence V θ can be viewed as the absolute sensitivity traded by the market regarding to the parameter θ. Note that this quantity depends only on the universe the day d. Therefore all the θ-metaorders executed the same day will share the same absolute sensitivity. Hence we define V θ (ω) V θ as the θ-daily participation rate. The sign of ω will be noted (ω) (i.e. = 1 for a positive sensitivity options metaorder and = -1 for a sensitive negative one), the sign of (ω) is also the sign of S θ t 0 (ω) , S θ t 1 (ω) , ..., S θ t N (ω)-1 (ω) which is invariant during the life of ω. Clearly, most of the quantities introduced in this section depend on ω. For the sake of simplicity, we chose to omit this dependence whenever there is no ambiguity and will often write T , N , V θ , instead of T (ω), N (ω), V θ (ω), (ω).

The market impact curve of a metaorder ω quantifies the magnitude of the relative θ-variation between the starting time of the metaorder t 0 and the current time t > t 0 , θ being a parameter of the implied volatility model. Let I t (ω) be a proxy for the realized θ-parameter variation between time t 0 and time t 0 + t. We use the variation proxy defined by

I t = θ t -θ t 0 , (1)
This estimation relies on the assumption that the exogenous market moves W t will cancel out once averaged, i.e. as a random variable, W t should have finite variance and basically satisfy E( (ω)W t (ω)) = 0.

One can thus write

(ω)I t (ω) = η t (ω) + (ω)W t (ω), (2)
where η t (ω) represents the market impact curve and W t (ω), the exogenous variation corresponding to the relative move that would have occurred if the metaorder had not been sent to the market.

Equity Metaorder Options Metaorder

Object of study stock market options market Quantity of interest stock price P implied volatility parameters θ Effective size quantity Q sensitivity S θ Effective variation

P t -P t 0 θ t -θ t 0 Effective Sign 1 {Q>0} -1 {Q<0} 1 {S θ >0} -1 {S θ <0} Market impact proxy × (P t -P t 0 ) × (θ t -θ t 0 )
Table 1: Comparison of the quantities of interest between equity and options metaorders

Implied volatility model

In the previous section we have introduced the following proxy

× (θ t -θ t 0 ) (3)
to measure the market impact of θ which is a parameter of the proprietary implied volatility model used to conduct this study. Although the approach used is model dependent, we think that for the two parameters presented here (at the money forward volatility and skew) the results must be the same for any implied volatility model fitting the market. This is also one of the reasons why we limited ourselves to study the market impact on the at the money forward volatility and skew in our article as every reasonnable implied volatility model must have at least those two variables. We have respectively sketched on Figures 1 and 2 the expected market impact variations of the at the money forward volatility and skew. For sake of clarity let us recall that the terminology at the money forward refers to quantities function of the strike evaluated at the forward.

Data

Data description

The data set we use for this analysis contains trade orders executed by the BNP Paribas options trading desk for the 2-year period from June 2016 through June 2018 on the KOSPI 200 options.

The KOSPI 200 Index is a capitalization-weighted index of 200 Korean stocks which make up 93% of the total market value of the Korea Stock Exchange. In order to perform rigorous statistical analyses we need to be able to calibrate the parameters of the implied volatility model as often as possible. This is necessary to observe the variations of the parameters at a frequency similar to that of the transactions. To make this possible with sufficient accuracy, we have only considered executions trading mostly short maturities options.

Because of the high frequency in the execution of the orders, we only consider options metaorders with at least 5 completed transactions. This underlines the fact that we want keep only metaorders that reasonably act as liquidity takers and could impact significantly the market. Indeed, while the equity metaorders studied in [START_REF] Said | Market impact: A systematic study of limit orders[END_REF] had an average time life of several hours, the options metaorders presented here last a few tenths of seconds.

Data representation

For sake of confidentiality numerical values of market impact, daily participation rates and durations are given in percentage of reference values -related to the market and BNP Paribas options desk -that cannot be disclosed. Nevertheless we will give when it is possible the value of the ratio of these reference values in order to make feasible comparisons between the at the money forward volatility and skew metaorders. For instance the reference value of impact is the same in Figures 9, 10 and 11 for the volatility metaorders, so it is possible to make comparisons between these graphs. The same applies for skew metaorders in Figures 12, 13 and14.

In what follows we will use the following convention: All graphs relating to volatility (resp. skew) metaorders will be drawn in blue (resp. green).

Options Market Impact -The Liquidity Taker Mode (Aggressive Orders)

Let us now focus on the market impact generated by a series of aggressive executions on the options market. More specifically, we consider the two kinds of metaorders: namely the at the money forward volatility and the at the money forward skew metaorders. We recall that aggressive limit orders are limit orders that cross the spread in order to trigger an immediate transaction.

It is important to understand the fundamental difference between these two types of metaorders. For example, if someone wants to buy at the money forward volatility (Fig. 1), the simplest strategy is to buy options with strikes close to the money forward. Whereas if one wants to buy at the money forward skew, a simple way to do so consists in buying out of the money options while at the same time, selling in the money options (Fig. 2).

A simple way to check that the previous intuitions presented in Figures 1 and2 are not totally irrelevant is to study for the two kinds of metaorders the relation of the θ-market impact curve as a function of the (unbiased) standard deviation (weighted by the θ-sensitivity) of the strikes of the child orders pertaining for each metaorder denoted by σ K/F (Fig. 3 and4). Hence for Figure 1: Market impact on the implied volatility surface for a given maturity T under the effect of buying options with strikes near the money forward. One can notice how this could increase the at the money forward volatility parameter as emphasized by the red arrows.

Figure 2: Market impact on the implied volatility surface for a given maturity T under the effect of buying options with strikes at the right of the money forward and sell options with strikes at the left of the money forward at the same time. One can see how this could increase the at the money forward skew parameter as emphasized by the red arrows. each θ-metaorder ω the expression of σ θ K/F (ω) is given by

σ θ K/F (ω) = N (ω)-1 i=0 S θ t i (ω) K i /F t i (ω) -µ θ K/F (ω) 2 V θ (ω) -1 , (4)
where

µ θ K/F (ω) = N (ω)-1 i=0 S t i (ω) K i /F t i (ω) V θ (ω) (5)
is the weighted mean with K i the strikes of the corresponding child orders and F t i the forward prices at the instants t i . Figures 3 and4 are obtained by following the methodology presented in [START_REF] Said | Market impact: A systematic study of limit orders[END_REF] and briefly recalled in Sec. 4.1. A careful scrutiny of Figures 3 and4 shows that in the case of the at the money forward volatility metaorders, the less diversified the strikes are, the more important the market impact. It is the opposite in the case of the at the money forward skew metaorders: the metaorders with the higher values of σ K/F match those which present the most important market impacts. Besides as expected, one can note that the average values of σ θ K/F are higher for the at the money skew metaorders. Those observations are in line with the intuitive predictions presented in Figures 1 and2. The first points appearing on the Figures 1 and2 actually correspond to the shortest metaorders in length and duration, so they are just noisy points.

The ATMF volatility metaorders

In this section we focus on metaorders impacting the at the money forward volatility parameter of the implied volatility model. One can observe that, in agreement with the intuition, metaorders with shorter durations are more frequent (Fig. 5). This histogram is quite similar to the one of presented for the equity aggressive metaorders in [START_REF] Said | Market impact: A systematic study of limit orders[END_REF].

Daily participation rate distribution

The ATMF skew metaorders

In this section we focus on metaorders whose at the money forward skew parameter is the parameter of interest in the implied volatility model.

Data

• Study period : 1st July 2016 -30th June 2018

• Order types : Aggressive Limit Orders The distribution of the durations of the at the money forward skew metaorders (Fig. 7) is quite similar to the distribution observed in Figure 5 for the at the money forward volatility metaorders. One more time we notice that metaorders with shorter durations are more frequent.

Daily participation rate distribution

T AT M F -vol ref ≈ T AT M F -skew ref and |V| V AT M F -vol ref ≈ |V| V AT M F -skew ref ,
we can say that the duration and participation distributions are quite similar for the two types of metaorders. Also the two kinds of metaorders present closed number of orders and metaorders during the same period as disclosed in Sec. 3.3.1.1 and in Sec. 3.3.2.1, so we can conclude that at the money forward volatility and skew metaorders must behave in the same manner. Hence the differences in terms of market impact between the two types of metaorders could be only explained by the way the market reacts to each type of metaorder.

Notations

Notation Definition

ω A metaorder O(ω)
Option of the metaorder ω d(ω)

Execution day of the metaorder ω t 0 (ω)

Start time of the metaorder ω T (ω)

Duration of the metaorder ω N (ω)

Length of the metaorder ω V θ (ω) θ-Sensitivity of the metaorder ω V θ (ω) θ-Sensitivity traded the day d(ω) on all the options of the universe θ (ω)

Sign of V θ (ω) O(ω) Price of O(ω) Ω θ
Set of all the θ-metaorders identified by the algorithm Ω θ n * ⊂ Ω Subset of the θ-metaorders with N ≥ n *

Table 2: Notations and definitions

Remark 1. As we only consider metaorders that have at least 5 executed transactions, Ω θ = Ω θ 5 .

4 Market Impact Dynamics

Market impact curves

The main results related to market impact dynamics are now given, namely, the market impact curves for the at the money forward volatility and the at the money forward skew metaorders.

In order to plot the market impact dynamics, a similar bucketing method as the one presented in [START_REF] Said | Market impact: A systematic study of limit orders[END_REF] is used: Let x, y being two arrays of data and consider for example that one wants to plot y as a function x. First one starts by ordering the couple of values (x i , y i ) according to the values of x and then divides the sorted (by x) distribution (x, y) sorted into N bucket equally-sized buckets. This procedure yields N bucket subsets of the distribution (x, y) sorted , (x i , y i ) i∈I 1 , (x i , y i ) i∈I 2 , ..., (x i , y i ) i∈I N bucket , and for each bucket I k the means values (x k , y k ) is computed. The last step of this bucketing method is to plot the points (x 1 , y 1 ), (x 2 , y 2 ), ..., (x N bucket , y N bucket ).

To study the dynamics of the market impact, one plots ( (ω)I t (ω)) ω∈Ω,t 0 (ω)≤ t ≤t 0 (ω)+2T (ω) . The first sub-interval t 0 (ω) ≤ t ≤ t 0 (ω) + T (ω) corresponds to the execution of the metaorder, whereas the second t 0 (ω) + T (ω) ≤ t ≤ t 0 (ω) + 2T (ω) corresponds to the relaxation. The study of relaxation presents a degree of arbitrariness, since a choice has to be made as to the elapsed time after the metaorder is completed. For the sake of homogeneity, the relaxation is measured over the same duration as the execution. This choice seems to be a good compromise to cope with two antagonistic requirements, one being to minimize this elapsed time because of the diffusive nature of prices -which affect also the parameters -, the other being to maximize it so as to make sure that the relaxation is achieved.

In order to perform an extensive statistical analysis involving metaorders of varying lengths in physical and θ-sensitivity time, a rescaling in time is necessary, see e.g. [START_REF] Bacry | Market impacts and the life cycle of investors orders[END_REF] and [START_REF] Said | Market impact: A systematic study of limit orders[END_REF]. With this convention, all orders are executed on the time interval [0, 1] and parameter relaxation occurs in the time interval [1,2]. For each metaorder ω, one considers [0, 1]

instead of [t 0 (ω), t 0 (ω) + T (ω)] [0, 1] = [t 0 (ω), t 0 (ω) + T (ω)] -t 0 (ω) T (ω)
for the execution part of ω and [1, 2] instead of [t 0 (ω)+T (ω), t 0 (ω)+2T (ω)] for the relaxation part of ω, and then averages using the bucketing method previously described on the time-rescaled θ-sensitivity quantities.

In fact the θ-sensitivity time here plays the same role as the volume time in [START_REF] Said | Market impact: A systematic study of limit orders[END_REF]. Indeed, traders in the options markets buy and sell more sensitivities than the products themselves in order to reduce the expositions of their portfolio i.e. they think and plan not in terms of options (e.g. calls, puts), but rather in terms of sensitivities of their portfolio.

The time variable t ∈ [0, 1] in the Figures of Sec.

The ATMF volatility market impact dynamics

The blue points correspond to θ-values during execution, θ being the at the money forward volatility parameter, and the red points correspond to θ-values observed at identical times starting from the end of the metaorder.

Figure 9: Market impact dynamics in the case of the at the money forward volatility metaorders (θ ≡ ATMF volatility, set : Ω, 1,026,197 orders, 149,441 metaorders, temporary impact: 0.34, permanent impact: 0.17)

The analysis clearly yields an increasing, concave market impact curve. However, on can observe that the curve has a linear behavior at the beginning and becomes more concave towards the end. This is explained in particular by the fact that the duration of the metaorders is quite short. The decay observed in the last points (in t = 1.0 and t = 2.0) of the curve is an artifact, already discussed in [START_REF] Said | Market impact: A systematic study of limit orders[END_REF], inducing a bias towards the end of the curve. It can be explained by the larger number of metaorders of smaller lengths and with lower impact. Also note that on the three figures 9, 10 and 11, the larger the metaorders, the higher the impacts: 0.34, 0.60 and then 0.86 for the temporary market impact.

Figure 10: Market impact dynamics in the case of the at the money forward volatility metaorders (θ ≡ ATMF volatility, set : Ω 10 , 215,274 orders, 17,286 metaorders, temporary impact: 0.60, permanent impact: 0.39) Figure 11: Market impact dynamics in the case of the at the money forward volatility metaorders (θ ≡ ATMF volatility, set : Ω 15 , 54,203 orders, 2,958 metaorders, temporary impact: 0.86, permanent impact: 0.57) Figures 9, 10 and 11 clearly exhibit the concave shape of market impact during the execution part, followed by a convex and decreasing relaxation. Simply by eyeballing Figures 9, 10 and 11, one can safely assume that relaxation is complete and stable at a level around respectively 0.17, 0.39 and 0.57. However, on Figure 11, relaxation does not seem to be quite smooth. This behaviour for those larger metaorders is essentially due to the fact that the set Ω 15 contains much less metaorders.

A conclusion to this section is that the concave shape of the temporary impact and the convex relaxation curve concerning the at the money forward volatility metaorders are in line with the empirical results observed on the equity markets and higlighted in [START_REF] Bacry | Market impacts and the life cycle of investors orders[END_REF], [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF] and [START_REF] Said | Market impact: A systematic study of limit orders[END_REF]. Also, and more interestingly, the market impact and relaxation curves confirm the theoretical findings of [START_REF] Farmer | How efficiency shapes market impact[END_REF] that the impact should be concave and increasing, and that the final impact after the execution is performed should relax to about two-thirds of the peak impact. Indeed while on the Figure 9 the ratio between the permanent market impact and the temporary market impact seems to be much closer to 1/2, one can observe in Figures 10 and11 -which correspond to larger metaorders and therefore more significant -, how this ratio gets close to 2/3.

The ATMF skew market impact dynamics

The main results related to skew market impact dynamics are now given, namely, the market impact curves for the at the money forward skew metaorders. To this purpose we use the same bucketting method introduced in Sec. 4.1.

Figure 12: Market impact dynamics in the case of the at the money forward skew metaorders (θ ≡ ATMF skew, set : Ω, 1,304,714 orders, 174,091 metaorders, temporary impact: 0.26, permanent impact: 0.10)

The green points correspond to θ-values during execution, θ being the at the money forward skew parameter, and the red points correspond to θ-values observed at identical times starting from the end of the metaorder.

The results show an increasing, concave market impact curve, with a linear behavior at the beginning. The curve becomes more concave towards the end. As mentioned in Sec. 4.2, this is due to the fact that the durations of the metaorders are quite short. The decay observed in the last points (in t = 1.0 and t = 2.0) is the same effect observed in the at the money forward volatility metaorders. Also note that on the three figures 12, 13 and 14, the larger the metaorders, the higher the impacts: 0.26, 0.48 and then 0.77 for the temporary market impact.

Figures 12, 13 and 14 clearly exhibit the concave shape of market impact during the execution part, followed by a convex and decreasing relaxation. Simply by eyeballing Figures 12, 13 and 14, one can safely assume that relaxation is complete and stable at a level around respectively 0.10, 0.32 and 0.51. However, on Figure 14, relaxation does not seem to be quite smooth. This behaviour for those larger metaorders is essentially due to the fact that the set Ω 15 contains much less metaorders.

Figure 13: Market impact dynamics in the case of the at the money forward skew metaorders (θ ≡ ATMF skew, set : Ω 10 , 405,918 orders, 30,932 metaorders, temporary impact: 0.48, permanent impact: 0.32) Figure 14: Market impact dynamics in the case of the at the money forward volatility metaorders (θ ≡ ATMF skew, set : Ω 15 , 136,410 orders, 7,260 metaorders, temporary impact: 0.77, permanent impact: 0.51) A conclusion to this section is that the concave shape of the temporary impact and the convex relaxation curve concerning the at the money forward skew metaorders are in line with the empirical results observed on the at the money forward volatility metaorders. Also, and more interestingly, the market impact and relaxation curves confirm the theoretical findings of [START_REF] Farmer | How efficiency shapes market impact[END_REF] that the impact should be concave and increasing, and that the final impact after the execution is performed should relax to about two-thirds of the peak impact. Indeed while on the Figure 12 the ratio between the permanent market impact and the temporary market impact seems to be much closer to 0.4, one can observe in Figures 10 and11 -which correspond to larger metaorders and therefore more significant -, how this ratio gets close to 2/3. 5 Square-Root Law

Volatility and Skew metaorders square-root laws

The results presented in this section are certainly the most important of the article. They confirm the consistency of the Square-Root Law already observed in the equity market [START_REF] Almgren | Direct estimation of equity market impact[END_REF], [START_REF] Bershova | The non-linear market impact of large trades: Evidence from buy-side order flow[END_REF], [START_REF] Gomes | Is market impact a measure of the information value of trades? market response to liquidity vs. informed metaorders[END_REF], [START_REF] Mastromatteo | Agentbased models for latent liquidity and concave price impact[END_REF], [START_REF] Moro | Market impact and trading profile of hidden orders in stock markets[END_REF] and [START_REF] Toth | Anomalous price impact and the critical nature of liquidity in financial markets[END_REF], the bitcoin market [START_REF] Donier | A million metaorder analysis of market impact on the bitcoin[END_REF] and more recently in a short note [START_REF] Tóth | The square-root impact law also holds for option markets[END_REF] the authors highlighted that the Square-Root Law also holds for option markets according to their definition of the implied volatility metaorders which is more global. The method presented here which relies on the definition 4 tends to be more local focusing on the local deformations of the implied volatility surface through the variations of the parameters of the model which are in fact the projections of those local deformations.

The Square-Root Law is the fact that the impact curve should not depend on the duration of the metaorder. Indeed, almost all studies now agree on the fact that the impact is more or less close to be proportional to the square root of the volume executed. Considering options market, which plays the role of an executed volume is in fact the θ-sensitivity executed regarding to a θ-metaorder. However, the so-called Square-Root Law states much more than that. It basically claims that the market impact does not depend on the metaorder duration.

In what follows one plots the θ-market impact normalized by a volatility factor θ × I θ σ θ as a function of the θ-daily participation rate V θ V θ for both the at the money forward volatility (Fig. 15) and the at the money forward skew (Fig. 16) metaorders, σ θ being the daily standard deviation of the parameter θ and I θ t = θ t -θ t 0 as defined in Equation ( 1). One observes that a power-law fit gives exponents for the θ-daily participation rate close to 0.5 (≈ 0.56 for the at the money forward volatility metaorders and ≈ 0.53 for the at the money forward skew metaorders). In both cases, the analysis shows that our options metaorders, present a market impact following a theoretical curve of the form σ √ R with σ a volatility factor and R a participation rate factor. Those findings support the idea for a universal underlying mechanism in market microstructure.

× I σ AT M F -vol ref ≈ 2 × × I σ AT M F -skew ref and |V| V AT M F -vol ref ≈ 0.7 × |V| V AT M F -skew ref ,
show in comparison to skew metaorders, that lower participation rates is needed to obtain larger normalized impacts in volatility metaorders. Hence altough volatility and skew metaorders present similarities as underlined in Sec. 3.3.3 the market reacts much more easily to metaorders of volatility. This can be explained by the fact that trading volatility is more common than trading skew in options market.

Fair Pricing

In this section we deal with the fair pricing condition of our options metaorders. First of all let us define the S-WAP (sensitivity weighted average parameter) of a metaorder ω as the quantity defined by

θ S-W AP (ω) = N (ω)-1 i=0 S θ i (ω)θ t i (ω) (ω) V θ (ω)
where t 0 (ω), ..., t N (ω)-1 (ω) represent the times of the transactions of the metaorder ω and

V θ (ω) = N (ω)-1 i=0 S θ i (ω).
Hence we want to compare θ S-W AP -θ t 0 with θ t 0 +2T -θ t 0 (Fig. 17 and18). The red line represents the perfect fair pricing condition as it corresponds to θ S-W AP -θ t 0 = θ t 0 +2T -θ t 0 . It appears from Figures 17 and18 that the fair pricing condition can reasonably be assumed to hold. This is in line with what has already been observed in the equity market and mentioned in [START_REF] Said | Market impact: A systematic study of limit orders[END_REF]. One observes also that the greater the absolute θ-variations, the more one moves away from the perfect fair pricing condition. In agreement with the intuition high variations are generally associated to longer and larger metaorders that are therefore more affected by the diffusive nature of the prices.

One can note that the fair pricing condition is more effective for the at the money forward volatility metaorders. One reason for this is that trading volatility is more common than trading skew on the options market as already mentionned previously. where t 0 (ω), ..., t N (ω)-1 (ω) represent the instants, |Q 0 (ω)|, ..., |Q N (ω)-1 (ω)| the quantity (positive) and O 0 (ω), ..., O N (ω)-1 (ω) the prices of the transactions of the metaorder ω. Hence we want to compare P -P t 0 P t 0 with P t 0 +2T -P t 0 P t 0 (Fig. 19 and 20) where P t 0 and P t 0 +2T are respectively the prices of the same portfolio at t 0 and t 0 + 2T . The red line represents the perfect fair pricing condition as it corresponds to P -P t 0 P t 0 = P t 0 +2T -P t 0 P t 0 .

To conclude this section, we observe that the fair pricing seems to be also hold on the options market. This confirms the fair pricing hypothesis introduced in [START_REF] Farmer | How efficiency shapes market impact[END_REF] as a universal mechanism concerning the metaorders and their interaction with the price formation process.

Conclusion

This work is an empirical study of a large set of metaorders in one of the main Asian index options market. A new algorithmic definition of an options metaorder has been proposed. Our study contains two distinct groups of metaorders using aggressive limit orders: a set of at the money forward volatility metaorders, and a database of at the money forward skew metaorders. The statistical results based on this definition show a pretty good agreement with some observations already highlighted in the stock markets: Square-Root Law, Fair Pricing and Market Impact Dynamics. In both cases, the analysis shows that the temporary impact is increasing and concave, with a convex decreasing relaxation phase. More precisely, the price reversion after the completion of a trade yields a permanent impact such that its ratio to the maximum impact observed at the last fill is roughly two-third, as predicted in the paper of [START_REF] Farmer | How efficiency shapes market impact[END_REF] and already highlighted empirically on equity markets.

  Market Impact: A Systematic Study of the High Frequency Options Market (published in Quantitative Finance, Volume 21, 2021 -Issue 1)

Figure 3 :

 3 Figure 3: Market impact as a function of σ K/F in the case of the at the money forward volatility metaorders (θ ≡ ATMF volatility)

•

  Study period : 1st July 2016 -30th June 2018 • Order types : Aggressive Limit Orders • Parameter : θ ≡ at the money forward volatility • Filters : metaorders ω ∈ Ω θ • Number of orders : 1,026,197 • Number of metaorders : 149,441

Figure 5 :

 5 Figure 5: Duration distribution of the at the money forward volatility metaorders

Figure 6 :

 6 Figure 6: Daily participation rate distribution of the at the money forward volatility metaorders

•Figure 7 :

 7 Figure 7: Duration distribution of the at the money forward skew metaorders

Figure 8 :

 8 Figure 8: Daily participation rate distribution of the at the money forward skew metaorders

  4 is actually the θ-sensitivity time, i.e., the ratio between the θ-sensitivity of the metaorder already executed at the time of the observation and the total θ-sensitivity of the metaorder -of course, at the end of the execution part this quantity is always equal to 1. The market impacts disclosed in Figures 9, 10 and 11 in Sec. 4.2 (resp. Figures 12, 13 and 14 in Sec. 4.3) are given in percentage of the same reference value making possible comparisons between them.

Figure 15 :

 15 Figure 15: The square-root law in the case of the at the money forward volatility metaorders, power law fit: y ∝ x 0.56 , R 2 = 0.965.

Figure 16 :

 16 Figure16: The square-root law in the case of the at the money forward skew metaorders, power law fit: y ∝ x 0.53 , R 2 = 0.977.

Figure 17 :

 17 Figure 17: Fair pricing of the at the money forward volatility parameter in the case of the at the money forward volatility metaorders

Figure 19 :

 19 Figure 19: Fair pricing of the portfolio value in the case of the at the money forward volatility metaorders

Figure 20 :

 20 Figure 20: Fair pricing of the portfolio value in the case of the at the money forward skew metaorders