 I will try to get back to points which were mentioned. But I don't know, you have questions? Yes? I mean, if it's a conceptual or a bigger question, we'll discuss it here, maybe at the tutorial, I think. We're wondering what's the actual definition of our market maker? Perfect. I want to discuss. I want to get back to this, because that I see, that it's not super clear. OK, let's discuss this. And OK, so actually, this is a difficulty of all these scores that we have to introduce many concepts. And some are OK. OK, I'll try to make it clear. So to start with, there was all this story that there are essentially two types of orders in a financial market. In today's financial market, there was limit order and market order. I don't know if these are clear. I will write them up, what they are. So because it's OK, what you say is, OK, there is LO is limit order, MO is market order. And the way you define them is there are a few properties. So let's say you're trading, you have a direction. You buy or sell. So you have a direction of this. There is a quantity that you want to buy or sell. And there might be a price where you want to buy or sell. Depending on what you do, you have in mind some type of price. Let's say price wished for. OK, I will call it like this. And OK, so there is a hidden column here, which is what are we discussing, so the product. But I can always write it there. But we are talking, in most of the cases, we are talking about single products. That there is one given product. It may be the stock of Microsoft or whatever that you're discussing. So in this moment, you're not choosing between Microsoft and Apple. It's one given product. So what does direction mean? Direction means buy or sell. So if you want to buy or sell, whatever type of order you send, you have an idea of if you want to buy or sell. And I mean, this is logical, but we are defining here things. So you have a sign for both cases. Usually, we use it plus or minus one, because it's simpler in an equation, but it's buying and selling. You also have in mind what is the quantity you want. And actually, it's defined for both. So you have a total amount of these things that you want to own after the trade, so you want to buy or you want to sell. However, the difference, OK, and you have a product in all cases, however, the difference between limit order and market order is that in the case of limit order, you define a price which you wish, if you're buying to pay for this product. I will try to give a better example, but so this is the definition. Well, in case of a market order, you don't have this. You say, I want to buy a given quantity now. And OK, of course, you want the best possible price that is available in the world, but you do not define yourself. So this is a question of definitions. And it's clearly true. OK, so this should be known. This is the way markets are made. I mean, you could have other type of orders that you define further things that I want to buy. I mean, this could be some dynamic quantities, whatever. But this is the definition to simplify. What is obvious from this is that in the case of a limit order, you're more constrained than in the case of a market order. It's not ensured that you will find someone immediately. So you might have to wait or in the language of a limit order book where we are listing these things, you could usually say you're queuing, but it doesn't matter. You're waiting. While in the case of a market order, since you have less constraint, but given the fact that there is someone who wants to say, if you want to buy, so someone to be on the opposite side with this quantity, it's typically immediate. But of course, strictly speaking, you can only say that you have to wait less. You have less constraints. So that's the definition. And this seems to be complicated the way we put, but it's essentially the same if you go into the market. It's always hard to say good examples, because as we started, these products are well-defined. And buying one stock of Microsoft is the same as buying another one. So they are identical. There are no differences. There are several flats that you can buy in the market, and they are the same. It's a new set of flats, but for some reason, they are not exactly the same price that they are sold. You can do different things. You can go there and you say, I want to book a buy. The direction in this case is defined. It's one flat, but you say, OK, well, I don't want to pay more than $100,000. People are selling between $110,000 and $150,000. So you can be there waiting, or you can come and say, I have money in my pocket, and I really want to move out of my current flat, and you want to buy. So these are, we'll get back to the example. So this is a question of limit order and market order. And so the word market maker, I think, is not a super good word, actually, because we are confusing everyone here now. So why is this word market maker? Because traditionally, in a market, limit orders were not possible by everyone. If you went to this market, it's to a stock market in the 80s, for sure, I think also in the 90s, you could send market orders, you could send the market orders to buy or sell. And limit orders were defined by these people who were called microtechers or specialists or something who were designated. They had a contract to do this. They had the right to be there posting limit orders. They had also obligations of how they have to do. So that's where the word comes from, market maker. Today, anyone can post limit orders in the book. OK, sorry, just one step back. So these market makers that were designated, of course, there are two directions. So they were there to buy from you if you want to sell and sell to you if you want to buy to be an intermediate, intermediate. Today, anyone can post limit orders in most of the markets. Actually, there is no one clear rule there. Different markets can have different regulations. But typically, anyone can post limit orders. So it can be you if you don't need any contract. But it doesn't matter. So it's different. Market maker is not very well defined anymore. But why we talk about market makers in these models is because many of the intuitions that come up are valid also if there are several people that can place these orders that are queuing for a while and then are being executed against or are canceled. OK, is this clear? And so most of the models where this came up, so there was this economics type of models, what we called Keil model and Gloucester and Milgram model. The idea was that you try to describe all people that are trading in a market with a very fine, so very simplified model. And to get intuitions of how different people should be acting. And we had in mind that there is only one person here posting limit orders. I will say what's the difference if there are many. And so I think the intuitions of this model, I won't write up these models again, of course. But the intuition that came out is that in a type of model like this, typically those who put limit orders, so those who are the market makers, gain something because they can define their price, so they pay the price which they want. And if they are on both sides of the market, they are gaining the difference. But they are afraid that the moment, so the moment when they execute these guys in the first line is chosen by the second line. So they are afraid that these are not good moments. So this was one of the intuitions. It's actually, we can come back, getting back to the example of buying flats, you go to the market and say, OK, there are all these flats between 110 and 150,000. You say, OK, I'm ready to pay 100,000. And you say, OK, then go home. They're waiting there. They know that you wanted to buy for 150,000. In three months, someone calls you who before wanted to sell for 150,000, calls you, OK, I'm ready to sell you for 100,000 now. So what does this mean? You start thinking, OK, maybe there is something. He knows something. The fact that he sends me 150,000 now, which is much lower than it was before, might be because there is some big problem with this flat. So you might decide that you're afraid of this problem, right, of this situation. Now, in a market, if you are there in the market, when someone wants to sell you at this price, you have to buy. So you cannot change your mind at that moment. But it's clear that you want to choose your price in a way and change the price, maybe dynamically, in a way that you try to avoid these type of issues, OK? So the problem of being, what was the good for this? So the person who chooses the moment will have more information than you, is logical in any everyday example. And in these type of models, you can get an intuition about it. So what this type of model said is that, OK, several different things. But one was this, that for the person putting limit order, he has to, well, get an idea of what's the distribution of the information, whatever it is, of the people on the other side. And how many people have information, and how many people you don't have to be afraid of because they are choosing their movements. Let's say, if they are putting market orders, it's essentially random. Is this clarifying a bit? Or it's, and OK. And actually, there are other things that we learned from these models, it's important to have, for a financial market to function, actually, in this type of stylized models, it's important to have people who are there trading because of not perfect information, at least. So this we call noise traders or uninformed. And this is also a trivial claim if you say that, yeah, everyone is perfectly informed in the same way. And their conclusions from the same information is the same that no one will trade. It's in the limit, it's clear. But of course, if you write up a proper model, you can try to understand what quantified them, what noise means, what information means, and come up with different regimes of the market where it functions in a different way. And so one more comment on this. So this was the definition of market maker. Why do we put in these models only one market maker and not 72? Because first of all, because these models were made in a period when there were typically one market maker, so it's a historical question. But the intuitions do not change if there are three market makers there. They are all afraid that someone has more information on them. So they are gaining on, trading against people who are stupid and losing against people who have some superior information. Of course, actually, in a computation, the model becomes more complicated because they themselves will be in competition with each other. But the intuitions do not change. The quantity, the final solution, the numbers can change. And this is actually what we have seen. There was this Gloucester-Milgrom model where there is a spread as a function of the dispersion of some variables. And there was a regime where there are two solutions. Of course, if there are several market makers in competition, you expect the lower level solution to win in that case. If it's only one person here, it's not necessary. OK? Thank you. It doesn't really change the thing. So the idea that you can, so how did you discuss it? Yeah, today's markets, actually, what we discussed, let's say on Friday, and we'll discuss today is more to these continuous options where these are defined. But so the idea that there are these two types of, or it doesn't really change in the two situations. And I've discussed this whereas in auction. It's more because it gives an intuition of, OK, it gives two intuition. One is that, how can you put orders in a market? How can it function? And there is also the intuition that, OK, you're not happy with the type of market where the origin of Arasium, where there is no information passing through. So you put an order, you're blind there, and then they say, OK, were you able to buy at this price or not in a continuous update. But main ideas do not change. OK, in that type of model, no. There is a market maker who is, but actually Cal will be discussing this tutorial. So I don't want to go into detail here. It will be discussed again and properly looking at the calculations. What you do is, you have a market maker. So by definition, he will decide these limit orders. And there is an, OK, how to convert it to this language. It would be, we'll have to think how to convert it to exactly this language. It will be discussed in the afternoon. Maybe it's better discussing after. But it's the same world holds. OK, we continue. So getting to new things or, OK, first let's start with a bit old things because half of the group wasn't here on Friday. So what we discussed is a type of model. I mean, we won't go through everything again, but there was this idea of coming up with some type of linear model for the changes in the price. So we had this propagator model. I'll just write it up. But I won't do, of course, the calculations again. What you say is that there is some type of linear model for prices. So you say that the price at t will be some, OK, you always have to put some initial condition. But it doesn't really matter. Times the following. So where epsilon was the sign of the trade at time t prime. And there was this propagator. So you said, OK, let's assume a model in which each order has the same impact, but it propagates through time. Somehow there is a time dependency. And what we did is, OK, let's solve this problem for us. And what we had, you can write it up properly, but I don't remember exactly. But what you can do is, well, OK, actually, I won't write it up in a clean manner. You go to get to expect that you want to get measureables to measure. So you say that the response will be some type of convolution of this propagator and the autocorrelation. OK, don't write this right properly. The idea is this. We said it yesterday. It's in the slides. I don't try this. It's a bit longer. But what do you do? There is a convolution of propagator and correlation. And you can solve it. You have everything, OK? The way I put the indices here is incorrect. But what was the important thing? So actually, a question came up there. So OK, so what do we care about this type of models? We postulated something that is a linear model. What can we do with it? OK, one thing that we saw, what we can do with it. OK, of course, you can solve what behavior you expect for G, even the behavior of R and C. So you can try to come up. If you have some analytic form for R and C, you can come up with an analytic form from G. Good. And what we have seen is that, OK, if this is the analytic form from G, what does it give as an impact for a meta-order? We decided, so several connected orders. You are trading in the same direction all day. And we saw that it's not really good prediction. OK, this is what we have seen last time. Is this clear what I'm saying? You remember a bit. Those who are here, I don't know who was here. And so the question that came up, so actually what this said, is that if CL is something like, is a parallel with some exponent, then you can say, then you can find how G will behave with another exponent. We'll be very, very, very, I don't know. So I think it's this. I don't remember exactly the form. I think it's this that you find, right? Yes. So we had a form for the propagator. But of course, the question, OK, this is given by us. I mean, we defined a linear model. We say that they should govern the prices. But we know that prices are diffusive, which means that this R will become flat. We know how the correlation behaves. It's all defined. We don't get new information from this model. So actually what I want to discuss is what extra one can have in this type of model. So first of all, what does a propagator mean? This is only a mathematical way of writing. But what would this mean? Yeah? Well, you said the value of, yes. I mean, the way the model is given, it's something that we have to set the value of beta. But you know how R and C behaves. That will define how G behaves. It's a definition. And then you calculate in the market, it will be the same. So what are the two things that, what are important here? I think, OK, two and a half things. So one is get an intuition. What does the fact that you have this G here and it is decaying? So what does this really mean? And of course, what you want to see is can we look at other predictions that we didn't put in by hand? And do we get good results about those? So first of all, one thing which is what does the fact that you have this propagator here which actually decreases. So what would this decrease mean? Of course, you have a market here where you're modeling only the trades. But there are many other things happening in the market. Limit orders can be put and canceled anytime. So there is a lot of activity. For any two actual trades, I would say that a few tens of events are happening in people deciding that they want to change their price a bit. They cancel their limit or they put it elsewhere. So the effect of a G decreasing in a simple linear model would be that all these other things that you're not modeling make your impact decrease. Because what are you doing? You're deconvoluting the price from the autocorrelation of the signs. If you had all events here, every possible event in the world would be modeled, then a linear model should be just a constant kick in the system for everything without any decay. Because all the decay would be encoded in the further events. If there is no event coming in the market, nothing can change. So is this clear what we're going to do? So first of all, in a type of model like this, we get a bit of an intuition that, OK, so if you measure on the trade level, this G will have some decay. So actually, it's here. Decay, this has to be effect of everything that we are not modeling. And OK, what is the prediction of this? One is that if we model everything, all events, then you can define a G with, let's say, actually, this was a homework problem. So that you can have several events, solve the problem. You will have all the types of correlations. You have, I don't know, let's say you have six type of events. You would have six times six different correlation functions and six response functions. You can solve the system. What you would expect is that if you model all events, then this GP should be somehow constant. So the impact of an event should be like this, that there is an immediate jump. And then it stays there. It's only this. You model that every thing else has to be in the propagator of something else. I mean, it's simply, and there's no finance here. So one thing that we can test is this. Is this true? And actually, it's from a paper. Take with a grain of salt, but it seems to be OK. So there are people who looked at this. I will define. So they define six events. I will say it in a second what they are. So by chance that I said six. And they seem to be all flat. There is some propagator for each of them, which is flat. So it's, again, it's the same thing that we have seen for trades that was decreasing. If you enlarge your space, it seems to be flat. Take it with a grain of salt. I'm not sure that this is completely to be trusted, but it seems to be the case. Actually, the six type of events that they did is, so MO is market order, CA is cancellation, and LO is limit order. And everything which has a zero means things that didn't change the best bid and best ask. And things have a prime, which means that they did change. So this would suggest that everything that doesn't change immediately by definition have zero effect on the price. Because all its effects, of course, will be due to the correlated events to them later. They didn't immediately change. And all those that change immediately, so the prime events have some level, which will depend on the structure of the market, and they would be flat. So there seems to be a sign that indeed, with a finite amount, of course, one possibility would be that, I say, if we model all the events, this has to be the case, but it might be that you have to, every type of event is different. So if you put a limit order of size one to the best price plus three, that would be a different event from putting a limit order of size two. Then you have to model really a lot of events. It seems that with six events, you can encode and you get OK results. You can measure between, I mean, you can go in an event time world for each of the case you do and say, OK, did it change the best bid and best ask? It will change the state of the book, the entire limit order book that you can measure. It is before this event and after this event is the best bid and best ask the same. You can measure it. If you move to an event time so that your time ticks when something happens, then there is nothing simultaneous anymore. Of course, there comes up the questions that in difference. So what would be this in real time? But we don't care about real time very often because what you, OK, I don't go here. We can discuss later. There are many questions that can come up. But if you define your time to tick when something happened, there is always a finite time difference between any two events. You can, it's your definition of time that makes things. Yeah, so you don't have fixed windows. It's continuous time. And yeah, OK. But this is sort of a trivial claim here and the trivial result there. But you should think a bit about it because there is no finance here. And I think the other thing that comes up is, OK, can we use this model to calculate something new that we didn't put in the model by hand? This can be several things. So of course, the question is, OK, can we predict something that is measurable and can we confront it to actually the measures? Of course, the main thing that we want to predict is the volatility of the prices. Because, OK, we have an idea of how the average behaves here. How does volatility look like? And actually, that's where these type of models do not fare too well. So here is one example. I mean, two examples. But so for two different stocks, what you see is, first of all, so this is, OK, this is DL, which we call signature pots. So it was this guy. So the variance on timescale L divided by the timescale. So first of all, this is what you see on data. So this type of curve here. It's good to back you. So what would this mean for actual data? So what does this suggest? How does the price behave? Oh, it's cheating there. Yeah, so no, OK. But others, I didn't discuss yesterday about this. So would it be clear for everyone here what was these points here, the way they behave, so the way this DL behaves as a function of time lag? What does it mean? So I wait for an answer. It's clear what this is. So this is Pt plus L minus Pt. Can someone tell me where? You're talking about this here, this curve, yeah? So where is it? Subdiffusive? Exactly. So it's D. And exactly. So the fact that it's decayed, it's going down, means that this increases less than linearly. So it's sub-diffusive. The fact that it's flattening out means that it's diffusive. So what would it mean? It's sub-diffusive on short scales up to, I don't know, 20 events. This order of magnitude, diffusive after. I wanted to say something about this. OK, yeah, sure. So of course, there is also a numerical value. Just looking at the slope of it to define sub-diffusion, super-diffusion, whatever. There is also a level here, 18 ticks, whatever. It's essentially the diffusivity of the diffusion constant. Diffusion coefficient would be. So now we see what it does. So actually for real price, it's sub-diffusive for short scales and then diffusive for long scales. Diffusive for long scales is always the case. Actually, it is indeed usually sub-diffusive for short scales for reasons that one can understand, which we won't go into very much detail now. But the simple propagator model is also sure that what is called one event is the one event propagator model. So this stuff here, you can see that it's, it doesn't have the same behavior. So it doesn't predict what it will do. It's, we are not very happy with this model. It's good for some things, but it's not perfect. So what is the difference? First of all, it is super-diffusive in the beginning. Then it gets diffusive. OK, that's not so surprising. Everything will get diffusive if all correlations die out. So two things that you can see. One is that, well, the behavior is really different here. And the level is different. Now, you can fix these. You can, of course, see here we have something about that is noise. We can try to, we can try to model this noise as different noises to trigger the system. But then it will be a model that is described much less general. So actually, one could define this noise as having a white noise part that you can tune so that you get with this level there, so that you can play with. Well, the initial diffusivity you have to subversive superdiffusivity, it's harder to trick. I will say a word about it in a second. And OK, so can we fix this? You can add two. I have the same stuff here for two events as well. So this one here is added. It didn't really change that much. OK, the actual level of the diffusion coefficient changes. But still, there is a super diffusive behavior in the beginning. So it means that, no, these models are not good enough. There are things they don't model. There is a positive correlation, which will make, on short scales, a super diffusion. Supportive correlation between epsilons and each of them behaving the same, that you have super diffusion on short scales. And OK, one solution to this that actually you could model this. OK, I don't want to go into it. In the noise term, of course, you could put that you put some short-term, mean-reversion, noise term. So you can play with this, but it becomes ugly. You can tune things to fit the data, but it won't be a very general model. Actually, I wanted to say one word about why this is so, which I mentioned. Why is there is this subdivision for short scales? I don't know if we discussed it before when we were discussing signature plots. So it's not just one trivial explanation, but I think the main explanation is that, so this is on very short scales, from one trait to the other. You have some type of mean-reversion. So because of the fact that you have a, in the limit-order book, you have a best bid, so the best ask, and the best bid. Price is moving here and there. So there will be some trivial mean-reversion on short scales which do not contain much information, which is some structural effect in this type of market. Is that OK? It's OK. Actually, I wanted to talk about something. OK. I want to say one more thing here about this type of models just to sort of close the loop with those models that we had last week. So actually, you can try to do, so these are completely mechanical models. And there were these models like the Kyle and Gloucester Milgram that were more economics-like. So one, you can do something to match the two. What? I should leave it. Or what? Correlation between? The question is so large. So I was just trying to understand what's the deviation in this model, which we are considering to be small so that we can use the linear model. Well, OK. First of all, it's good to keep this in mind because it's a simple model. So just the question of can we use it or not. We have these type of models because it's easy to use. It can give predictions. They won't be perfect, but sort of give an idea. So that's why we have linear models. Now, if you have extremely long correlations, I mean, long correlations in what? In epsilon. For example, in epsilon, you see that you have long memory. Still, a linear model can work. For me, that's not the problem. You could have a linear that some things are long range correlated, but there are linear effects. The problem is that we see that there are several things that they don't explain about the non-linear effects. So epsilon, which is what I didn't get back to you. So epsilon is the sign of the trade, which is the order flow. We saw that it's actually l to the minus half, roughly. Yeah, but it's a correlation. So it's not a distribution. So I don't say heavy tail, but yeah. It's the correlation. So this, OK, one-half is sort of one-halfish. It's below one, which means that it's extremely long range. In theory, in principle, it means that it's the case to zero, going to infinity. Of course, in a real market, you cannot define up to which you can measure things. It is extremely long range. So there is one more thing I wanted to add here, which is a bit too confusing. And to answer one thing that we looked at before, which is possibly all this shit, which is another model. I will very briefly discuss it. It's very similar to what was there before, but another intuition to get, which connects it to what you're saying. It's called to write of the name in its Madavan, Richard's on, and Ruhman's, a model which comes from economists, but somehow it's very similar to what was this, let's say, Gloucester-Milgram type of thing. So assume that there is a, so in this Gloucester-Milgram, this type of things, we had a fundamental price. And there was an issue that came up. So something non-measurable. This is the economics type of model. Something that came up that this is a single. It doesn't evolve in time. So it's that also makes these models non-realistic. So you can assume that this does evolve in time. What is this fundamental price? And actually, for simplification, we will call it simply Pt, which is the trade price. So we will say that, OK, there are no strange definitions of prices. It's the price at which you trade on which people can have information, et cetera. And so exactly in the same manner, it's similar to what we discussed last time, which were these history-dependent impact models. So you could have a model in which you say that Pt minus Pt minus 1 is this type of model. So I won't go much into detail. Those who were not here, so this is really fine. OK, so there was this type of model, which actually, in a very simple case, maps to the propagator that we had. But in general case, it does not. It's that you say, OK, the price change will be a constant. There is no lag dependence here, times a type of surprise. So the sign that appears at t, the direction of the trade that appears at t, minus the best expectation of this direction, one time step before. OK, we discussed this. You're not happy. So we discussed a type of model like this on Friday. So those who were here saw it. Those who don't, we can check in the slides. But we won't go into much, much detail here. But so what I just want to say is that you, OK, this is a model which is very similar to what we had in propagator models. Because so if this epsilon is some linear combination of past epsilons, you can map it one to one. And also in a model like this, you can get to the type of description that we had in this Gloucester and Milgram type of thing. So you say, OK, so if there is a market maker, he wants to protect himself from the others. This is the way prices evolve. So what he will want to do is it's exactly the same equation that we had a few lectures ago. So he wants to do this, to set the ask price which is ready to sell as this. So he says, OK, I want to set my price to the point which is the expected price at t plus 1, given that someone wanted to buy. It's exactly this, which will become OK. So he will have an end in the same way he can set his bid price. I won't write it up, but it's the same equations that we had three lectures ago. So you can come up. OK, so there is a trivial solution that this can have. So this will do anything I just wrote things in. But so for this very trivial type of structure of the market, he can set his quotes A and B. And what is good in this case is that, OK, so there is a dynamics in it. It's not that he will behave. He does a one-step trading and he can get out of the market. But things evolve. And it's on the long term that he has to break even, so not losing the market. So I just want to say two things in this about this thing. And then we go to things which are more interesting. So one thing is that, OK, so I'm just going to write up the predictions of a model like this. And we don't have time to discuss it. And it's not essential, I think. So what this will do is give the following predictions. It will give, for example, a prediction for the spread, which is obviously this, which if one writes up this stuff here, which will be simply 2G star in this model. And another thing that you can get as a prediction, exactly in the way that we had before, you can get the type of prediction following. So this is, let's say, one. And the response of the price to an action on long scales, which I really just write it up. It will be in the slides. But it's super simple to calculate if one wants. We will be something like this. I assume we'll be done with this type of models. I'm going to go to more interesting. So these are, OK, these are simply very simple to calculate from this dynamics. It's a super simple market. But OK, it gives some. So what is the new thing in a model like this is that it's, well, it's somehow in the same old setting. But it gives, for example, a relation. You get a relation between the spread, which is what this market maker will gain immediately in the market, or half of it he will gain immediately. And the response of the price, which is exactly what he's going to lose. He's trading against you. And he's afraid that actually that he sells to you at 100. But actually, the price will be going up later. So he's losing on you because he could have sold later. So actually, there is a relation here between. So this is somehow the loss of the market maker. And this is the gain. This we see. So first of all, one can come up with analyze. This is how, in what regimes, how does this market work? We don't really care. But actually, what we can do is it's a very, very simplified model. But it gives testable predictions. So these in a real market are, so of course, CL is the autocorrelation of the order flow as always. So this is a testable prediction, right, the second line. So actually, if one looks at it, exactly, it's here. So on the left-hand side, actually, such a simple prediction can work well. So what we do here is, well, it's exactly this thing, just the way it is done. It's a bit reorganizing the terms of this equation. But it's the same that we are plotting. So what you can see is that, actually, OK, it was a super trivial model, this economic style, but putting some dynamics in it. It seems to work OK on a daily scale. So what do we see here? Tip average spread on a day, and average of this thing on a daily level, so putting all data from a day together. And it seems to be OK working. It's not perfect, but not bad. The problem comes, actually, if you look for shorter timescales. So that's what you see. On the right-hand side, it's exactly the same things that you are plotting, but not daily level, but one-hour statistics, anyway. So in smaller windows. And you see that there it breaks down. It's different, actually. What you would see is that when this quantity here is small, spreads are larger than they should be, according to this model. And here, they are below what they should be. So what this means, that this type of simple model, it's a bit the same message as there was in a profigator model, that overall, they can give some good prediction. But if you look at the dynamics on short scales, they are not able to describe what's going on. Different correlation. OK, so I know I was fast on this. And I think it's not completely clear. But I don't want to spend more time on this model. The calculations are clear. And then there is not much more insight than what we had. But anyway, if you have questions, after what you think about it, if you have questions, you can discuss, but this model won't go to the exam. So that's it for me for this type of super phenomenological models. But what you do here is you come up with something like this. You can test its predictions. Even if the propagator was perfect, you wouldn't understand why is the G behaving exactly like this. It's not explaining really the underlying microscopic behavior. So actually what I want to get to is another family of models, which I would hope that it's closer to your heart. It's closer to my heart for sure. So it's about modeling the effect that we have seen. Here is there was this curve so that the impact increases the square root of the volume of the meta order. So it was something like this. This is OK. This we discussed. So Q is the size of your meta order. Volatility is the volatility of the price on a scale T. And V is the volume traded in the market on the same scale T on which you are acting. And your impact, so the change in the price in the direction which you are trading. So this is PT minus P0 times epsilon of your trade. Somehow this. So the final price minus initial in your direction will behave like this. And so we saw several things here that actually this T doesn't really matter. You can put different timescale. It doesn't really matter because it's OK. I don't think I have to. Tell me if I have to return to these questions. Or it's clear. It's clear to you, but this is your internship. So OK. If you have questions, ask me, please, because some. So anyway, this is what we are trying to model. And just to come up, just to mention so that this is also these are relatively new things. So this is OK maybe since the end of the 90s, people started measuring this. But it's not a century old phenomenon. And there are several type of models for these type of things, which I think do not work, which are contradictory. And OK, actually, I have a slide about it. I won't write it up, but I just want to briefly explain. I put some slide today because I have a slide from a talk that I did some time ago, actually, because I worked on this model a lot. So there are three types of models in the literature for this type of behavior. And I will just quickly say, and I'm not sure if you just read it, I don't think the language is not super clear. One is that these economic types of models that OK, so if you're trading a quantity Q against someone, so someone is buying it, if there is only one person in the market against you, so who bought this from you, then he will own this quantity Q at the end, and he wants to get rid of this position. And you can come up with arguments that OK, so the time he needs will be linear in Q to get rid of this position, which means that the typical movement of the price, the volatility of the price, would be square root of time, and it would be somehow square root of Q. And so to protect himself from this risk, he moves the price. So there are these very, very economics type of models, which there are several problems with it. I won't go into detail. I'd rather want to go to the third type of model at the end. But just to say, there is another type of model which is actually similar to the economics type of model that we discussed, that there is some this type of fair price conditions, so no one can consistently gain, no one can consistently lose in a market, which is based on, which is actually done by physicists, but based on these things. But again, there are super realistic assumptions. So there are several behavioral models. And actually, the third type of model that I want to discuss is, OK, let's just try to be, let's not try to understand everything immediately, but just let's try to see what can be the case. So if you say that impact increases the square root of your quantity, that in a very simple world would mean the following. So if you imagine an order book which looks like this, so this would be the best bit, this would be the best ask, and this is the volume in the market that is available. Volume. OK, if there is a linear profile, it should be the same linear just that I cannot draw, which is exactly this equation here, right? So the volume in the book goes as the difference from the P0, from the best price. Then of course, if you start trading in a book like this and you trade a lot, if you start eating the volume, what you eat would be the area here. This would be, if you trade a queue, this would be the queue that you eat. And the change in the price would be exactly this thing here, which would be somehow square root of Q. OK, so it's simply geometric reasoning. It's clear, right? And so the question is, OK, can we understand the model in which, so OK. Empirics suggest that it should be somehow like this. The problem is that actually markets do not look like this at all. So if you look at an actual market, the way, I don't know if I have colors. But the way the volume looks like, so this would be an idealistic volume from this equation here. But actually, if you look at the market, somehow you see something like this. So pretty different from what you see. So the question that comes up in, if you want to do just some sort of, OK, this is a completely mechanical approach to this. We don't want to, we don't see what's going on. But this would be N. So actually, we do not see. So why should be this profile linear? We do not see that at all. But also, this is super simplified stuff that I showed here about the linearity, of course. Because what happens is that the price itself is moving all the time. So this, let's say, let's go talk about the mid price. So the mid here is moving a lot. And which you see actually here that, so this was discussed, the orders of magnitude, is that the typical movements of the price is this. So the amount by which you change the price is a small fraction of this, is 1% of this or something. So it's a very idealized view. Even if it were linear, you are adding some diffusive behavior, which has much larger movements than what you're measuring. How could it work in any case? So because this is not what we see. But we will try to understand this type of models. So what can we do in a mechanical way? So actually, even if these are very mechanical models, in this sense, what we want to come up is some type of what is called agent-based model. Which is, I mean, in a physics language, it would be microscopic models. So what we try to do is describe the dynamics of the system on the micro level. That could lead to something like this, and then think about the effects. So agent-based actually seems to be a different world. Of course, an agent-based model, it has a microscopic behavior in it. So micro-syncing dynamics. But it can be also a very simple dynamic. So it's not given that it has to be something super complicated. So this greenish stuff is what we see. Actually, it's even more pointed to the left. So I think it's actually maybe even more like this. Most of the weight being close to the best price. There is an understanding about this. So why is it like this? And how does reasoning go? Well, the reasoning goes the following way. Okay, this is the green stuff that I see. But of course, the green stuff is very much conditioned. So people, you only put your limit order in the book if you think that there is any sense in putting it, right? So we said that you can put a limit order to any price. But of course, if you put a limit order here, it's not very useful because the price, no one will trade with you now, there. I mean, here to sell. No one will trade you there, but people will know that you want to trade there. So you're giving them information, but it's no use of it. So actually, the reasoning goes the following way. Well, one thing that you do know is that even if this is green type of curve that you see for the actual volume in the limit order book, you know that as you go further away from the price, the available volume, which you do not see, should increase, right? Why does it mean, why is this? Okay, so if the price is 100 now, there are people who are, okay, so this price is, I don't know, 105. There are some people who are ready to sell for this price, but surely there are more people who would be ready to sell at 102 because it will contain everyone who wanted to sell here and those in between. And the further you go away, the more people you want to sell, right? If you go up to 200, then everyone who owns this stuff will be ready to sell it to you and go home. So there is a trivial idea here that even if you do not see it, there has to be a volume there somehow, which is called, we will call it latent. So there is a volume that you do not see, but you believe strongly that it should be there. And what we will try is to model this latent volume in the order book. And I will try to, of course, what can one do? Okay, you can come up with models that model the latent volume, then you will have stepped, okay, so why is this? How does this become visible? So how to connect, let's say in this picture, the white and the green, but that we'll leave for later. Is this clear what I'm trying to argue? Someone should answer, right? Yes, is it clear? Is it clear? Because if we should discuss it, if it's not clear, we should discuss it now. So what I say is that for very simple economic reasons, you know that the available volume further away from the price has to increase. You don't know that it has to increase linearly, but you know that it has to increase, so what you actually see in the book might not be the good quantity to look at. So we try to come up with models to see this. And so we will come with a super simple model to start with, so okay, we call it an agent-based model, but we will come up with a sort of deposition, what's the name of it, it's deposition, evaporation type of process to model this because that's what we like, these type of models, is that, is the following, I don't think I have slides about this. There's a question. So this legend order book is in the first, so we will essentially look at two versions, but a simple version is to say that you have some reign of particles that fall on the price, on the price axis. So you say, okay, I don't care about people, I want to simplify world, I say that these are particles falling here, they have some microscopic dynamics, but that's it. So what we'll say is that, okay, this deposition type model, and the assumptions will be very few, so one is that you say that at time t, you have a price which is called pt, and what you say is that there are, okay, some particles or some investors that decide to, okay, so I have to put it actually, I could have made a slide on this, so put a limit order at, what is it, pt plus minus u, so somewhere a given distance from the current price, with some probability, well, I call it lambda minus plus u, so what, what is it, it's okay, at t, okay, so okay, I write it in an ugly way, but I think this should be, is it coming from physics, so what you say is, okay, there are, let's say, particles falling with some rate, lambda, which has a plus minus subscript which we'll actually remove later, it could be different in different side of the book, what you say is that now the price is pt, there will be particles arriving at pt u distance from this pt with a rate, so in a given window, t plus the length of dt, it will be this rate, time dt, the probability of a particle falling somewhere, and sorry, and okay, to get back to the origin, we call this limit orders, but okay, yes, okay, so okay, actually we will also, even forget the subscript as well, so I say here is that, you could in theory say that, so those that arrive above the price will be orders to sell, right, because people want to sell at an expensive price, so above the price it will be selling orders, okay. It's the sign of the trade. It's the sign of the wish that you want to, yeah, so it's, but okay, we write it like this to be general, but we'll simplify life, and actually if you want, I can just simplify, okay, no, I'll simplify in a second, and so this is one, you say that okay, there is a, okay, in a financial language, they can also cancel what they do, so if I put a limit order, they can cancel with a rate which I will call nu, I don't know why. Any distance from the price, again, you can have a rate of cancellation, evaporation in a physics model, right, which will define, so this defines somehow a lifetime of each particle staying there, being somehow one over nu, okay. Why is there a, okay, why is there a U dependence here? Of course, you would guess in a first approximation that the further you're away from the price, the, these rates should change, could change, okay, in general something. So this is okay, so these are particles falling, but they don't do anything for the moment, so in this model, what you say is that okay, let's add something a bit by hand, so we say that, so there is this price fluctuates, so the middle is fluctuating, and in a diffusive manner, so which will be effect of, no, sorry, effect of market orders, which will have some sigma diffusion coefficient, and sorry, we only say that they fluctuate, we don't know what really happens, because okay, we'll see that it does. So it's a very simple model, okay. So what does this mean? In a physics language, it would be okay, there is some particles falling, so some which have a finite lifetime, but they stay there, so there is some media, medium, in which there is a price which fluctuates, which moves, okay. By price, I mean with respect to this price, let's say here. This is one price, actually here, so I mean, at this point, okay, the price is, okay, with respect to the mid, in this drawing, so here, there is no clear, this, I mean, you don't know how it will look like, so actually, your first guess that it should be, that these three price, it should be the same in this situation, if you don't put anything else. So you add in, by hand, you say that, okay, there are these things falling, they don't do anything for now, so we will actually, we'll get a more complicated, you could say that, okay, they interact with each other, but for now, they don't, they fall, they have a lifetime, and then there is something, which is a price, which diffuses, yeah? Yeah, so exactly, indeed, we will go step by step, so here we have something which is not agent-based well enough, we say there is a diffusion for a God-given diffusion there, but yes, we'll get to this, actually, this will be a difficulty, this will be a problem, and we'll try to, but okay, if we try to be step by step. So okay, so this type of models, I think, this you remember, the idea here, I can clean. So this type of models can be analyzed and solved, so one can write up the density, so we'll move more to density of the order book because we are in continuous, so we'll define, okay, so row, we still keep, okay, for the indices, so we'll define the following, so okay, the density of the book at that point u in time t, we will define simply as the volume there, so we'll do some type of average over this, so the volume, you can look at how these things look like in any moment, okay, it's trivial, so over, okay, so we can define the density in this way, so you look at all the different parts and you have everything over, and you can write up an equation for some, a master equation for the dynamics, we'll write it up, but it's not too complicated, so it will be something like this, write it up and then we discuss. I think I didn't make a mistake, so what I simply do is write up the dynamics of this density, so how to say it, motion. Oh, what's the word for this? Equation of motion. So, and there is, okay, let's forget, okay, so we simplify life because it's better. So is it clear what I'm doing here? I say, okay, so the dynamics of this density, so the change in time, how will it look like? Okay, there will be things falling, so it increases with lambda, things evaporating, which is with a rate, so things have to be there to evaporate, so it's a new times roll rate, total rate, and there is a diffusion that we put in by hand with diffusion constant sigma, so there will be, okay, so it's sort of a diffusion equation. So it's, yeah, so it's a diffusion, yeah. So we added a diffusion here, so we have to do something with that, it's a diffusion, okay? So, okay, one can solve this if one wants to, what do you want? Okay, okay, no, one cannot solve this so simply because you have to have some boundary condition, but actually what you have is that how do you define the price, which is moving? Well, the price which is moving, so what does the, okay, sorry, what is the fact, something that I didn't really define so well, so this price diffusing here is eating into, so there is still a financial model, so what he does is that, okay, price moves somewhere, he eats up all the volume going there, right, this is the way a price is defined. Is this clear? Is it clear? So, price is given by a, so price is given by the point where this density will be zero, and okay, so you can come up with some, okay, this we'll get back to in a second, but okay, so some very simple cases, in quite general you can give the following, the following things, so, yes, so okay, if you define rho zero as zero, so the price, where the price is, so we are in distance from the price, where the price is, there is no density zero, and that lambda zero and nu zero is finite, so close to the price, these are well-behaving quantities, there is no infinite flow coming on the price, and no infinite diffusion, then one can solve this, and okay, sorry, so if this is the case, then you will have some linear behavior actually in a model like this, but we will get to an explicit, so okay, it's a general thing, if things are, so if this is the way you define zero, and things are regular around, then it will be a linear, locally it will be linear somewhere, it's in any of these diffusion-deposition processes, but actually you can look at, so okay, if you define, so let's say example for this, you can say that you remove most of the dependence, you want to say that okay, let's say that lambda u is lambda nu u is nu, so there is no, nothing depends from the distance, okay, so it's actually, it's not super realistic, but even in this case, you can solve this equation, and then you will get that the rule, so the stationary shape of the book will be something like this, some shape like this, okay, so this is the solution of the, if you put in it, you find it, and okay, so what do things mean with this u star being, well I write it up because it's important to understand a bit what the structure is, will be something like this, probably this thing will be, I think this will be something like this. Yes, so this model doesn't give a, okay, there could be a region, but, and then you have to, no, sorry, no, in a model like this, if things are falling, then it shouldn't give a, I mean, if everything is continuous, then it should be point like, so there are things, again, it's a model, there are things now that you're not modeling, you're modeling the price, the mid, one can come up with more, you could discretize, but if you say that you're on a discrete grid, already you will have a difference between the two, so okay, you can come, this is the solution, and so what does this mean, that you will have some type of order book that looks okay, so in the middle it will be somehow linear, okay, and you know that far away it will have some level, so this will be infinite, and okay, and you don't know how exactly you go there, your guess is something like this, okay, but what you know is that, okay, very far away where the diffusion doesn't matter, it's only lambda and nu which matters, so the higher the rate of falling, the higher it will be, so you will have this depth far away, and you will have a linear behavior close to this regime, which actually this U star should give the widths of this, so actually what you would say is that this is U star, so the scale on which you can assume that this book is linear, okay, so what did you say, we came up with a model in which some things are microscopic but there is a diffusive price which we don't all like and we will discuss it, but what you can say is that you get some locally linear profile for something which we didn't say that this is really the order book because you will have to think about the numbers, you will have something which is linear around the price on this scale and so what is this scale really, well, one over nu, it's not here, but so one over nu is somehow the lifetime of an order, right, the time it stays there before evaporating, so this U star will be something like the volatility over the time scale, one over nu, okay. So the execution in a model like this is modeled by the diffusion in the middle, so there is, and it's put in as an extra thing, it's, we say that it's diffusing, we assume that there is a diffusion in the middle that heats up the existing volume which will be an effect of executions which is the market order, so it's simplified, but so what it says is that okay, okay, you get some weights of the linear regime which is essentially the volatility over the lifetime of an order. Now, what is the lifetime of an order? There you have to be careful because here it's the question that is this the real order book or a latent order book? Well, what we are assuming here is that, so that this latent order book is the real intentions of people, so not the things that you really see in the actual order book, this is what people would do if there was no information leakage. So what you expect is that this should be order of, okay, so this is an assumption, but you think that this is order of days, that the actual real decision of most of people, so the average of one over nu will be defined by the slow players in a market. So this life's not. And you say that you expect that this should be dominated by the people who are slow, so who really have an idea of how the price should move on longer timescales and not just locally at the best. But this is also the way one has an intuition about this model, this is not written in the model, yeah? No, you don't know anything about their information. Here, you say that they behave like this. Actually, one would say that, actually, this type of model, since we said that row is, so there are people falling, right? To prices where we want to trade. Actually, this type of model, well, I would call rather like zero intelligence. Actually, that's the way you call it. There is no really, you put a microscopic behavior and we will put some x-train in. But there is no real intelligence in it, right? You say that, okay, one approach is like, can I, okay, if you come from physics, the question would be asked a bit like this. Okay, can I reproduce parts of the things that I measure with completely non-intelligent, so as if they were particles who have to obey the rules of course of a market, so there is a structure given. It's okay. So you get a linear regime, okay? This we found, so we are happy. There is some underlying linearity that we should have close to the best price. And actually, this is quite general. So I just, okay, it has actually several consequences that I want to write up and then I want to get to a bit enriching this. And then, yeah. So yeah, it's up to, it's a family of models that are called usually zero intelligence. So it's a quite usual approach or at least on physics is to say, okay, can I reproduce the things that I can measure with all of these great ideas of how people act? But okay, if they were simple parts, simply particles, what would be the microscopic prediction from that? So just to say, okay, so here we gave an example, but it can be generalized quite well. So in a general result, you can say that linear book profile is extremely general. So I wrote up what you want is, okay, there are two things. What you want is rates that are regular, well behaving, whatever, around zero. So no infinite flow at the best, close to zero also. So I mean, of course, you can do some strange stuff in a system like this, but if it's not the case, which is the wide behavior and you say, and the diffusive price, what we call price here, in that case you have linear profile of the book for some region, okay, it's not super. I mean, of course, it's sort of general because you can do an expansion close to zero. You, well, as I said, the second is, so width of this linear regime is related to the lifetime. So it's related to the volatility on the lifetime of the diffusive process on the lifetime of an order. So okay, this is what we said. Another thing which is important to think about which is here, but you can say that, so the volume here will be vanishingly small. So what I will call, what I call the best, so the volume at the best selling or buying price is vanishingly small. Actually, okay, it will be absolutely vanishing if you're a continuous world. You can put a discrete tick size in the system to make and then you can give an exact calculation, but indeed you will find that what is there very close to the best is I don't write up the orders of magnitude, but several orders of magnitude is lower than other meaningful volumes. So actually, in a very simple model like this, what you can say is that there are essentially two type of consequences, so this is the general result and the consequence is that one, is it visible, that I write it up and then I explain. So there are the two consequences in getting back a bit to the financial language, but of all this that there will be a vanishingly small volume available in the order book at any given moment. So if you're trading in a market like this, you have to cut up your trades. You have to really trade only incrementally, so it will give an explanation why there is this long memory order flow in a real market, so you have to split up your large trades into small pieces. And we will see, we see it from here. So there will be a square root impact of the meta orders, which is, well, essentially we see it here. We will have to see if it really works in this model, but it should work, right? If the profile is linear. Is it clear what I'm saying? So you have the feeling that in a simple model you get these two things out as consequences and then somehow this should be related. This should be the two sides of the same thing. Now the problem is that, okay, so this is super nice that we wrote up, but the question is, okay, can we really trust all these results? And well, I think no, because there are two things. So one was already pointed out, so that we put in a diffusivity by hand in this model. So first of all, you cannot really call it microscopic and why should be their diffusivity? I will discuss it in a second. And the other point is what I discussed before. So sure, if you do the calculation, you will find that locally this is linear, but what is not clear that if you enact, actually if there was a market which had this behavior and then you come to trade your metorder in a market like this, so you're an extra flow pushing the price, why should this square root picture dominate if there is a price which moves like crazy around, diffuses much more, okay? So this is what we'll try to see. So there are two approaches here. We'll go both of them. So one is to, okay, let's try to enrich this model a bit. So let's go from this model. So we have an idea of a model. Let's go to a numerical version to see how it would behave and to see at least if the claims are true and how things can be diffusive. And then we'll get to another model which is slightly more consistent. So you give more dynamics to it. I think it will be tomorrow morning. Okay, we'll have to discuss about it. So okay, so actually here I have slides which is good. So the problem is this type of model, it's again very general. So there is no financial issue here is that if you have a, let's say, local linear profile of a book and you're trying to diffuse in it. So let's say you put a simple random, okay, let's go to, sorry, let's go to a bit finance language. There is someone there who is doing market orders, who is eating volumes from both sides and he's doing, let's say, a random walk in the two directions. They actually will never have, even if you wanted to have diffusion, in a simple picture, you wouldn't have diffusion. So what would happen is that, okay, you have a moment like this of the order book. So, in the beginning, so that's what's there locally. But if there is a walker who starts to go in one direction and the other and eating up the volume, so the way you would think in a market, you would, for easy, so let's say, you start eating here, so what you will end up with, okay, maybe there is some differences, but you will end up with something like this. Okay. So you will have a higher wall here if you ate into it and things didn't change. So in short times, you will have a higher wall in front of you in the direction than behind you. Is this clear? So you're doing this and you start walking into it, always taking unit quantities, okay? It's not clear. It's clear? No, no, but tell me if it's, it doesn't have to be clear at all, but you have to ask me. Because I see many people here saying sure, and then I don't know if there are people who don't. Okay, so either you ask now or you ask later. So the problem is that in a process like this, actually in a simple manner, you will have super sub-diffusive behavior because what would happen here? There is someone eating in a book in the two sides and by pushing it into eating into one side, he finds a larger wall in that direction than in the other. So the probability of, if it's one half probability, he will eat unit quantity from one side of the other. It's more probably that he will be going back. He will be moving the price back because it's easier to eat a level here than to eat a level here, okay? So actually it's not at all trivial how to, so there is some confining effect. If you put a walker in a random media and it will be a confining effect. So to do actually a numerical, so now we go to a numerical version of this model, you can do two things to get to a diffusive process. And the two things are the following. You will add an autocorrelate, so you will add someone who is walking here in the middle, so effect of market orders going, eating into the book. But he will have persistence in his directions, in the direction that he tries to eat in the book, okay? So that's what we see here. We see it's in a numerical setting. You do this because also it's a realistic thing in a market, but especially because what you want to achieve is can we have a diffusive behavior in this media, in a natural setting? So you add a persistence in the direction, which actually doesn't really, still the model is sub-diffusive in most of the cases. So there is another thing that we'll add is that, so again for this diffusive point in the middle, we'll add some type of conditioning on the volume. So there is this point here, I mean this, I don't know, what's Pac-Man going around and eating into the volume in the two directions? He won't just have persistence in his moves, but he will also do something which is actually something in markets that you see, that he will take a typically a given fraction, he will eat a given fraction of what he sees in front of him. So if there is a larger wall in front of him, he will have a larger bite of it to make the process diffusive. So what we do here is we add two things which are, which in a real market exist, which is empirically measurable, and we try to tune the system, try to see if we can make the system diffuser. Okay, so it's only a numerical version. I don't describe exactly this, so it's here about how this function behaves. It's not very important, but you condition on what you see, you condition your eating on what you see, and actually you get to some, you can tune the system, so okay, we had a gamma which was the autocorrelation exponent two slides ago, and we had a zeta which was this guy here. Sorry, there shouldn't be, there is one extra parenthesis here, I think. Sorry. So you have these two parameters of your system and you can look at the phase space and what you find is okay, there is a sub diffusive and a super diffusive regime, and there is a thin line only between them. So this, I suggest it here that actually it won't be that trivial to have diffuses, but you don't have a region when you're diffusing, you have a line where you're diffusing. Give it to Cheyenne. And if you want to be very strict, there are some conditions which we won't discuss here. So what you see is that okay, you have a diffusive boundary, it's clear to understand sort of the behavior in the corner, so intuition tells you that it has to be, the behavior in the extremes has to be like this. And what we have is okay, so we added these two extra parameters to our model, so we really made things diffusive, but we had to become numerical and had to add just a second extra parameters, and we can check numerically that indeed we get to an order book, so a profile of the book, which is described by the equations that we had in mind, and which is linear locally close to the best. So what I do here is a simulation of this model, empirical simulation, so simulation results and then the theoretical prediction in blue, but they match well, so we didn't break the things, and indeed we can get to linearity, yes. So what I do is we discuss the model here for the deposition and the operation. We said okay, we want diffusivity, but if you want to analyze a model further than this, you can see okay, how can, what's the way to put diffusivity? So let's do, give some microscopic definition of my behavior that can lead you to diffusivity. What will be these two that, okay, so this guy which is in the middle, we say okay, he also has a behavior, he's going in one direction or the other, eating something. We added two parameters, unfortunately we needed two parameters, we couldn't do it with less in this case, so one that he has a persistence in his directions in his moves. So he's walking in this between these two walls, his persistence in his moves and he sees what's in front of him and the conditions, so I mean it's a finite fraction of what's there, okay? Meaning that if there is a huge wall in front of him, it's non-zero probability that he will still, he's not confined by it. If he was always eating unit, okay? Yes, so what I'm simulating here is exactly I have a price axis, okay? So there is some discretization in it which I don't remember but it doesn't really matter. There are things falling with lambda, with lambda being constant. There is a new constant set, so an average lifetime and you add these two further ingredients and you see how does this market behave. So what do you care about? First, can you achieve a situation where indeed you have diffusivity in this model and then you want to check, okay, so that was one issue with this model. The other issue was, okay, does the square root impact, don't, does it still work if you have fluctuations, if you have a diffusion in the middle? So we try to look at this numerically and then actually I think tomorrow we'll see a model, a more consistent model where you don't need to be numerical. Okay, so okay, so this we discussed and actually the three points here is that I will show three points but it doesn't really matter. You can look, yeah? Yes, I didn't understand. No, it's a line, I mean, yeah, it's a line. So yeah, but it's a diffusive behavior but it's a line, it's a critical system. You don't have a region, a wide region where you have diffusivity in this. Well, you have to tune yourself to this which actually gives questions because, okay, in a numerical you have, you have to tune yourself to this but that also means that, okay, it was a super zero intelligent, super simple model. It would suggest that, okay, the real world, it's not a exact proof but what you would get from this is that, okay, so you want to be on this line, you know that things are, prices are diffusive, you want to be in this line but it would suggest that somehow the actions of all of these people organize together to get to some point on this line because that's what you measure. Could call itself organized criticality if you want. Okay, so what we have is we put this long memory opposing order flow, so this gamma and zeta and we get to a linear profile of the book locally. And so what one would be interested in this case, okay, do we, still staying in a numerical simulation, can we get to square root impact? And so for this you have to introduce meta-orders. It's good that I made some slides here. It's good, you have to introduce a meta-order. So, okay, well what do we have for now? We have sort of a background market, right, which is diffusive, which has a local linear profile and then what you want to do is, okay, let's test this system. I add an extra meta-order to it. It's defined in some way. Let's see how what's the response of the market, okay? So what we do here is the following is that you add someone extra. So you have the diffusive market and you add someone extra who is trading always in the same direction. So trading essentially a meta-order doesn't really matter in different ways. So of course in an actual model you want to test different hypotheses, but what he does is he trades in one direction and you measure how much he pushed the price when beginning to the end. And how it behaves. And actually you find something which is not that bad. So this is the type of curve that you find at the end. So what do we do is notations are slightly different but this here is impact divided by sigma, right? It's this that you want to check if it's true. It was this, okay? Well for some reason here, okay, it's delta, which is I here, but it doesn't really matter. So what you do, impact divided by sigma. It's a change of price divided by the volatility versus the Q over V. So what you see is that if the slope of this thing on a long log is 1.5 ish, then you have a square root behavior. And if it doesn't work, you would have a slope more like one. What you get is okay. So forget the different colors. It's a different type of settings. But what you find is that indeed you can reproduce a square root behavior in a model like this for basic settings. Now I wanted to show a figure here which what is when this type of model of course breaks down is that okay, what's the entire effect that we are doing? We have these particles falling, staying there for a while. So somehow there is a linear shape of the book and there is some memory in the book, right? It will be the lifetime of orders in the book that defines the memory. Once you wait the time that everyone is, every order has evaporated and other orders fell, there cannot be any memory in the system, okay? So actually I can show it tomorrow. Actually what you can do is add methoders that are there executing for longer time or for a time scale which is comparable to this thing here. So one over new. And indeed as you expect, things break down and you get back to a linear response because you cannot have anything else in that case, okay? So this is one type of model. So what does one learn from a model like this in the first place? So what you see is that okay, with some super simple settings of these things falling you get what is I think very important, some vanishingly small liquidity close to the price which will, this has to be like this, right? It's simply the effect of particles falling but eaten by some process which is diffusive in the middle. It has to be super small, the liquidity in the close to the center. Okay, which will close these two as we discussed. So there has to be some type of splitting in a market like this because there is not enough liquidity. And the linear response will break down. And okay, so of course what this means is what we have seen that you have a concave type of impact. It's quite square root, it's not exactly square root. But what is also important in a type of model like this is that if there is this very small volume close to zero which is available here what you would expect is that if you add any further fluctuations in a system like this you can have sometimes enormous jump. So if you go to a, in equilibrium things are okay but if this volume here is super small then you can easily have because of a small fluctuation no volume available close to the best price if you go to a financial setting here. And these liquidity fluctuations must have a very crucial, very important role in the dynamics of the price. I don't know if this is clear. Okay, so what I'm saying here is that well it's just very simple claim. Given that the volume close to the best is very small you expect that any small fluctuation in it the fact that there is instead of a very small volume there is zero volume there for some reason can easily happen in a real setup and will have huge effects on the prices. An example for this actually which is a very important question is that okay we're discussing some medium here which is the latent order book. So it's the real wishes of what people would be doing. But of course what we have shown in the beginning that the actual actions of the people what are visible so what they really do doesn't match this. What does this mean? Yeah sure there are people who want to let's say on this price there are a lot of people who want to sell but only a small fraction of them will be there really in the market having their limit orders. What the others what you expect are doing is that well I will wait for the price to come closer to this point and then I will place my order here. Right, so you have to have some type of mapping from this latent wishes to actual actions in a market. So things that you can measure in a market directly. And of course if you have some small fluctuations one you could define is that if you go back to leaving the position type of model but for real people okay you can say that now the price is 100 if the price is 102 I want to sell but I will define when it comes to 101.5 if to put my order, okay you're waiting. But if the price comes too fast you might not have the time to wake up and you can have few and if others behave in the same manner as you you can have huge swings in the price. Which actually are things that one can measure. So okay and there are some other important consequences of some types of model like this. What I suggest is we stop and so I want to okay tomorrow I want to finish a model which is related to this. So it's the same idea but both further. So instead of adding numerical things try to add some more behavior. It will be quite fast but I think it's important. I have the feeling that we spent a lot of lectures on concepts and now we get two models that are easier to understand. I don't want to stop without discussing them. But of course there won't be other new stuff tomorrow. I'll try to wrap up a bit okay so from there where did we start and where did you go because I have a clear idea but I'm not sure everyone has and that's it yeah. It's an announcement I'll tell about the tutorial session today. Okay. So today we're going to have a tutorial session on this course at five, okay about an hour of tutorial. Half an hour. Yeah, well an hour and a half. Sorry. So five and a half. As you like. For me it can be half an hour but then. Okay and sorry and if you have questions come I don't know write to me come to me or go to the tutorial but. Give. I am giving the example. Your turn. Yes. It will be closed book. It's closed book. It won't be hard. It won't be hard and the idea. Okay so. Okay so the idea is for of course there won't be definitions. So I mean it's obvious. Won't, will not be. I mean so of course if you understood nothing from the definitions it will be hard to do the rest but of course it won't be defined what is a market make. If you did the, I mean if you followed some calculations that we did, I mean and then the tutorial that could be enough, some of the exercises and papapam. What else should I say? That I don't know. That I will have to decide. I mean it's also ill-defined like a group questions together. And yeah so the idea is not to ask some stupid detail but to get, did you follow? Did you get the intuition? Did you, okay so so looking at the models that we had and the ideas of the calculations. I mean may even be able to reproduce the important things and being okay it's important to know. I mean we had a lot of empirical results and of course some of these are important and it might be important to be able to read a figure.