 Welcome back. As usual, I'll try to discuss what we had yesterday. One thing is obviously that tell me things are not clear. You should tell it always, but I'm not. OK, I think things are clear. I think you understand it because we take a lot of time explaining, but if not tell me, because I don't have other way to know. So I'm less after the exam. So OK, what we discussed yesterday were continuing this question of models in which some stylized models of how information gets built into prices. So this economic type of models, which are OK models, so simplified use of the world, but give some good insights. So one was this figure that discussed. So there was this Gloucester-Milgram model. It's written there. I don't write it up. And so what we have seen is that we had two parameters in this world, several parameters. But what were interesting was this W and sigma. One was the dispersion of the way uninformed traders trade, which is this W. And sigma was the dispersion of information of how well informed traders are. OK, so we had these two measures. And what we saw that already in a stylized model, we can come up with different regimes. One regime, which we said that is on the right-hand side here, where this dispersion of these uninformed people is very large. There is only one solution. So we had the bidas spread on the y-axis. There is only one solution. No, on the upper one. So what happens here? This is the spread in the first approximation. This is the money, or proportional to the money that a market maker makes in the business. So in this regime, what we say is that there is a small spread. There is one solution. It is relatively small spread. But SW increases, the dispersion of these uninformed increases, so they continue to trade always. So the price is now 100, and some want to trade at one, to start selling at one. And others want to trade at 300. So super disperse. Many people, so small gain on each trade, but everyone always wants to trade. It's a money machine. I mean, the uninformed. Here you have two solutions. So there is a middle range of this W over sigma. You have two solutions. If there is a single market maker in the world, what he would say is that, OK, it's better. The upper solution is more gain for him. And if he's alone, he might choose that, though. Again, at every transaction. Yes, so this is per transaction. I mean, the units are. We don't have units here, but yes, for each transaction. So he can choose to be up here if he's alone. But if we are two of us, and you choose to be there, I might choose to be here. Everyone will trade with me. I will make a smaller gain than if I were here. But you will make zero gain because nobody trades with you. Of course, we can agree that we are stressed here. So I mean, there are things which are not in a model like this. But there are two solutions. One in which he's making a lot of money. One in which he's relatively OK. If you have competition, you expect to go to the downward curve. But that becomes a field which is not studied by the model. And there is the third region, as we said. So below some critical value of this W. Simply, there is no solution to this. Meaning there are so, so the uninformed people are so non-dispersed or the informed people have so super information. So the sigma is OK. So the sigma is maybe the W is easy to understand. It's dispersion uninformed. The sigma is the dispersion of information. It's how different the true price or the final price is from the price I have now. So if it's a very big difference, then there is a very good information of those who are informed traders. And so there is a region where the ratio of this is small for this market maker. There is no way to set any spread in which he can make up for the loss that he had against the informed traders to make up for this loss on the uninformed. So it was, yeah. Yes? Physical meaning. I would say that no real physical. In this sense, there is nothing to take off from it. And we can analyze it. So it's easy. The way we, I mean, from then on, we have one solution. So this solution diverges. And one remains for larger W. But no. So there is no special case there. Numerically, there is a divergence, but not very important. OK, so in a super simple model, you can get some intuition about how a market can work. We discussed all these problems with the noise trader models, among which I think there was a main. So the issue was that, OK, can all these models assume a one-step game where there is a final price and then this final price is the price that the market maker can get out of the market and put in his pocket his game. And this is not obvious. If then you think that, OK, the market maker boasts from you, but then he has to exit this position against someone, maybe against another market maker, then the game continues. So he will also, with the fact that he wants to exit his position, he will impact the price, changes all this picture. We'll see a model, I think, tomorrow where, I mean, a very short model where this is taken into account. Just to exit the position, that's a clear claim, right? If you have something in your pocket, you sell it, that's OK. But if you sold something, then you have to re-buy it. So to be at zero, at the very end. So OK, so there were these discussions, and then we went to more empirical approach. So there was this question of, OK, there is this way claim of fundamental efficiency that prices contain all information available in the market. So it's already ill-defined, but maybe because I made, I claim it like that. And so we discussed a bit these questions of, OK, can this be the case? This is what I called liquidity paradox. There was a question afterwards, why is it a paradox? It's not a paradox. It's a contradiction. That if you understood, so what did we see there? Do you remember that actually, if you look at empirically the liquidity that there is in the markets, any moment, what is the volume available at the best price for selling or buying, it is several orders of magnitude lower than the total daily volume traded and even lower than the total market capitalization of the stock. So if you want to trade anything which is a meaningful quantity, 1%, 0%, 1%, 10 to the minus 4 of the total value, which is a possibility you have to trade for days or weeks. So this was the claim being a bit vague still, that if this is the case, indeed, you cannot have all information in the prices because nobody knows your true intent. You're trading a small quantity now, but in your mind, you have something much longer term. So it must be only on a longer term that maybe this gets built into pricing. And to see this empirically, we got to the question of, well, indeed, there is a very long-term persistence of the signs in the trades. So there was the correlation of the trade sign, which is a sign that people are doing some, OK, for a long time, the order flow is persistent. And we discussed very briefly this question of, OK, if you call this the correlation, you might try to write it up in a term. If you know the identity of traders, which is a complicated question in itself, we won't go into detail. We assume that we might know this. You can write up this thing in somehow two terms, which we call, so here we were a bit in a hurry yesterday, so you can write up in just the sum s. If you know the identities of people, you can write up this correlation as splitting and herding term, where essentially these were off-diagonal terms. So cases where the actor i and the actor j wouldn't be the same in the correlation and when it's the diagonal term, OK? I don't write it up again. We said it yesterday. There are in the slides you find it. OK, and what we found is that, well, indeed, what we assumed or sort of hinted on is that splitting is, so this is the correlation. The sum of this is what we have seen here. So that the splitting part dominates over herding on essentially all scales that we study here. These are intraday timescales. So let's say 1,000. This is in trade time, as we discussed. So 1,000 trades is order of a day in this period, OK? So this is just to catch up with yesterday. And OK, then there are two things that I wanted to discuss about here, this. So actually, but only going very deep into a very, very slightly a point that I wanted to mention is that actually what I have very briefly written up, it's a very simple calculation that one can somehow define this type of correlation simply as the following. So if you for two agents or two traders, I, sorry, trader I, you can define his autocorrelation. So which will be C split, OK? Sorry, let's be cleaner. So you can do something like this. So there was a sum over I, so the splitting. So it can be written up as a sum, OK? You sum over different traders. But there is a term which is the activity. How often does, OK, here we have I, I. How often does the same trader trade two times apart? How often does this happen divided by a number of points, let's say? So it's a probability of being active two times later and the actual correlation of the signs, OK? And actually, one can look at the two terms. These are related to things we have seen before. So this activity is, OK, how bursty you are behaving. And actually, one can look at the behavior of this. OK, let's, OK, maybe rather look at this curve here. So what we see here is this Pij, so this term here. So how, probably, them to be active, but here we plot it only for leg one. So given one I traded now, how probably is the J trades in the next time step? This is real data, so people. As a function of Pij, meaning Pij, it would be on the identity line if they are independent, right? And what we can see is that, so the black ones are the off-diagonal elements. So when I is not equal to J, the blue is the diagonal I equals to J, you can see that the off-diagonal is roughly on the identity line. So agents seem to act independently from each other. But if you look at the same agent, OK, at least for leg one, you see that they have a much higher probability of acting. So if I traded now, I have a much higher probability to trade in the next time step than an independent model, I would say. So it's these bursts of activities that we have seen on other scales before. And of course, we plot it here for tau equal to 1, but it's for any time scale. So essentially, this Pij has a power-lowish decay in time. So but this is just to know. So that's it for this question of persistence. I guess it's clear. I mean, it's just a simple thing. OK, now the question is, is what I discussed all the time. I mentioned it all the time that we will get back to this. So the question of price impacts, OK, so how does, how do actually these trades move the price itself? And my claim was that instead of these models on complicated economic models and information, we will try to come up with mechanical models of how simply trades, which means that there are some fluctuations in the supply and the demand, how they mechanically change the price, and how this can lead to diffusivity. So that's it for persistence. And I wanted to discuss more about the impact, which I almost started yesterday, but then didn't have time. So I said it millions of times price impact is the correlation between the order flow, so correlation between trade. Yeah, actually, I didn't explain it much, because I forget that it's assumed that it's essentially PII that we had. It's a slight difference, but it doesn't matter. I just want to say that the probability, so the persistence in activity for the same person has a parallel HDK. So it's, again, a long-range correlation in the, if it's just a 0 and 1, if I am active or not, there is, again, a long-range correlated possible. But we didn't discuss it in detail. I mentioned it. It's good to know, but we won't. OK, there are many of these things that come up that we don't have time to go into detail, but just I mentioned. So this is one of them. So OK, so this is price impact. I write it up again. So it's essentially the correlation of trade direction and subsequent price change. OK, so you'll write that properly. And OK, here I said trade direction. It could be any event, right? You could measure the impact. So what is this? This is the response of the system to what you're doing. It could be anything that you will only discuss trades here. But of course, it could be the question of limit orders. It could be cancellations. Any type of event can generate a response in the system. And of course, the problem is that, OK, so the correlation of the trade direction and subsequent prices, but we call it impact. So we think it's due to the trade. But how do we know that the price wouldn't have changed without us? So this is always the difficulty in trying to measure the response in a system like this. You don't know what the system would have done without you. You cannot easily do experiments. And so we'll see how to study this. And actually, this price impact, so what we do today, we will discuss, I think, only empirical results of price impact to be able to get to models afterwards. And we'll divide this in two. So we will, OK, actually, it's important to think to mention. So we'll divide it in two. So first, it will be the question of individual trades. That's clear, I think. But the question will be a notion that actually I mentioned yesterday, but I have to define it properly. So given that people are auto-correlated, as we have seen here in the red curve, meaning that you make a decision now which you're fulfilling on a long time. You want to buy 1,000 shares, and you're doing it for two days. The big decision is called meta-order. It's just a question of language. So meta-order is the big decision that you want to do, which has to be cut up into individual orders or trades. So the existence of meta-order is essentially seen from the red. And we will divide the question of impact into two, individual trades and meta-orders. And OK, so there are some comments on this that we made before that there are two different beliefs of this impact. So if it's really you changing the price or you forecasting the price, this is two different schools. So is the price there existing without you, meaning that there is a price discovery that you're making? You want to understand what the true price is, which is a bit like this model that we discussed in the previous talks? Or it's a price formation. The price actually moves just because you're doing something. Well, I believe in the second one, and that's what we'll see that I think that's the case. But there are several schools on this. So what do you want to measure? OK, just to be a bit more clear on this. So what one would like to measure as impact? At a leg. OK, let's say this is the, I write it up and then I explain what I want. So to be formal, one wants to measure something like this. So what do you want to measure? You execute at time t. Execution means trade. OK, it's a bit of a language question. Executing at trade. You're executing at t. There is some state of the world which can matter for you. You do not know what actually these things are. It will be the state of the limit order book. It will be some states of the market. And so your question is, what is your impact at time t plus l? How does the price change until t plus l? OK, so this is this type of thing that you want to measure. We'll write it up in more proper terms. But the issue is this, that of course what you want to, what you would like to measure is the following. You define it like in the following manner. You say, OK, what is the expectation of the meat price at t plus l? Given the fact that I traded and the state of the world was f t, OK? Clear? But of course what you care is what would have happened if I wasn't there. So actually what you would like to measure is somehow this. I'm writing a simple stuff, but it's good to think about it once. Yes? Is i stands for impact? Yes, i stands for impact. This is a definition. The way you would like to define this, sure. But how did the price move given that I traded? And of course there might be some conditioning variables. But of course you want to subtract somehow what would have done if the world was the same, but I didn't come to trade. Is this clear what I'm writing here? So what you want to say is that at time t you're trading. What you care is what will the price become at t plus l? Given that you traded in this state of the world, OK? f t is ill-defined. Everything, I mean you can imagine that if your trade happens during Christmas it has a different impact because everyone is at home at Christmas, I don't know. Then if it's on a day where there is a huge market crash happening in the US and everyone is there. So there are some variables that might control this dependence. We will forget later, but in theory there might be variables that it's OK. Imagine most of you are physicists. OK, so even then you're doing an experiment in physics. But you don't know, don't yet know what. It's not that they reproduce this experiment. It's a new thing. You don't know what are the variables that come out. Maybe it's the temperature in the room has an effect on the critical point in the system, whatever. But what you want to subtract, of course, in this case you want to care how much the price moved because I did something. So ideally you would want to subtract what would have happened if I had not done this execution. So this is the dual of this. OK? So essentially you look at the response of a system to something you did, but you want to subtract the expected move of the system without you. So you really want to look at the response to you. Yes, but the difference between these two is actually the response of the system to what you do really. Yeah, so I want to know what happens to a ball if I kick it. But the ball is already moving. So then I kick it, I measure what happens, but I want to subtract what the ball would have been doing if I hadn't kicked it, which was already moving somewhat. So I think it's an OK concept. It just never comes up in physics. In physics you can do experiments, typically. In physics, you can't really know what the system, I don't know, you know that the system is in equilibrium and then you kick it. And then you can reproduce this, then you run next time. It's in the same state. Somehow you try to govern the state of the system and then you don't kick it. And then you measure this one. It's not always trivial to do, but in most of the physics experiments it's a doable thing. Is this clear? So this is what you want to measure, but you cannot measure this, of course. So actually what you can in practice measure, so let's say this is some ideal thing. If I write here, it's visible. So what you can actually observe instead of this, so let's say let's call it an observed impact at t plus l by in the same language, so pom, pom, pom. Instead of what you can measure, it's simple. It's simply what happened to the system. OK, it's basic when I'm writing up. So the first term is the same. OK? And you can just, what you can measure is really subtract the state of the system in the moment you acted. The price, what you want to do, the variable you're measuring at the time you acted. OK, you cannot do much. So exactly what is the difference between these two? Is this? I mean, is this minus this? I don't have to write it up, right? So the difference between this and this will be the difference between the second term here and the second term here, which is exactly what the price would have done without me in this language. Well, OK, which you could imagine that it's not very important. Actually, we'll see that often you can forget, but most probably people trade it because they expected the price to move up without doing anything, right? So it's a dangerous assumption to use this instead of this. But this you cannot measure. And OK, so what is a lucky case actually that one finds out that, so what is my order of writing here? Without us, nothing would have happened. We don't know what really to do. And OK, luckily, so the difference which will be actually, we often call it prediction. This is pred, but it's not really important. So just the difference between this will be exactly where the price would have gone if we haven't traded. So the change of the price if we hadn't traded. OK, I'm not doing anything here. So OK, actually, what is luck that in practice one finds that this is not a very bad approximation. But just to get an idea of what you can do in a market like this, of course, possible measures. Well, one is that, OK, you hope you can do an experiment. Just it won't be cheap. Or it depends. So you can do a random experiment. And this is things people do. And I analyze data of this. So what you do is a random trade. Essentially, OK, what you hope is that, OK, if I'm just by chance choosing moments when I'm trading, on average, this thing should be zero. You hope that the average is out. So it's doable. The problem is that, of course, it means that you have to go to the market. You have to do trades in random moments, which will cost you. It is not free to do, or they will let you. In some way, they will cost you. We'll see how. Because so what does this mean? If I hadn't traded now, what would the price have done, going up or down? If I'm choosing properly randomly, we have seen that the price is essentially diffusing. So on average, it should be OK if I have a large number of data points. On average, it should be OK. You can try to find trades of which you know that there is no information in it, which is a bit harder. So let's call it to be more clear in random moments. So it's similar to what it is before. But OK, let's look at the data and try to identify maybe people who trade for other reasons, not because they have a prediction on this move. It's possible, maybe, so it's harder to believe than an experiment if you come from physics. But it's a possibility that you try to choose moments. So it's at least free. Or you can try to model this thing separately. The other possibility is that, OK, you have some separate model of what the market would have done. If you have a lot of data, you might be able to understand the structure of this second term. Yeah, so exactly. Now the question here, are these the same? Of course, this is the question. Yes, exactly. So the question is, are there these three things that you can do to, I mean, are there you can forget that this exists here? But that is a bit dangerous. You can do three types of measures, do random, or do I go to people who trade it because they had an idea of what this is and ask them what it was. They might tell me if it was in the past. They might tell me just the number, what they expected the price to move in that moment. Or if you work on this, you can do it yourself. So but there are several ways to do this. There is no trivial thing. It's, I mean, the problem is that, OK, you can experiment as physics like, but it's not like in physics in general. OK, so find. OK, that's a bit the hardest in a sense to understand. So maybe I understand that, OK, on average, I don't know how this behaves. But in the moment when you trade, I know what it does because I know that you're not trying to predict the price. You're just buying because for some other reason. Let's say you're, OK, let's go to a simple example. I know you're selling only because you want to exit the market and you want to do something completely different in your life. So I can guess, OK, so it's not because you see what the price is going to do in the next time that you're doing, that you're acting, but for completely other reasons. So I can expect that this averages to zero on your own trade. Yeah, but what I can hope is that the moments when you choose to trade are some random moment. So you don't have a prediction on this. This averages to zero on this moment. No, just that the state of the world being F. Yes? But I hope if they are random, that things average out. And so all these terms, so it averages out on F, but it also, this has a zero average. Let's continue and we can get back to this. So there are several ways to do this. Actually, what is lucky is that they lead to the same result roughly. So similar results. So you're lucky. You can measure things. But it's not obvious, you have to choose one of them. And is this OK? OK, so what do you want to really measure? So OK now, if you verify that, OK, what does this mean? That these are similar, that somehow this term here probably is non-zero on average, but it's negligible to your effect on the market, OK? On short time scales. Is this clear? And so you want to, OK, so what you typically want to, one possible measure that you want to do if it is to look at the response of the system. So what you can measure, one definition actually that we'll do, you define a response in the following way to look at the following average. So what you do is, OK, first of all, there are many simplifications here, but is it clear what we do? You trade at time, or someone trades at time t. There is a sign, so this we discussed. So this is the sign of the trade at time t. And you correlate this to the price change from t to t plus l, right? This is a response function. And OK, so there is an important thing to see that actually that what you have here is only the sign, right? So what you say is that this is only the sign. So it's not the volume exactly. I don't care if you traded 100 or 1,000. I look at the sign of it. We'll see that this is, it's not by chance that we have this approximation, but OK, this is the way we wrote it up. And then we'll look at the volume dependence as well. So OK, just let's look at the function like this. OK, so this is what we see on the left-hand side. So what we see is exactly this function, this function of l, time lag, in number of trades. Four different products. So there are, the different curves are for different products. So two of them overlap, actually, I think. Anyway, so let's forget the dashed line. You have three lines here, OK? So what you see, so this is, OK, this is the response of the market. So this is this measure, right? So given that there was a trade at time 0, we sign epsilon 0 in this sense, how where does the price go up to time l? And so what you see is, OK, we see a couple of things and we'll discuss this in details. But OK, you see, first of all, that you seem to see that the first point, so that there is some structure in it on short scales. So it is, it flattens out for long times. There is some increase on short times and probably not much more to see in the first approximation. So OK, let's also just to write this up. So what you see is that flattens out. So what is your idea of why this is happening? OK, this is just an empirical fact. The idea is that the fact that it's increasing on short time scales, it probably is due to the autocorrelation of the trades. So the fact that you saw a trade at time 0 in one direction is autocorrelated with the trades in the next steps. So this is our guess, why, why, why, why, why, why, why there is a time, where is the structure? I mean, we will show this, but for now it's only a guess. But if it's our guess, then it means that maybe we want to look for other measures of impact. Because if this response function that we are measuring actually is not just the measure of the response to this trade here, but the response to all the subsequent correlated trades in the market, then we have to do some deconvolution to see things better, but OK, we'll see this later. And OK, what does it mean that this flattens out here? That is easier to understand, I think, is the fact that if it continues to grow forever, it means that there is some type of predictability. Given that I know this thing here, I can know that the price will be increasing much later. That contradicts several things that we have seen. It contradicts this unpredictability. So you can say, OK, so after a few trades, so OK, the time scale here is an order of a few seconds at most, this flattens out. All predictability is excluded from market and it stays that forever, believe me, OK? Yeah, so there is some type of response in it. OK, we'll see this when we try to, we'll see an answer to this. But it's decaying, isn't it? Decaying exponentially? Yeah. So there is no correlation anymore. There was some correlation, you integrate the correlation essentially, and then the correlation is zero, so your integral remains constant. But we'll see this more in detail, try to understand what's the effect of a single trade. So what we do now is try to go through empirical results and then try to understand this. There are things that you cannot measure explicitly, you have to have some model for it. This is it. So this is entirely empirical. This is just empirical. We will try to understand them in the next. But we have an idea of, I mean, I can give you a guess why it is the case. So the first one is due to the correlation in the order flow that we've seen. The second is, OK, the second is not just a guess. If it weren't flattening out, you have the feeling that there is a really long term correlation in the price itself, which the price changes themselves, which we know that is not the case. OK? So I just show some other thing, which we won't go much into detail here. But actually on the right-hand side, what I show is the, OK, on the x-axis, we have some function of this response at leg one, OK, so the immediate response. So forget that it's proportional to this. And on the y-axis, we have the spread. And what we see is that there is some type of linear-ish relation between the two, which is, I won't go into details now here. It's just for you to think about it. So recall these models that we had in the other days, this Kyle and Gloucester-Milgrom model, especially the Gloucester-Milgrom, where you say that, OK, so the spread is what the market maker gains on you. But of course, the price response is what he loses on you. The fact that you traded makes the price go up. So the fact that they are, OK, we don't see the proportion here, but they are similar things, means that indeed there is a competition between what he can gain on you and what he loses on you. And it is not exactly zero, but they are close. So we won't discuss this in detail. Think about it a bit, OK? No, but because this is a correlation of, so this is essentially, this is the integral of the change of the price from t to L, right? So it means that the correlation of sine at t and the price change later in the time would go to zero. So this is somehow epsilon t m t prime plus 1 minus m t prime summing over t prime larger than t. Is this clear what time? Oh, sorry. Is this clear? So the price change from t to t t plus L is the sum of price changes in each time. So this is an integrated correlation. So the fact that it flattens out means that the correlation itself went to zero. Is this clear? Move on. I can discuss afterwards. So OK, so the right-hand side is just good to think about. So things make sense of the type of models we discussed to have some idea. So but one thing you can of course do, OK, here we have only the sine. But of course what could be interesting is, OK, sorry. One thing that we see here is that leg one and leg infinity seems to be different. Well, leg one, at least we know that nothing else happened just this trade up to then. So we will concentrate a bit on this for now. But of course, if you are looking for leg one, then this is just a number, not a function. That's very interesting. But probably what should be interesting is to, what I tell you is interesting is to look at the volume dependencies of this. So this is a sign here. But of course, you assume that it can depend on the volume. So if someone trades a unit quantity or a quantity of 1,000 in the same market, it should have different response. So what does one write up? So you want to look at the response at, OK, let's have L equal to 1 to a trade of size V, absolute size V, so which has a sine epsilon, so which we'll define in the following way. I mean, it's very similar to what you have. So OK, nothing changed. But of course, you want to condition on the fact that the volume at T was this. These are all V's. I write V in several ways, but it's always V, OK? Simple. Is it clear? And OK, you want to look at this because you look at data. You want to understand the structure. You look at different things. There is also another point is that, OK, you mechanically would assume. So what does it mean to assume something as a mechanical impact? Well, there is this underlying structure. So this medium you could call, which is the limit order book that we defined. So you can assume that there is some type of mechanical impact. The larger your volume, the more you will clear out volume from this limit order. Is it clear? So in a super mechanical world, I mean it's nice models on this and moving in a random media. But also, if you want to be more in this economics language also, there is this question of information. Well, what you could imagine, maybe it's not true, but what you can imagine is that, OK, larger volumes should somehow contain more information. If you trade a lot, you know something. So even in an informational point of view, probably paraphrase should move more if you're trading larger. So let's see what it behaves. So this is what we see. So first of all, looking at, so on the left-hand side, what we see is this measure here as a function of volume normalizing some way, which I won't define now for a second. It's volume, just if you want to. And what you see is that there is some type of increase. It doesn't seem to be extremely increasing, but OK, there is an increase in it. Still, you have the idea that from here to here, there is a one to two order of magnitude increase in the volume itself. And the response doesn't seem to change that much. And actually, to be more explicit, we can see here the same type of figure on a log-log plot. So the price shift has a function of the volume. And OK, so you seem to have some exponent. Maybe I have to always tell you, so what is the exponent? OK, so this is simply what I'm doing here. I'm redefining a new response function. First of all, I put l to 1. I don't really care about the leg 1 behavior. And I put a v here, and I define that this will be simply the same type of response. Sorry, this should be, of course, 1. So l equals 1. But it's a conditional expectation that the volume was vt. And what I consider mechanical impacts is one has to do that. So you don't only see ideas of the volume in the limit order book. We plotted the limit order book. So you can have some ideas of what is information. But you can also say, OK, so there is the quantity in the book. So we should always keep in mind this type of figure. So we saw that there are some volumes in the limit order book. Actually, I can tell you what the typical shape is, just to keep it in mind. It's important. So the typical shape is somehow something like this, empirically. So this is the best. Is it OK? We've seen it. So this is the typical shape of a book. But the mechanical impact, you can imagine that it's simply, OK, so if I'm trading, if I'm buying, I'm trading against these people here. The price move, I mean, in a very naive way, should be like you integrate the volume, you move the price up to there. So if you want to trade all this volume here, then you will move the price to up here. So this is what I call a mechanical thing, but you can, OK. But we'll get, again, back to this. So it's OK. This is what we can see. Maybe I've calculated the exponent in the meantime of that thing there. So you see that it's a power load dependence. And actually, one can write this up that this will go somehow. I won't go into all the details, but it will have some type of behavior of this. We can see there are some, so it's more complicated. But I don't want to go into all the details, but it's the volume dependence we care about. So why it's more complicated? Of course, you need an overall scale. So we are talking about prices here. You have to have some variable which defines the proper scale of the response. I mean, I think this is clear. This is more physics-like issues. So you should be used to. So there is an exponent here. So if one looks at the exponent, it's low. So let's say you see here you move three orders of magnitude. On the y-axis, you move less than one. So I actually did this guy here. So this exponent is between 0 and 0, 3 empirically. So it's a super weak dependence, right? Is it OK what I'm saying? So it's an extremely concave function. You could feed a log on it if you don't want a power low. It doesn't really a low power low, which is similar to log. So what you see is the very weak dependence. And the question is, why is this dependence so weak? So one tries to understand this. I think it's important also to have an idea of this mechanical picture. And so one way to do this is, OK, let's be explicit. So the question is, why is this dependence so weak on volume? Why, how to look at this value? We can look at, OK, let's look at the shape of the limit order book. And let's look, OK, how does the impact compare to this shape? So what I'm saying is that you could imagine some virtual impact, which I define now, which is simply really the one-shot mechanical component. You say that you integrate, so x will be the ux volume in the limit order book price x. So it's this function here. It's writing it a complicated way. But this function here, you have something there. And you can say that you can look at the integrated volume up to a price change r will be this. I'm not doing so. This is, let's say, cumulative, accumulated volume. OK, so I'm just writing out explicitly what I was saying. Here, you can look at the function of the typical shape of the book. You can integrate it up. And then you say, OK, how should impact look like from this, which is the following, then the virtual impact of size. So a trade of size, I don't know. Let's use another letter just to make size q should have the following impact. OK, so to use the same language, sorry. So this is a capital V. I'm sorry. So this is the same V as here. OK, I think I'm saying it complicated, but it's super simple what I'm saying. It's the integral of the volume in the book here where you invert it. And so this assumes that, what does this assume? So what you do here is the average, yeah? This is the same, exactly. And what is important to keep in mind, of course, what you do here is this is the average. So exactly this integral. You look at the average, and this can fluctuate around this shape. You're looking, OK, what is the average shape of the book? So what should the trade of size q push the price? How much it should push the price? Vr is the cumulative volume in the book. So I'm integrating this function here, the function I found empirically. And so OK, so this assumes that the trades are unconditional, and what we care about, of course. OK, in the first person, let's compare how this. So let's compare this to what we actually measured before. And this is what you have. So OK, it's a bit on another type of curve. So what you say, you have the blue, which is called true impact here, which is exactly what we have seen before. Just to honor, we are on a Lin-Lin scale, so things look different. So the blue, and there is a fit on it, is a low power behavior. And the red one, which is this virtual, seems to be linear, right? And it seems to be linear, which we won't discuss. One can understand this. But especially what you see is that they are very different. So this idea that, so what is the assumption here in this model, so in this virtual impact, is that it's OK to look at the average shape of the book, and that defines the typical impact. And the fact that they are different tells us that it is not at all a good way to look at the market, is to look at the average shape of the book. Because the suggestion of this is that the trades that arrive are very much conditioned on the volume that is in the book. Is this OK? So they are not in random moments. How they are conditioned, of course, we don't understand from this. One can analyze the curves. But it's clear that there is a strong conditioning in the moment when a trade of size V comes. It's very conditioned. It's not a random moment, which, OK, one can expect maybe. But it's good to be able to measure it. And one more thing that I wanted to discuss on the same issue is that it's a bit in the same direction. So how things are considered. So I will just put on a figure. I won't write up much details about this. But what can one think about is, OK, so the trades are conditioned. So there are moments when there are large price changes in the market. What causes them? So is it OK? Again, in this picture, what you would think, that, OK, price moves a lot. If there is a huge volume coming and eating up, what's on the opposite side? It's a simple one. Somebody wants to buy a lot. He will move the price a lot. So what you can do, of course, here is the following. There will be a mechanical part that, of course, if you trade a unit quantity, so something extremely small, then you will, of course, never push the price if there is some liquidity there. So what you can say is, OK, let's condition on the case is when there was some price move. So the volume that came was larger than the size on the best. And let's try to see if there is any size dependence afterwards. So I will show you the figure and try to explain. So what we show here is the distribution of price changes. It's what we have seen before. So the probability that there is a price change larger than x as a function of x. And the four different lines are five different groups of trade sizes, OK? So five groups of V here. And as you can see, the distribution is super similar for this. So very large trades, on average, change the price by the same amount as very low trades, which is a way to say that the moment when large trades come is the moment when there is enough liquidity on the other side not to push the market too much. Is this OK? Someone should say something. Is it not OK for someone? So this is what you see. So what do we see here? Price is OK. There are some, OK, all these things are, I think, interesting. So the volume dependence, the conditioning on the volume in the book. But the most basic thing that we have seen here is that this thing at leg one moves it. So if you trade, it moves the price. This we have already said, but we have seen it explicitly. Now, so R1 is positive. Price moves in the direction of your leg. So the question is, OK, but we have seen it yesterday that the signs are extremely auto-correlated. So if these things are auto-correlated, so how does this not cause inefficiency in the market? And so I will just brief. I mean, one can come up with a guess, but we will be explicit. This I leave here, OK? So OK, so we have seen persistence in epsilon yesterday. We have seen, so OK, this response function I consider, no, no, so I won't write R1 larger than 0. The question is, predictability in the price? OK, so and the question is, so the answer to this, yes, the price remains efficient. So OK, I don't have to explain this. Is it clear what the issue here? Because of course, we can write up explicitly that given that we know the correlation function that tells us, given that we saw a buy, we know the extra probability that there will be a buy in the next second, all these things. But what we can actually look at is the following. We can write up two versions of this response function. So given that, so this is a plus here, one can look at this. So what I'm doing is writing something very similar that we had before, but I'm doing a conditioning on whether the price moving is OK. I'll let you write up, but at least one. Absolutely, sorry. I mean, of course, I could define it for L, but yeah. So this is L equal to 1. So I'm not conditioning on volume anymore. I'm conditioning on a fact that is the sine of theta t is the same as t minus 1, or it's the opposite. So sine is plus minus 1, so we have an easy five here. And which, of course, given the stuff that I didn't write up, of course, the probability of this event is very different from the probability of this event if you're autocorrelated. And so what you can guess if I wrote this up that actually in practice what you can measure is that empirically the first one is lower than the second one. So while there is a, I don't think I will figure about it. So I think you can see that, OK, so there is stock persistence in the order flow, but there is a different response in the market if the prediction was good or not. If I saw a trade of sine epsilon t minus 1, I have a good prediction that the sine at t will be the same. And actually, if my expectation is fulfilled, then the price change will be so smaller than if I'm surprised by the market. So this is how efficiency eventually gets restored. The only way that it can happen, one of the ways that it can happen, OK? So there is no inefficiency, and there is some type of asymmetric response in the market to what's happening. So if the expectations are fulfilled, there is a much smaller response than if there is a surprise. So I wanted to show some figures about the question of, OK, we have seen there was this question of the response function becomes flat for long times. Actually, I wanted to show a figure about this, which is a bit in a different, well, different colors and slightly different language, but the same type of information in it. So what we see here is the blue is the response to a trade at time 0. Forget for now the difference. So the left and the right is the same type of figure. There are differences, but it's not important for us. So let's look at the left-hand side. So the blue is the response function as we have seen it before. So it increases to some level, and then it's flat afterwards. And actually, yesterday we discussed today about the persistent order flow. We discussed, OK, but you can have knowledge about who is doing what. And actually, in practice, if you look at this, you can write it up as being the sum of a response coming because of your own later actions. Oh, sorry, because of the later actions of the same broker detected at time t0. So same is the same detected at 0. And you can look at actions of the other brokers. So this sets all the sum of the two should add to the total response. So what you do here is essentially do a further conditioning here. And what you find is that, OK, it was nice to say that there is a flat response on long times, but actually it's the sum of two very different behaviors. So you continue to trade, and you would be pushing the price up constantly. And others are coming in the market and are either trading against you or putting large volumes to trade with you. So either with limit orders or market orders, they're trading against you. So what I wanted to say with this is that, OK, so there is an equilibrium that this blue curve is flat, but it's the sum of two very strong positive and strong negative dependence with the difference being the blue, which already suggests that we will see in other cases that, OK, there is an equilibrium somehow, so the price response is flat, but it should be somehow easy to break down this equilibrium. If this weak flat response is the sum of something very positive, very negative, you can expect that it's easy to break down this subtle match of the red and the green lines. Is it OK? Another thing that I wanted to mention, what's the time? OK. So all this that we discussed here is on single products. Of course, one can define these type of functions, of course, on cross products. You could also say that, OK, given that I trade one product, what does the other product do? We won't go into detail here. There are some, it's an ongoing, actually very much ongoing field. There are some very nice papers. Visually, actually, it's a similar matrix that we saw for the cross correlation, but it's a different structure. There is a very interesting different structure. So a matrix like this would mean that, given that I trade this product here, how much does the price of this one go up or down? Of course, it's a non-symmetrical matrix in this case. So it's OK, but do you want to discuss this? It's OK? Yeah. So what your condition here is you, you, you. So at time zero, there was someone acting. It is the right guy in the case. And you look at all his further actions up to like 1,000. And you consider and you look at the immediate, essentially you are summing up the responses of each of his further actions. And you're summing up the response of the actions of others. OK, so let's move on. So that's more or less about impact of single trades. So what we see is that it exists. It's very much concave in volume. So it's weakly dependent on volume. And then what I want to discuss is this next thing is to discuss meta orders that I defined before. So as I said, OK. But it's good to repeat. So meta order is a series of trades which originates from the same person and from the same decision. OK, this we discussed. And so you can be a bit formal. So it's a decision to trade a quantity Q, an absolute quantity Q in a direction epsilon, and over time horizon. OK, T. OK, so it's to formalize it. What does it mean? I want to buy. My epsilon is positive. 2,000 stocks over one day. OK, and then, of course, what I will do is just cut this up as we discussed several times. And OK, so the difficulty with this, which you should think about, is that this is not a type of information that it's easy to obtain. I mean, no one will tell you what I mean the actual trades in the market you can look at. But what was part of the same big decision that is no one will tell you? It's hard to obtain data. So either you can go look around and ask from people. You can obtain data like it. You can have some data of your own if you're in a company, which is the case for me. But it's hard to obtain. And actually, I just want to mention that, of course, here I say, OK, it originates from the same decision. Actually, in practice, what you could say, yeah, sure. I just look at all the trades. And on some time horizon T, I want to aggregate all the trades in the same direction, and that will define my Q. So I could say that, OK, I look at all the trades in a day. I sum up what are buying trades, and I say, OK, that as if it was one big metal. It's a very bad proxy. It would give something completely different from what we see here. So it's really important that it's the same decision. It's hard to understand why exactly, but it's important. So this is what we want to look at the impact of these. So on overall time horizon, there are two types of impacts you can look at here. So you can say, OK, I want to look at the impact at the peak. So at the very end, when I executed all my trades, you can define this thing here, peak of a metal that are size Q and T, S, well, essentially this. So usually, obviously, what we expect, typically, that there is a symmetry between buying and selling. So you don't have to look at separately. You don't have to condition on epsilon and do separate functions. OK, sorry, this is an equal. So this is a trivial measure. Sure, you want to see how much the price changed on all this time scale, but the important is there's one decision. Another thing that you can do, actually, is to look at the past dependence of this. So what you can say is that, OK, I write it up, then I explain. So it's clear. It's past dependence. So what was my impact? Given that I wanted to trade capital Q and capital T, over capital T, and for now, I traded small Q in this time scale, in this time that I've passed. We are at time T now. What is my expected impact? So what is the expectation of epsilon times the price change up to small t? Given that up to small t, I traded small Q, and in total, I want to trade this. Simple. In practice, actually, well, again, there is some type of luck in the world that the two give very similar. So actually, the market doesn't know why when you're going to stop trading. So the fact that you trade up to small Q up to now, good, but the fact that you want to trade capital Q is not known. So typically, what one can write up is the following. So this stuff here that we had. OK, so it's easy expectation. So luckily, the market doesn't know the fact that you traded small Q until small T. The market doesn't know that if you're continuing or not on average, so it will be this measure. It will be S if you wanted to trade this in total. What is lower case Q? Lower case Q is the quantity you traded up to lower case T. We have put another index there. So this is what the two types of impacts one can define. So peak and the past dependence. And there will be, of course, we will discuss a bit later what happens if you stop trading. When you stop trading, you push the price. But after the peak, there might be something happening. We will discuss a bit about this at the end, I hope, today. So what is the rule for this? So how does this IP look like in practice? Well, it looks in the following way, which is, I think, one of the most intriguing issues, at least for me, in much of this in this course. It has some type of behavior like this. And I will define all the numbers in a second, all the parameters in a second, something like this. So what do we have here? The impact of trading Q over horizon T will be, OK, this is a number. Let's say, let's consider that it's one. It's order one. It's not exactly one. In practice, it's important for us. It's not very important. Here you have the volatility of the price on the same time scale, time horizon T. How much the price typically moves in this time window? Here you have VT, which is the volume of others in the same horizon T. So what percentage of the total volume are you trading? To some exponent delta. So that's hard. So how does it look like? Actually, here there is a curve. OK, so again, so what's the exponent? It's always a good test. So what's the exponent of this? What's delta? Actually, there is even a hint on it. So what's the delta? What's the delta and why? Yeah, but how would you argue that it's written on it delta? Exactly, I didn't remove it this morning. So how would you argue that it's one half? Yeah, so it's good to have an idea. So indeed, here you move one of the magnitude, while there you move two. And it's two exproximates. So there is OK. And in practice you have, so this delta, in practice is maybe between 0, 4, and 0, 7, close to one half. Actually, so the low is called square root impact law. Square root impact of meta orders. So what do we see here? Actually, this is for, so the two curves are slightly different contracts. It's for futures contracts. It doesn't matter to us very much. What we can see is that, essentially, on four orders of magnitude, we have this parallel usage behavior. For us, on a log log being linear is a parallel, but you can test for the power on S properly of these curves. And OK, it's important to know that you need a lot of data to measure this. So to measure this type of thing, like 10 to the power of five of meta orders, at least. So you should say, if there is one per day, typically, it's a long time that you need a lot of data. And OK, what I wanted to do for discussing what this means, I wanted to just discuss other two other curves, which are very similar visually. So the first one that we've seen was for futures contracts. Here we see one on options. I won't discuss exactly the units here, but the same idea, right? You see the same type of behavior with a power, maybe a bit lower than 1 half. Again, how much is the volatility in the case of options or the price of the option impacted by your trade? And on the left-hand side, you can see it actually on a completely different market, which is on the Bitcoin, which is, again, very similar. So buy and sell separately, you see that the same type of behavior can be measured. OK, so it's interesting that it's universal. If it weren't universal, you wouldn't discuss it much here. And so what is the consequences of this? I mean, everything is seen here, I think, that should be understood, but it's good to explicitly write up. So the consequence is, so what is the consequence of the power law? Well, that it's not linear. I mean, the power law is the square root, that it's not linear. OK, it's trivial. So in practice, what does it mean that the first half of your trading, so you want to trade 1,000, and the first 500 will push the price much less, much more than the, so you want to trade 1,000, the first 500 will push much more than the second 500. OK, this is what it means that it's not linear. It's concave. It's OK, obvious, concave, which, in a sense, one can write. So there is some, anomalously, high impact of small meta-orders. OK, so obviously, mathematically, this is diverging it for quantity being 0. In an actual system, you have discretizations. It's not diverging exactly, but there is a very high impact for small meta-orders. Concave, so what does concave mean? There is some memory in the system, right? So if the first half of your trades have more impact than the second half, then it means that somehow the system remembers that you did the first half. So there is some memory in the system on some probably finite scale, so not forever, but there is, it seems to be an interesting system. Is that clear? And everything is clear, I hope. And OK, so what does this mean? We will get back to a bit order of magnitude soon. But what does this mean? That anomalously high impact of small order as well? OK, so if Q over V is 1%, so you're 100th of the market, which is big, but OK, it can be. This would mean that I is 10% of the volatility, right? So you're moving, by being 1%, you move the price by 10% of your typical daily fluctuation. So it's a big number. It's a big effect. OK, I mean, it's the same. All I'm saying is seen here. And OK, one more thing, which is, I think, important, which might not be seen explicitly there, that, OK, we wrote T here. There is sigma T and VT, but actually, if you look closer, there is no real non-explicit T dependence here. So actually, in practice, what you can assume is that sigma T will behave like the square root of T. And you could assume that VT, so what is VT, is the total volume traded in the market in this period. Well, that is on, to a first approximation, could be linear in time. So in two times more time, there is two times more volume traded in the market. So we should say that there is no T dependence of your impact, OK? Is this OK? If you do the same thing in one hour or in two hours, it will have the same effect. This is an empirical fact. Of course, it's not. I mean, if you do it in a large volume in one second, then it might be. So I mean, of course, it's for well-defined T's. Of course, well, all this that we write here is on an over-never, which works on relatively not infinitesimal windows. Yes, yes, yes, sorry, yes. Did the exact T dependence depend? Yeah, so I wrote delta here because I showed that it can be slightly different than square root. But even if it's 0, 4, it's a very impractical. Actually, there are papers that claim that there is some T dependence to the power 0, 1, 0, 2, which is, OK, you believe it or not, it's super hard to verify. But it's very weak, if there is, it's weak, yes. But those were single trades. Those were also in the moment when single trades arriving. And what we have seen is that it's essentially a conditioning effect. It's a conditioning effect. So we compare this virtual impact, so just the integral of the volume inverted, and the actual impact, and we saw that it's essentially due to a conditioning. Large volumes arrive when there is large available liquidity in the book. But what we are seeing here, these are on longer time scales executed. And here we see that there is a quantity dependence sublinear, but there is, OK. And so I said that this universe, I showed here, some of the university, but I just want to mention clearly that it's true for different type of markets, or equities, futures, options, papapam, for different time periods. So, OK, here these are quite recent data. We did it. Since the mid-90s, there are many results showing this. So in different periods when markets had completely different structures and different execution styles. So it really seems that it's a universal effect, so some minor things in how the microstructure works. If there is one market maker or seven, or everyone can be market maker, these do not affect this result. It's quite universal. And, OK, and so I can say claims that I think in physics, you know this, that the fact that in most system what you expect is the response to be linear, so small kicks have small effects. The fact that this is not the case is, OK, it's usually some being close to some critical point and having long range correlations and things like that. So we will see actually what critical means in the case of liquidity. So why is this susceptibility to the system infinite to small trade sizes? So we will discuss in models, but I mean, there are all these things which I think are clear from this that you can think about. And I wanted to discuss two more things. What's the time? OK, it's perfect. So I wanted to discuss two things. So one is, OK, actually I will switch the two things. So first I want to say I also wanted to give you a bit orders of magnitude for this. So because, OK, I gave an order of magnitude here, but I think it's interesting to understand really the consequence of this type of impact. So what would it be, when does it matter for you? And then I will discuss briefly the decay. So I wanted to discuss two things. One is about costs. OK, so we are discussing about all these impacts so that you are yourself changing the price. There is a response to you, so it's more in the physics language, but it's also an actual cost. So why is it a cost? Because you want to buy, but the actions, because you think that the price will go up. But actually you are pushing the price up and you're buying much higher than what you wanted. And this is a cost for you, right? You had some idea of gaining, and that's why you trade, and it's a cost, is it pure? And so one can, OK, let's forget about this why. I said it's order of one. Let's imagine that it's exactly one, if it's not the case. And OK, so let's assume that the cost is exactly defined in this manner, which in practice won't be the case because if you're moving the price on a square root, then the average price that you pay will be the integral on the square root, which will be 2 thirds of the highest point. So there are some order one effects, but one you can imagine that these do not matter. And so let's go back to the previous question. So when we look at the limit order book, we see that the fixed cost of trading, so the question of cost, so there is a fixed cost of trading if you're not the market maker. So if you're really trading as market orders, which is the half spread, OK? So the half of this difference we discussed here. And actually, just to give an order of magnitude, so these two for relative liquid market, I would say this is 10 to the minus 4 of the price of the product. OK, we will argue everything in the price of the unit price that you have to pay because what you carry is that if you buy something for 100, which is worth 100, what you carry is, compared to 100, what is this value that is in the spread? It compared to 100, it would be 100. Actually, this is called base one basis point, but it doesn't really matter. So this would be the fixed cost. And what would be, OK, so I said this stuff here, so what would be if you trade, let's say, trading 1%, how much we can, OK, we have to assume some things. So trading 1% of the market, so 1% is Q over VT. Let's say VT, let's say T is a day, OK? So it's on a daily scale, you have to have an estimation. So let's say that sigma is 2%, which is a typical daily fluctuation of impractice. So just to have some numbers, so what would this mean? So if we forget why and some things, it would mean that your, let's say, cost, so impact cost, you can calculate from this would be something like 2% times the square root of 1%, which will be 2 times 10 to the minus 3, I think. So it's, let's say, 20 times 10 to the minus 4, OK? So trading 1%, which is OK, it can be something. Actually, what it costs you, that you yourself are pushing the price, is 20 times higher than what the fixed cost was. What you saw in the market, what you really have guessed in a simple manner, up to some factors. So this is one way to look at it. Actually, another one, someone asked this question at lunch at, I don't know, someone I chatted, so yeah. Yeah, so the fixed cost simply means that, let's imagine a limit order book like this. You want to buy, so it's this price that you are going to, you send a market order, you want to buy, so it's this price that you get. And if you just look at this, then your best guess of the price would be probably here. So the difference between the two is exactly the half spread, which is order of 10 to the minus 4 times the overall price. The price of the product. You have a product which costs 100, and a stock on Microsoft costs 100. In empirical, the typical size of the half spread would be 10 to the minus 4 times 100, 0.01. And why I'm arguing, I want to be without dimensions. Everything is in the same units. What you care is that, OK, I obtained something, and what fraction of this do I have to pay in a cost? So another type of question that came up, which I think is good to think about, is that, OK, so this is 1%, but that's big, right? So at what size can we come up with something super hand-waving equation of, OK, for what star over v, which I will, for simplicity, I will call x. So what size does impact dominate or becomes larger than fixed costs, which is the spread, OK? So I'm going to do the same calculation, just with an unknown in it, OK? And, OK, so anyone can, so just taking the same numbers, one can write up. I hope I didn't make an error, but what you would write up is the same thing. So you want to say that, so what I think would come up to me, OK, check this. So it would be the following, so x would be something like 0, 25 times 10 to the minus 4. So what I'm doing here, I'm looking for the x for which this thing becomes 1 BP, equal to this, OK? So what does it mean? So x is your fraction of the market that you're taking. So you're slightly larger than 1 in 100,000, OK? So it's a small number, at least it seems, relatively. So this is the point where actually these impact costs start to be bigger than the fixed costs that you would have, OK, why am I saying the fixed costs? You see in the market, you can look at it, and you can guess for these impactors, you have to have a model for you, and that you have to. And so what does this mean? Just yesterday I looked on the internet what this means. So let's take the apple, which is a super liquid stock, the stock of apple, I'm not talking about fruits, super liquid stock, just the first example that one looks at. And what I saw that the day before, so probably it was one day leg, the daily volume on apple. OK, this you wrote up. So what I found, but OK, it can vary. But I found that daily volume, so for example of apple, daily volume, OK, let's put a D here, was the order of 10 to the 7 shares times, so in the price I think was like 175, OK? You can put VD, divided by P, I mean, you can put all this into the equation. What you get is the size where this starts to dominate, is that if your trade is order of $50,000, OK? If you trade $50,000 in the market, you start to, your impact already, so you're super small, your impact starts to, OK, seems a lot. Of course, you don't go and you don't go trade $50,000, but that's a small number. It's a company can easily trade these amounts and much, much more. So it's an imaginable number, right? So it's even there you start to dominate it. OK, already below that, it's a non-negligible number. And just to give a, I just wanted to give a why, actually, this is a super, these areas studied a lot actually in companies, which is OK. There is the answer to this, why it is. But just, for example, in the company where I work and studying impact, the order of magnitude paid in impact in this type of cost a year is, I think, $250 million or something like this, which means that, which is just your loss because of trading, right? Which means that, OK, if you can make it 1%, you can have a model which decreases it by 1%, then it means a lot. Because of course, we wrote up this square root equation. But of course, one can work on trying to decrease the pre-factor or change the system. OK, and one more thing I wanted to say then, we stop is what we discussed up to now is, OK, what does the price do while you're executing, while you're trading? It goes up as a square root, OK? There is also the question, what happens afterwards, right? So does it relax back? Does the price stay there? If the price stays at the level where you pushed it up, OK, you changed the state of the market, you might have gained less. But you still, you bought and the price is higher than before. You also bought at a higher price, so you gained less, but still it's there. Of course, if the price then reverts, then you're not happy. If you have a picture like this, if you have a picture like this that you start trading, let's go into, we are in time now, it doesn't really matter, we could be in volume. You start trading, you push the price up as a square root, OK? Actually, on average, this is the price that you're paying. You can do the average on a square root. OK, if the price stays here, great. You still made some money on it. But of course, if the price then decays, then you're really in shit. You pay this price and afterwards the price comes here. So it's an important thing to understand. Actually, it's not at all well understood. And I don't even have the figure. Oh, I thought I have a figure about it. I'll have it tomorrow. But actually, what happens in practice is that, OK, it's very hard to measure, because the longer you look in time, the more the volatility dominates everything. So you seem to have a decay. And it's not clear. So with some power low, some exponent, some low exponent. And it's not clear if it goes to zero or to a final price. And just one thing to think about in this, I won't discuss it here, that of course where the price decays afterwards is very important for something which is called price manipulation. So something that I didn't discuss at all here. But of course, if your actions change the price, is it possible that you just do trading in a way to change the price to gain on it? So can it be the case that I start buying, I push? OK, since there is a memory in the system, it could be the case. So what you want to say is you do a round trick. You buy 100 and you sell 100. You own nothing and you have nothing afterwards. Can you gain money on this? In a simple world, if you only have the square root, you would say, OK, you do this when you push the price up. You buy here. And you do this when you push the price down. And you sell here. And you lost money. But of course, mathematically, you can come up with dependencies on the way you're impacting here and on time of type of decay, which would allow you to make a money machine out of this. If impact was linear, you're doing exactly this. And you don't, you're on this graph. Well, I didn't define much time scale here. OK, I'll answer this in the question. So we won't discuss it at all here. But of course, it's a very important question. You don't believe that the market can be easily manipulated in a way that you just buy and sell, buy and sell. No idea what the price will do. And you're just gaining because of your own actions. Or if it's the case, you want to change the rules of the market. But probably, it's not the case. But it's a very interesting literature to understand this. And the case is on any scale. You don't want to be able to do this, of course. It's on a single trade as well. You don't want to just buy, sell, buy, sell, buy, sell. But you also don't want to be able to buy now, wait for 10 days, and buy during two days, wait for 10 days, then sell. What you expect is the same. It's that you expect the price not to be able to gain on this movement in itself without having a prediction on the price. Sure, but if there is the economics behind the price, we consider that here what we consider in any cases, as we discussed in the beginning, is that your expectation is that the price, you don't have a prediction on the price. A long time, sure, you can have it. Of course, you can make money because you have a good prediction, but not because of you pushing the price there and back. Also, actually, it's illegal. It's called price manipulation. Of course, illegal to trade in a way to just push the price to some artificial level. But that's another thing we don't discuss. But actually, mathematically, it's a very interesting problem of what type of decay is allowed. For example, if you have square root impact, what type of decay is allowed to have a non-manipulatable price? I think, for example, the result, if you have a power law increase, you probably have to have some other exponent power law decrease. If you had an exponential decay, you can prove it mathematically that you can manipulate the price. Let's stop. Well, also, the increase is hard to know. Is this true? I was a bit weighed in the definition of, so you have a meta-order. But is this true if I decide to buy for a year all the time? Will I follow this power law? You cannot test it. And you think that there is a memory in the system, but it won't be an infinite memory. But there are things that empirically is hard to test it. But what is more important, I think, is that, OK, you trade it in one direction. There you can measure things. But then you stop trading and you just wait for the price to do what it does afterwards. And you have strong ideas of how it should behave. You can measure certain things, but it's not a well-decided question. I will show tomorrow a figure. I wanted to show a figure, but I forgot. No, but there are regulators whose job is to try to understand this. The way it works in a market is that actually someone can come to you and ask you, OK, so why did you do this trade? OK, you can always come up with a nice story. But no, it's a hard. So it's a bit surely there are many people who try to manipulate the price. Actually, it can happen also without wanting to do bad. You're optimizing. You're trying to optimize all that we saw on impact. And by chance, you fall into it. It's not obvious to know. But so it's, and then there are the policemen trying to come after you. But you're OK.