 The boss said one minute. Ancora un minuto. Ok, so I'll come back. Ah, yeah, there is a paper. So, first row here is very noisy. So, ok. I think we have to maybe a bit discuss, wrap up what was yesterday or I don't know if it was all clear. So there are things which after the lecture I discussed with some people, so it might be good to get back. Ok, so what did we discuss yesterday? Essentially there were only I think two main points. One was this what we call the apps effect. Readable like this? I don't know who but someone was unhappy with it. Anyway, apps effect which was the time scale, the dependence of cross correlations on the time scale on which you measure your changes. So I think, I guess it was a simple concept we discussed a bit. The reason for this and the reason for its time scale being characteristic time scale does not not changing with the frequency of the market. I hope this was clear and it's in the slide. I think there was nothing new in this point. Ok, we started to get to discuss new models a bit. So more economic style models which we actually will stop. We'll discuss a bit more today but then we won't have this style of models anymore. And there was one model which we called CHI, after a person. And so I wanted to discuss two things about this. So one is that, ok, just getting back because I had a feeling that maybe what is this market maker exactly these notions weren't completely clear. So we discussed these in the first lecture but now things should be clearer. So to get back to a figure that we take often we look at. So to understand again, in a traditional picture of a market so let's say this is a market there are two types of orders that can be posted, people who are patient and put their orders in a place they say I want to buy at this price I'm waiting for someone to trade with me. And there are people who are impatient who we sell market orders they say ok I want to buy now at the best price. Of course they can see what's the best. They might also decide they don't want to trade but they don't want to wait in a queue. So in a traditional picture ok, so another way to say this so these are limit orders those are market orders another way to say these are providing liquidity and the others are taking liquidity it's just a question of language liquidity is, we can define it in several ways but I think it's sort of clear. So in a traditional picture which is not of today but let's say until the 80s when the Bay Market was worked it was people who could place liquidity place limit orders were special people these were called specialists or market makers so it's simply there were two types of people those who were somehow designated by the market to place limit orders and others who could go to the market and trade with market order ok? And so the thing to realize from this it's a very simple claim is that those people who stayed in a market of course you might get a good price so these people are wanting to sell they can sell at a price higher than if somebody trades immediately he will sell at this price against the trader so putting liquidity in a market may allows you to get a good price but of course the moment when you are going to execute is not chosen by you you are there waiting and you are executing someone comes to trade with you which means that ok you can expect that since you are there typically just waiting you don't have some big information about you might have information but if you don't have much information what happens is that sure you want to sell at a good price but the moment when you trade is when someone else thinks that this is a good price for him in the opposite direction so this is what we call probably adverse selection so these might be a very conditional moment when actually the price will move even further up you sell here you think it's good because you didn't use this price but actually the price much later comes up here or pretty soon comes up here then it wasn't a good one so the idea of all these models is that you can place limit orders you can get better prices but you risk some adverse selection and in practice in these designated market makers in this market what can of course they do is they put orders on both sides so even if they don't have any information they don't want to buy or sell they just want to be providing liquidity there for you giving you a service but in each trade they are going to gain they are going to sell here and buy here they are going to gain the difference or then if they have to get out of the position then they do two trades on average they will gain half of this which is great which is a good thing to gain on but at the same time as I said they have the risk of being the moment when they trade they are expected adverse to them so this is the traditional picture so this is what we call market makers are people who provide liquidity and typically we think about people providing liquidity on the two sides of the limit order book the way markets function today is slightly different but the idea is the same anyone can go to the market and place liquidity here or bomb both sides at the same time in most of the markets in the big electronic markets but it doesn't change the picture that still you are there and people trade with you is the moment when they want so they choose it so it might be better for them and so the first what we wanted to discuss I just wrote up Kyle here but so what this first type of Kyle model what we tried to discuss in this type of model is okay how can a market maker in the situation calculate the price to trade at in a way to minimize so he wants to gain but he won't be able but he wants to get the best price for him not to lose at least so this is the idea and so in this first Kyle model there were some calculations some simple model but the thing that we understood the message was that the market maker is changing his price accordingly to the volume that he sees we had some lambda I don't know how we wrote I think it was like so he has a price which is the original price plus so this is the volume that he sees overall and this is his susceptibility to this volume so how much he changes his price and what we saw is that that that for the market maker lambda grows is the amount of information of others which is okay, makes sense I think tell me if things don't make sense for you and the delta will decrease with I would say noise traders all those who are there trading you will gain what you gain but you won't lose anything on them what's the meaning of this lambda does it have to do with how much risk lambda has to say so what it says is that the price now is 100 and I see that you all want to come and buy from me so I have to decide at what price I am selling to you so I might say okay if everyone wants to buy then surely there is something you know something more than me so I say okay I won't sell you at 100 because probably you're buying price for go up I will only sell you at 110 in this language you have to risk but lambda will be the value to get from 110 to 110 given your volume at the same time if you come here and on average you want to buy the same as you want to sell so half of you want to buy half you want to sell I might not be afraid I still want might change my price to a certain degree but I won't be that much afraid so the way the lambda is calculated the idea was that he wants to he wants to have the expectation of the final price so it was this he said okay he sees delta V and he said so essentially it's this what he's doing he wants to set his price in a way that in expectation you will get the final price that people have as information I don't know if it's it's sort of clear yes there was a question there ten minutes ago no market makers place limit orders all these you see here so those who are waiting are market makers they are providing liquidity to both sides typically the idea in the language of market maker they don't have a view of what the market will do it's a service they are providing of course sorry yes the market maker is waiting in a traditional pictures some people the market maker is waiting let's say it's only him and he posts these two orders and any of you can come and trade against him but all of you if you all come and trade want to buy from him push this price upward yes this picture would be more modern but he could decide that he puts all these orders here said yeah I'm ready to sell two to you at this price but afterwards it will be this so he could do it everywhere or there could be several of these market makers having different models for lambda and they one puts here one puts there but you could also imagine this picture that forget everything which is outside the first they just say at this price exactly is an intermediary between exactly exactly traditionally this was a service the market somehow gave I gave you the opportunity to come you will gain the difference here always but you have to give this service and ok so just to finish up say something about the market maker more so there are these two things which sort of make sense if there are more yeah if there are ok if the amount of noise traders so on whom you can make money grows then you can relax your your lambda and if instead the amount of information of those people who gain on you increases you have to be more and similarly ok just to finish up for this other person who was informed who called Alice it's what we said is that he's gain her gain so the expectation of her her oval or gain which we defined grows with SS is the amount of noise traders so this is which we call actually liquidity so all those people are trading so the amount of noise traders in this type of model is good for both of them for the market maker because he can gain on them for the inform trader because she can hide among them ok ok so this was was the was the model from yesterday ok and sorry if lambda is negative ok lambda ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok Sì, ok, quindi questo è l'esercito, sì, quindi abbiamo visto che è andata oggi... Ok, cosa che abbiamo avuto oggi è che questa cosa qui è... era qualcosa di così, credo. A continuare, credo che sia una settimana di questo, oltre. Quindi, l'esercito di noi, è contente in questo. Ok, quindi questo è... per riparare un po' di... quindi questo è quello che chiamo il market making. Ok, quindi, per capire il momento, storicamente c'era uno personato designato per tradere qui, quindi si può fare il quale vuoi. Si può mettere questo molto alto qui, questo molto alto qui, e poi si gira molto a ogni tradizione. È una maschina di pieno. Se ci sono diversi persone, quindi oggi, quando qualcuno può fare questo, quindi c'è una competizione, ovviamente, quello che aspetti, sicuramente, puoi mettere i tuoi risultati molto fuori, ma c'è qualcuno altro, che puà mettere più vicino e poi ci sarà lui, che traduce in il market. Quindi è molto più competitivo e molto più difficile per fare la piena di questo, ma ok, questo non è molto più. Quindi, abbiamo parlato di un altro... un altro modello di questo, che è chiamato... ok, che è chiamato... perché, ok, quello che abbiamo avuto in questo modello, in realtà, sto discutendo tutti questi diversi risultati, i risultati che hai scelto e scelto, ma avevo solo uno risultato, avevo solo uno personato, io mi dico, avevo già un volume overall che avevo, e ha detto, ok, ho messo uno risultato per tradurre con tutti i tuoi. Quindi, c'è un altro modello che voglio discutere, che ha simile idee, ma in realtà, è più realistico. È chiamato Glouston Milgram. Questi sono due persone. E è soltanto dalla stessa parola della scala, dalla stessa stessa, il modello, quindi, che esattamente è per l'existenza della spedda bidas, quindi è ok, la lingua, la spedda bidas è capita... Questa è la stessa parola. Quindi, è per mostrar l'existenza, ok? E il set-up è abbastanza simile, ma non è che abbiamo uno informe, uno informe tradutore, ma c'è un set di tutti questi. Quando io dico informe e no, sono synonymsi, ok? Quindi, c'è un set informe e informe. Non so quanti. E si vede a tradurre con uno market maker. È un set-up simile, per esempio, per un market maker. Ok, quindi, è più generoso, quindi, quindi, chi vuole tradurre tutti questi informe e informe tradutore hanno qualche PI, come lo chiamo? PI-HAT valutazione. Quindi, c'è un P0, ovviamente, in la stessa maniera, c'è un P0, ora, e c'è un final prezzo che si vede a credere. Quindi, che si vede a credere. E ogni di loro ha un'idea di cosa va a essere. Che va a essere, hanno questa valutazione, hanno tutte queste idee, differente. E, ovviamente, ok, cosa fa la differenza tra loro? C'è un attimo triviale, è che quelli che sono informati, quelli che hanno PI-HAT, che è correllato per l'outro actuale, che va a passare. E sono informati che sono correllati. C'è un po'. Quindi, in quel senso, c'è... In questo caso, si può sempre complicare cose, ma ok. Quindi, è quella semplice e ovviamente, cosa fanno? Sì. Sì. Ok. Quindi, che fanno? Quindi, che fanno? Quindi, che fanno? Sì. Quindi, l'attimo è il seguito, se PI-HAT è above an ask price in the market. Quindi, se dicono, d'accordo, sono contenti di asked to buy this or above. Quindi, they will buy. PI-HAT is below B, then they will sell. Is it visible here? Ok, and otherwise do nothing. So, otherwise do nothing. Ok, B and A will be these. Ok, it's not so. So, B is the best bid, A is the best answer. So, it's the price that the market maker is offering to sell at or to buy at. Is it ok? Or it's very unclear. Here. Yes. So, each of them has an idea of what the price will be in the future. We just define informed who have an idea that is correlated at least, somewhat correlated to the actual outcome in the future. Yes, in practice what we will do is that they know he knows it exactly, but he could also just know that price will go up. He doesn't know how much. So, and it's correlated but not exactly. It's a general, but yeah, in practice to make it simple we will say for informed it's, he knows the final price. And for uninformed it's uncorrelated. You think you have information but it has nothing to do with the real world. I think that tomorrow stock will go up because I don't have any real information on that. It's just, yeah, it's the same, exactly the same. Yeah, it's exactly the same. Sorry. So, okay, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, , then it's, it's the same as in the type of model before that what he has to say is a Dr the marketmaker will have to set the price at which he, in so è che l'espectazione del prasente finale è il prasente che sto vendendo. Quindi, dovremmo imparare da me e in un modo simile, vogliamo questo. Quindi, l'impasto deve essere come se fosse la stessa informazione. Ok, quindi è il stesso tipo di game che abbiamo avuto prima, non è molto diverso, ma in realtà è molto più ricciata la struttura. Quindi, abbiamo due prasette qui, right? He wants to set these two. Ok, una cosa ovviamente è che, in questo tipo di modello, non ha davvero informazione. L'unica informazione del prasente finale che il market maker ha, è quello che il flow fa. Se pensessi che il prasente è, se avevi l'informazione, si direbbe che A e B sono uguali al prasente finale, e questo è tutto. E non si chiama. E in genere, ok, si può aspettare, quindi, qui non abbiamo provato, ma si aspettano ad essere above B. Quindi, il fatto che le persone vengono, è un segno che il prasente avrà avuto. È in questo modo. Non deve essere. Potrebbe avere un modo ad avversare, ma questo è... Sì, quindi, queste sono come, ogni di loro vengono a me, uno per trade, uno per uno. Quindi, questa p-hat è la stessa... In questo momento, vedo come vengono a me con il tuo PI, e devo decidere qual'è il prasente. Quindi, è... È abbastanza ripetito, quindi, in termini di tempo zero e f, qualcuno può vengono a trade. Quindi, vedo ogni di voi che vuoi, uno per uno, quando vengono. Però non so, ovviamente, se avete informazione o non. Sì, io sto cambiando l'A e B, d'accordo, sì. Però, in pratica, non so niente di momento. Quindi, secondo che c'è un trade, ora, cosa faccio? Questo è... C'è qualcuno che c'è un trade, ora, cosa faccio? Sì, sì, quindi, beh, il altro prasente è un po'... Se vedi a me, se vedi a me, dici che vuoi vengono. Siamo preoccupati con il prasente a cui vengono a te, non ci preoccupi con il prasente a cui vengono per qualcuno. Siamo cercando di una sola parte della situazione. Voglio di non. Però, vengono... No, il PI, c'è un trader informato che ha questo PI, ogni di loro. PI, sì. Quindi, vedi a me, dici che vuoi vengono per 100. Sì, sì, sì. Quindi questo è in ogni momento che la persona che c'è, potrebbe mettere un T per ogni momento, ma sì. Sì, quindi discutiamo la distribuzione. Ovviamente, questo in solito, non può essere solito. Il market maker deve avere un'idea del mondo. E quindi, quello che ha è, ok, in generale, lui deve avere alcune distribuzioni per la seguida. Per esempio, il rischio finale, come deve essere questo PI? Se nulla non ha informazione, cosa si avverà? Gosci e attraverso il rischio per il current, etc. Quindi, vi lascio esplicitamente un esempio per questo. Il senso di l'idea, è il stesso senso. Il senso ha un'idea di questo, quindi, il senso ha visto il passato. Il senso ha saputo come le distribuzioni sono soltanto... e quello che lavia è l'opposito. per ottenere questo A e B, ma lo che importa è questa distribuzione. Se ha questo, può calcolare questo se è bene calcolare. Quindi l'idea è la stessa, e queste idee sono semplici, quindi è semplice per calcolare, ma è ok. Quindi, ovviamente, cosa che dovremmo fare è di determinare questa cosa qui, che significa che dovremmo scrivere un po' di basso, che è ok, si può scrivere, in pratica, cosa dovremmo calcolare è questa, lo scriverò e poi lo dirò cosa sono. Vado a vedere a casa che ho scritto un basso, ma ho integgiato qualcosa di nuovo, che è questa Q0, che è l'idea di... Ho detto che ha un'idea di questo qui, quindi, given the information, how do people act, so essentially the behavior and the number of un informant, and this is the prior that he has on the distribution of the information itself. Ok, are these clear? Ok, yes, it's not readable, it's not clear. Ok, what I did here is simply write up this as, what I did is a base in theory, so this should be e hat df, which actually he has an idea about, should it be this? This is of course closed here. Ok, so the ideas of all these models are simple, it might be hard to calculate, but the idea is that he wants to set his price according to this, so from this, of course, once he has this distribution, he can set an A and a B in a way to not lose in the market. Right, of course what you want to look at again is that, given the different distributions, how will he behave, what is the intuition that one can get from a model like this. And so what is the following, so this is known by, and so what he will be, he will have to calculate, it will be something like this, so for example A will be, by the expectation that we have there, which will be ok, it will be an agreeing, do I have to write it up? It's ok, so what he has is that, he has this distribution here. Ok, I'll write it up for one of them only for A, so we'll have something like this. Ok. I might write it fast, but yeah, yeah. The price shouldn't go from zero to infinity, like to the negative values? Good point, yeah, so it can be generalized, but you can rather think about it as compared to the price now, so negativism and the price is going now, but yeah, so yeah, so you're right. Just a second do you have to, not really, it's essentially, so the way you're working in actually, ok so what you're comparing yourself to is the price now, so if negative being that the price went down, so you can change your, the center of your, how do you say this, of your frame, that your reference frame that you're working in. So let's not change it, I'm going to distribution, you define it, it might be zero in some places, so you integrate something which is zero. So is it ok what I wrote up here? It's ugly the way I wrote, but PF is exactly the final price, you know what I said, the change in the price is PF minus the price now. PF is the final price, but he doesn't know it, and I don't want to go in much detail in this, so what I wrote up here is it clear, it's just an expectation, so I use what we have above and below PF here, I think it's ok, I hope, I don't know, from the faces. So but it's not really the calculation which is interesting in this, but to come up with some ideas of what can this give, similarly of course you can write up something for A and for B, but what you can get is, ok let's try to see an actual example, so I hope it's sort of clear up to now, so let's say that the trader knows the following thing, that this distribution, so which we just discussed there, you can imagine that it's, so to simplify life we had this uninformed which are correlated, some are correlated with the price, some not, you could have some model in the following way, I will say the notions in a second, so we have some new notions, I don't know, there are chores here, so one can write up a given distribution, we can come up with some distribution, what do we have here, we have by five we say that the ratio of people who are informed and what we say is that exactly what was suggested that they exactly know the final price, that distribution is a Dirac around the final, they know perfectly, and then there will be all those who are uninformed who will have some distribution that we can define, some type of symmetric distribution around the current price, okay, we said nothing deep here, and Delta is a Dirac and F is some symmetric function that we will define now, so what he says is that uninformed have a function that we will define uninformed whose rate is phi is no perfectly the future, okay, and okay, no, no, you won't be happy with this, I'll give something for F and something for Q0 to solve this problem, but to get an idea of what a model is simple sign, you'll have some normal distributions what you will get, but of course one could write up any type of distribution for this F, you want it to be symmetric because that's what it means that they are uninformed. So okay, you won't be happy, you can just write it up what I choose for F but I won't do any calculation on this, you can choose it to be the following. So it's really not important, unless you love to do calculations, we simply say that for F here we choose something which is decaying exponentially around this, if you put this in the stomach of it, with some, how do you say it, some noise, some typical deviation, so that's a good word for it, some dispersion of the distribution. So this is what the uninformed are doing, yeah? I'm informed, don't have information. So from a picture like this what he said is that I know that it's only 10% that people have any information, so my 5 will be 0, 1. And in this example, okay, but I know that this 10% have been known exactly, have been given the solution or the final price, the 1 minus 5, so the 90% instead have, in this case, this distribution around P0. Of course not, no. So these are very, very stylized models which will give some intuition and then we'll try, we'll wrap up at the end of the lecture, why we are looking at these because I have the feeling that some people are lost a bit. But no, but what you could say is that, yeah, sure, he's trading a lot. He might have an idea of, let's say, of fine, he gets an idea of how many people, if you trade a lot, maybe you know, I mean if you're always losing you understand that everyone is informed. So he can get, in practice, he doesn't know perfectly these distributions but in his head he might know the base formula and what he has to write. And okay, so this is what is the dispersion of the uninformed. He can have an idea. So if the price is 100 now, he might, from his practice, know that people are okay to trade at 102 at 98, but not at 120 and 80. So he can have ideas. No one tells him the distribution. Two sigma. But, ah, sorry, that's a sigma. But I mean, in case you want to play around with calculations, I will give the solution what this gives. Actually it is a caveat. If you want to calculate, calculate, but it won't, I mean, sure, if you calculate in a model, it's always useful, but I don't think it will be, it will be integrals of exponentials. It's tough, actually I did it, and it's not tough, it's time consumption, not tough at all. Okay, so anyway, but the main thing is the idea, what we have here. Is that okay? And so, okay, I just say the solution to this because I want to discuss a bit the phase space of this problem. So what, for the different things. So what do we have here? There's a dispersion of the, of the, of the uninformed traits, of the uninformed people, and we have a sigma which is the dispersion of the information. So how much information there is in the market. And so, okay, I write up the solution just to have something, not just to talk into the air, but, so the solution to this, if he does this formula and the similar formula for B, it will be the following, he will have an A, this is just the definition, he will, so I'm just defining an S, he will define a current price minus a half spread, plus a half spread due to put his prices, nothing given here, but this S will be the following ugly thing. Okay, it's something we don't really care. We want to look at its behavior, it's not really important what the final thing is. Actually, there is a caveat in it, which is anyone who is relapsed to do integrals can test it, that actually, officially the solution is this, which teaches in white, I did the integral, I think there is a, S divided by 2 in every place here. It doesn't change the behavior of the model, if you want to do the integral, do it and tell me what you find and we'll compare. But okay, it's not really important, so what does this mean? In this case, so in this idealized case, if the market maker wants to break even not lose consistently on the market, he has to put his prices to current price plus half of this and minus half of this. Okay? So, so, okay, we have a solution to this, but what we are more interested in is okay, how does this stuff behave? Because okay, it's ugly this thing here, right? Nobody knows what behavior it will have, except me, because I did a numerical solution to this. It's okay, the numerical solution will be this. Okay? Believe me, so what I did. But it's quite interesting to understand and threaten us. And so what do we plot here? Is we reset phi and sigma to some levels. Okay? We reset phi to zero one. So there are 10% of people who have information of the future and sigma is one. Okay? It's something, a number. And so what we plot here is what, in this model, what the spread should be as a function of this W over sigma. So it has a function of the dispersion of the of the noise traders and the dispersion of the the information, right? So as you go, as you go upwards, you have more and more noise. And as you go down, you have less noise traders and more and more stronger information. Is it clear with them? Okay. And so actually it's, I think it's quite interesting. It gives a quite good intuition about how a market can behave, of course. These numbers, the exact numbers cannot be calculated in a real case. But so what do we see? Essentially we see three regimes remind us of what the previous model actually said. So one case is that when W is larger than sigma. Okay? So we are here. And there is, so what do we see? That there is only one solution of this problem. There is one solution, there is only one solution of this. There is only one spread. So, so, okay. Which will be, okay, the value of this line. So, S, for small phi, it will be just to say, it will be something like this to a first order. Okay? So what does this mean? That there is a regime here, which seems to be a well-behaving orderly regime. There is only one possible spread that the market maker can set. Which will be this value. Which actually we can, we can get an idea of what this would predict for a real market. Because what does this mean? This means that, okay, so this means that if the spread in this regime, so in this well-behaving regime there is no problem. This would mean that, okay, phi is equal to S over 2 sigma. Or something like this. Or the order of this would behave like that. Which actually gives a, we can get, we can check. So these are numbers in the market. So sigma, of course, will be, will be somehow the typical volatility of the price, if it's the information. So it's how much people believe that prices can move and they are informed. So actually, you can get, for a real world, the order of this would be, let's say 10 to the minus 2. For a, on a daily level. Cos'of course, sigma is something that, the longer time you have it, it can, so, say, if you put here sigma daily. So it's, which would just, okay, we don't have to believe in this model, but some intuitions of the model seem to be okay. We don't have anything against it. It would say that, so, in, for a real world, this would be quite low, meaning what is 10 to the minus 2. It would mean that, okay, essentially 99% of people are trading randomly. And 1%, if you believe in a model like this, then 1% would be the, roughly the ratio of informed people. So it gets a prediction. It's hard to test, so, but that's what you get. The market maker doesn't know five, but he can calculate the, sorry, what does, what does the market maker know about five? Sorry, you're right. Yeah, sorry. The market maker actually knows this, but he doesn't know the, yeah, he knows this. Sorry, he knows five. So he, from his past experience, he knows five. Okay. So actually what he says is, if there are many, yeah, okay, so essentially going into this limit is if the dispersion of these, these noise guys increases, he knows how many there are. He's gaining. So what does this mean? Okay, going into this direction means that, that he keeps the spread at the level that he had, right? And his solution is this. Increasing W means that the dispersion of these noise traders increases. If it's an enormous dispersion, they will be always trading essentially, right? So what happens is it's really in this limit, he's making money all the time on these investors who do not have information. The other case is, is let's say, okay, still one can handle, so, so where W is, sorry, yeah, is below Sigma, so you're below one here, but above some level, above some W star, which is some critical value of this, which will be what is here, it's hard to calculate, it's not really important, but it's a finite value. So in this between, you have two solutions, which is okay, seems to be strange. What does two solutions mean? The market maker can set his spread to a low value or to some high value. And as he goes left, the two solutions converge, but there are two possibilities to do. Okay, so one can handle this still because you can say that, okay, if there is one market maker, of course what he would do is probably set it somewhere up, but if you had competition between several of them, then it would be, the system would choose the lower solution. If you choose this one, you choose this, and you will be just not trading. And there is, okay, and there is a third solution, a third situation, where there is no solution. So when W is below this W star, which has a numerical value in this model, then there is no solution to this problem. You can call it breakdown of the market. So there is no way for the market maker to set a spread and what we had here a bit, and not lose consistently on the market. So what does it mean? Going left here means that this W decreases. So the amount that you can, in practice, gain on the uninformed traders decreases. And there is a point where simply you cannot gain enough on the uninformed to make up for the loss that you lose on those who had information. And then you just get out to the market, you say here I'm only losing. So the message is that, well it's the same type of message that we had in the other type of model that for the market to work, so the market to work functions smoothly, you need to have uninformed, you need to have noise traders in the market. Otherwise, okay, it seems to be actually a real claim at the end of it, of course. If everyone has perfect information, there is no, why would there be trading? But you can prove it in some simple models with certain assumptions that it is the case. Okay, let's write it up to be clear. So a functioning market is one where there are enough somehow noise traders. Okay? So is the message of this type of model clear? Yes, yes, yes. So here it's clear the three phases, right? The first is this one, you have one solution, here you have two, here you have zero. In terms of, in terms of sigma and w, so this is only the ratio of sigma and w that you look at here. So one means, we have some certain units, we gave distributions and this lucky case it is one, but of course we have some other value. Above this, above a certain value, there are so many people, the stupidity of these noise traders, let's say, are so widely distributed, they are ready to take prices so far from the price now without any underlying information that it's very easy to make money on them. You can set a spread which is relatively small given by some factors because you make a lot of money on these people, you're not heard by those who have the information so those who are related to sigma, right? So this is a width of the distribution of the information and the noise essentially. Then there is a regime in between when it's not the case anymore so you, as you go leftward, okay there are two solutions but the low solution, so the low spread solution actually is increasing so as you decrease the dispersion of these noise traders you can gain less and less on them so you have to start increasing your spread to make even because you're losing still on the informed people. Of course this could be, the reasoning can be in the other way. If sigma increases, I mean this is the same, it's the ratio of the two that we care about. I don't think we have to discuss it more. So it starts to increase and also you have two solutions in practice but I don't think that for practical reasons the two solution is that important because if I'm not alone to be in the market then I will have to choose the lower one so there will be a breaking of the symmetry and then there is a region and as you go left there is a point where the upper and the lower part of this curve touch and below that it's simply impossible to, below a given value it's impossible to trade in the market because the market maker is losing too much on informed traders and gains too small amount of the noise traders. So it's more easy. This is the message that an intuition about how markets should be working, how given information to the market work of course everything here so the actual numerical values depend on these distributions and you could write up some strange distributions that have slightly different but for meaningful distributions so which makes some sense, so for example everything symmetric you have a type of behavior like this. So I want to say two things. One is still about these noise traders and then a bit more wrapping up on this part. So, ok, it was obvious that these models that we discussed this skyline and Gloucester Milgram are a simplified version of a market, a simplified version of how information and non-informed traders can behave but of course it's overly simplified so there are a lot of critics against this. Nobody thinks that these are realistic models. Why would people, how part of the people, so why would these people know very well something and these know nothing? You would expect that it would be some distribution and life should be much more complicated. One can handle it just not analytically but you can make more complicated models. Ok, I think the main critique ok there is also another critique that we discussed before in the first type of model that if you assume that the market maker only cares about setting price in a way to lose consistently but he is not afraid of having enormous positions of if you want to buy from him on a good price he says whatever quantity so there are some non realistic things I don't think these are important. I think a serious issue about all these is that you approximate that there is a final price this PF which some people know and the only thing the market maker wants to do is to break even compared to this. Ok, essentially it's only this what he does. So he says that ok if I'm not selling to you below the final price I'm happy but of course this assumes that the final price life is over and he can sell his own store I mean the market maker if he buys from you let's say a quantity of stocks then he will have it in the pocket and of course there is a life after this so what after PF happens he wants to sell it away because he wants to retire or he wants to get out of the market it's not obvious that he can really get this price later so the fact that it's on a very finite horizon is very realistic we will try to see a discussion of why this is the case I don't know if this claim is sort of clear or not Ok, so the point here is that the fact that it's this type of expectation that he solves means that the only thing that he cares about is to sell you below his expectation of the final price it's something like this but this final price is not final in the sense it's not the end of the world then afterwards he will have to get out of this one situation ok, I'm a market maker I don't have anything in my pocket I'm just buying from you I buy from you a hundred in a way that indeed in the next time the price goes to PF but then I have a hundred in my pocket what do I do with it eventually I will have to sell it if I can sell it at PF great, things worked ok but if by selling it I will myself behave like the informed and uninformed and impact the price then this doesn't work so this one step game is it can be misleading it's a very simplified version is that ok? they know a number yes, in this case yes, so someone came and told them that the process will go there it's unrealistic for some reason they know of course you can be more realistic that it's going up by a certain amount and then it's ok you can imagine that people know this they are making it move to that direction asking for that price it's the only thing the only point is that they all agree on a price which which things where the price will move where the market will move but the other side doesn't know this so he wants to guess what this price is the price will go there and he wants to set the price in a way that on average the market maker indeed to the price ok so I hope to be clear you can have some time to think about this the main idea of this model it's a bit more complicated for some things but the main idea is similar to what we have in a simplified model is to essentially you have an idea of the distributions you want to optimize yourself to on average not lose ok, so this is this is the second type of model like that we will get a bit back to real markets instead of these models I just wanted to make a bit clear I don't know is it sort of clear for you where we are going because what I'm afraid is that there are many things coming so ok, I want to clarify a bit so what did we discuss up to now this we discussed ok, what markets sorry we discussed a lot about how prices behave so there was all this question of what we call stylized facts but I think the most important for us in this moment is this somehow diffusivity of prices that prices statistically seem to be unpredictable then we discussed correlations which is important I think for its own sake but correlation matrices we don't really use here what is the next question after this ok you see how the price moves you want to understand what happens on the micro scale so why is it moving in this way how does the price get diffusive you know that people who trade have some type of information in practice of course it's not information like this so what is the notion of information and how can it get built into prices so this is somehow the idea you want to understand on the micro structure what happens which I think boils down to the question which we call market impact ok, how are the actions so the single trades of people in the market correlated to what the price is doing a stylized modeling of this is that there is someone in the market who decides on the actual price that you can pay but doesn't have information so how can he infer a proper price so it's him who impacts the price because he sees what you all are doing and decides where the price should go so this is a type of stylized model so the idea is to have a notion of being informed and have a notion of how this information gets built into a price and so this is the way that these economics models get some insight and we can see that in fact we get some ideas how a market should work what's the notion of information what's the importance of those who are not informed all these things and well the next things of course I said ok this is what we call price impact what I will try to discuss is that is the more the mechanical so here there are many notions which make sense but it's hard to understand what is really information how does the market maker know who can be informed who cannot be so satisfied all these things how can you guess so where we are going from now so that you don't have to yet know of course is more to get a chance of traders change the price so how in practice this works but try to come up with more mechanical models to see ok given that a market maker or anyone who is providing liquid index I cannot decide who has information who doesn't have information how in practice prices respond to actions so somehow this is the try to model this so try to see empirically and try to get models about this ok the others how much they gain well that he knows because he yeah so that he knows because he knows where so he sees the price he knows where the price was what he traded and where the price went so he can calculate of course he doesn't know per single person he won't know that you are someone who is always losing against me so I'll be happy to he will know in an overall so he cannot point out what he cannot learn is yeah this guy ha ha he doesn't know anything of the price well you can write up you can say that ok but there is a next step and he has you can put constraints in the system and the fact that ok let's say that the next step is that he owns the quantity he owns the quantity that he bought in total he owns this quantity and you can have a next step model ok but right after he wants to get out of the market and you can make a model for this so you can make several step models yes pi f will be moving yeah so it's not obvious but you can do so there is actually a dynamic version of this model but we don't discuss it here so for me it's more to give a taste of some intuitions of what's going on I don't want to go into this I don't think this model has solved the world ok so now I will try to get to discussing a bit of price impact I think we won't go extremely far so what do you so what is ok what is price impact it will be more general introduction because we will have ok the next three courses I think we will be discussing questions of price impact in general so ok sorry I actually wanted to mention one thing before getting to the price impact ok sorry about this so one more thing that I wanted to get actually before the impact is something which is called liquidity paradigm and it's related still going back to this question that we discussed yesterday so this economics concept of fundamental efficiency so there is some underlying true price in the market and then the price reflects this quite well that's the definition so one definition of liquidity would be we had several ways but one definition could be that in a real market given you trade a quantity how much would you trade a change the price right it's a sort of definition so again ok ok I do a stylized version of what we had before so if you have a limited order book you have some volume in the market so this is the ask this ok so this is what we had just another version what if I mount definition of liquidity could be ok I want to go to market I want to buy a quantity Q how much of this volume will I will I consume how much will I push the price ok and I think that's clear and one proxy for this could be could be the quantity available at the first level so you could say that ok you don't want to move the price you just want to trade at the price at the best price which is available so you just want to say ok what is available there that's the version of liquidity that's the measure of liquidity I care about so let's say one definition we'll have other definitions liquidity is a bit ill defined you can think but one definition could be what I will call you call volume at the best level so the best bit ok a few orders of magnitude to get an idea what we think of this what we discuss fundamental efficiency so let's say you the volume which is which is traded daily in the market is is roughly let's say the following quantity of the what I will call market capitalization market capitalization is simply the total value of this product available in the market so it's all the shares that are in the world times their prices it's just some overall value of the product so essentially what you can see that any given day if you look at all the trades in the market it's around 0 to 1% of this which is traded and and actually if you look at what is the what is this value here so these are absolutely empirical type of numbers it's essentially volume ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ready at do un per cento del valore totale di il mercato. Questo potrebbe essere una versione del valore totale. Se vuoi trattare un per cento della capitalizzazione del mercato, che è, ovviamente, grande, se vuoi trattare un per cento dell'Apple, è grande, ma per un grande fondamentale è possibile trattare un per cento. Non è molto alto. Questo significa che per fare questo, puoi calcolare quanto ho visto, ogni volta puoi trattare questa quantità. E per trattare un per cento di questo, quello che puoi trattare è che puoi trattare, puoi trattare come... Ok, quanto trattare puoi trattare? 100... Per l'ordine di 10 per 4 trattamenti, credo che sia calcolato. Per una grande quantità, per una quantità di trattamenti, puoi trattare per un lungo tempo. Sto dicendo un po' di trattamenti qui. Sì. Sì, quindi siamo qui. Siamo parlando di un compagno di due, che ha questo ordine di trattamenti. E diciamo che, comparedo il totale di questo compagno, il volume di trattamenti daily è 0,1% di questo numero. Ok? E, in realtà, in ogni momento che vedete, in questo ordine di trattamenti, è un altro due ordine di trattamenti. Quindi, questo è semplice. È un fatto empirico. Non è importante. Ma che è il messaggio di questo? Se vuoi dire... Diciamo che il trattamento di 1% o anche 0,1% di un compagno di trattamenti, che è ok per... per un fund di investimento, se può succedere, non puoi farlo in una trattamenti. Tu devi fare 100, 1.000, diversi 1.000 trattamenti con questi numeri, significa che è ok. In un giorno, l'ordine di numeri di trattamenti è, diciamo, 1.000, un po' di 1.000 per un liquido. Quindi, significa che, per un giorno, ma tipicamente, per un paio di giorni, devi trattare in ogni momento, se vuoi push the price by a lot. Se vuoi trattare solo che è disponibile in un momento, devi trattare per molti tempi. Ok? È semplice. Ma ha, secondo me, un grande conseguenza. Quindi, c'è questa idea che i trattamenti sono comunque, che contengono la informazione di cosa le persone vogliono fare, questo tipo di effetti, ma non può essere il caso. Quindi, se qualcuno è davvero raccogliere exactly what he wants to do, showing his information, what how much he wants to trade, because he has to cut this up. So, I know I want to buy 1%, but I will be doing throughout 10 days in small amounts to buy this. That means that prices cannot reflect the information that I have. Right? If I'm, if it's not in there. So, you cannot say that supply, this supply and demand curves will have all information, because people have to somehow hide their true intentions because of, because not enough liquidity available in the order, because at any given moment. I don't know if this is a, it's a simple claim, but I, so, so at best, this type of equilibrium can make sense on some longer scale. So, let's say if you want to trade 1%, one can write up these numbers. Maybe you will, but you will be trading for 5 to 10 days. Of course, you cannot do all the trades in the market throughout the day. So, you have to be, somehow you have to hide yourself among others. You'll be trading for several days. It means that it, maybe on these several days, there are somehow your information gets incorporated into the price, but at any given moment nobody knows what your intentions are to really buy. And this brings up to, brings us to, the fact that actually, if you look at the flow in the market, so, so, the number of trades and number of, number of buys and number of sales in the market in time, there is a very, very long-range persistence, which will be most probably due to this fact that people are doing the same for a long time. So, so what I want to show you is that, as a, as a, as an answer to this, so the liquidity paradox is that, that because of low liquidity available in the market, you cannot have all information at any given moment available built into the price. So, what I want to discuss is persistence. So, order flow, we will define, we will simplify our life. Order flow is all the trades and orders coming. What we will simply do is decide that this is simply the, the sign of trade at time key. Okay. We look at the the time series of all the trades and we only care about their signs. So, we will call this epsilon t which will be plus minus one, plus one if there is some, if the initiator of the trade is buying. So, the market order is buying trade minus one if it's selling. So, it's, we really simplify this. We throw a lot of information. And okay, we want to look at the correlation. So, what do we want to check? We want to look in time for a given product. So, we are, we think about one product and we want to look at the auto correlation of this thing. Okay. So, this, it's the auto correlation of this product that we want to look at. Okay. And, and what you get is the following curve. So, what you see on this is that it's auto correlated. So, okay. So, what we can say about this, I will have a question in a second but what you find is, this here is that it's somehow a correlation in the following way. You can, you can write it up something like this if you put some constant times some parloish behavior. So, timescale to the minus gamma in this case. So, can you, can you tell me what is gamma? I think it's important to be able to read figures. It's, it's on the figure. Hm? So, how much is it? So, what is divided by what? It's a good approach. It's a good approach. Hm? Yeah? Okay. The approach is good. I didn't, so how much is the exponent? 0.1 No 0.1 Log of 0.1 doesn't exist. So, so, so the right answer is one half. I mean, gamma is one half. So, it's minus one half of the exponent. So, how do you read this? It's exactly, it's important. I think that this is basically, you already had this. You knew the solution. Sorry? You didn't come up with the solution now. You knew how to look at far-lose exponents before. You're poor, but you're still in good correlation. Anyway, so how do you look at this? So, what you do is is take the logarithm of this. So, take the log of this and the log of this, what you will have is something like log of c is log of tau and you will have a gamma, gamma, how is it? A minus gamma here, which will be one over. So, what you do is simply do a log against log. So, what you see exactly as he said is that here you can see that you go down essentially, go down one order of magnitude here and here you moved roughly two. Okay? It's good to be able to read this type of things. Are you okay with that? I mean, I think it will happen if you do some statistical physics, you will often look at parlos. So, this will be something, gamma will be something like 05. Okay, so, it's a good approximation. For a statistician it's not a good way. Of course it's not a proof. The fact that it seems linear on a log log, it's not a real proof of parlo, but you can see that it's a very slow decay. Yes? Sorry? Well, what's happening here? Some finite finite size effect can happen. So, what happens here is actually, I measured this on intradate period. So, what you do, what are you doing? You're looking at the sign of trades, you're looking at the autocorrelation. So, we understand what the quantity is. And as you get, so, since we have, for this product which I've measured, you have order of a few thousand trades in a day, here you start to get an analysis. But actually, you can do this. If you put more data together, you can go further away. Okay? But so, what does this mean? So, you, the parallel continues just here on, visually on this figure, we didn't, it's become noisy. So, so what does this, of course, mean? So, okay, it's long-range currency. What does it mean that if I'm buying now, or sorry, if I see a buy trade now, I can predict the sign of the trade a long time from now, which is, I think surprising in the first approximation. If you have all these numbers. So, okay, one can, okay, what one can do is that, actually, the C tau can be, of course, written in the following way, as well. What is the expectation of the sign at tau from now, given that now, let's say, I see plus one. Okay? So, this can be written as a conditional expectation. And so, if you could, so, so this gum is order one half, is, let's say, similar order to one half. So, at leg one, let's say, you're close to this. What this means that actually, if you calculate this expectation here, is that even, let's say, what I wrote it up, which means that even, let's say, C tau equals 10 to the 4. So, 10,000 trades from now, you can, this will be, let's say, something quite low, but then on zero. So, you will have a, so, okay, the way to read this is, it's just, I put numbers in the equation, it's simple. The excess probability, so given that I saw a buy now, the excess probability of seeing a buy versus a sell, 10,000 trades from now. So, in this scale, it's almost 10 days or a week from now, it's still half a percent. Okay? So, it's a very strong predictability that usually, you don't expect in financials here. Yeah, it should be. So, epsilon is plus minus one, and you can write up, oh, okay, so, of course, if you forget this. But this is, so this is, actually, I wrote it like this, but this is on any, enough data, this is close to zero. The number of buys and the sales on overall, it should be the same. Because I think that it estimates it's also a time t plus down. Yeah. So, I think this can be written by itself. But no, epsilon t and epsilon, this is, but actually, this is a plus minus one. You have a process of plus minus one, on which you can look at the, it's similar, yeah? Okay, I think this is these two are the same. But anyway, so, even forgetting the numerical value for here, what you see is that, okay, so you have a autocorrelation which is super long range, which means that you have a predictability which is super long range and you can quantify it, it's over several days. Okay, so, this is, just to recall that this is, that this is what we see here. So, this gamma is one-half, is something that we discussed, just to, to go back to this, means that it's a long memory process. So, what it means that, in general, if you have some autocorrelation, which is, which is, let's, I won't be very, very clean here, but it's K to the minus alpha with, with alpha in between zero and one. This is what you can call long memory officially. And then, so, which means that, that, that, that's the integrated process, of course, will be, will be, will be super diffusive. Okay? So, if you, if you look at the, the variance of, of, so, let's say in general for a process X, if this is the case, then the, the, this guy here, that and, I mean, it's, thefaced example but, but that means, okay, it's positively correlated, will be a, will be a super diffusive integrated process. So, it's just to recall what we, what we discussed before. So, I wanted to discuss a bit of possible causes of this. So, why is it the case that if you look at the market for ten days or for any timescale, because it's parable, you have a very strong directionality of the table. Which, Ok, quindi questo ho messo qui come un ricordatore, ma quindi ci sono due esplorazioni tipicamente, è difficile dire che ci sono due esplorazioni tipicamente di cosa è qui, puoi dire che ci sono alcune relazioni in tempo perché una persona fa qualcosa per un'ora, diciamo che è in una lingua che si chiede order splitting, che significa che qualcuno vuole fare un'idea, è esattamente un esempio che ho fatto prima, per l'inizio di un per cento di il total value di un prodotto, ma non puoi farlo in un tradito, puoi usare split-up in diversi traditi, e per una settimana sei sempre there e in ogni tre minuti puoi fare un tradito. Ok, quindi questo è uno tipo di underline processo che può definire. Nel altro può essere qualcosa, ovviamente, un tipo di pericolore, attenti tra i traditi. Quindi è un po' di quello che sto scrivendo, quindi se c'è una correlazione autocorrelativa in il processo, che è due alle azioni, il processo è l'azione di alcune persone, la correlazione può essere due due persone singoli che sono autocorrelati con loro, o persone che sono correlati a qualsiasi altro, che si chiederanno pericolore. Il primo significa, come ho detto, che hai qualcosa di grande da fare e per un'ora puoi fare lo stesso, buying, buying, buying in the market, e good general auto correlation. The other thing could be that I buy and you guys look at me and you also start buying, because you say yeah, maybe. So you can see the sign of trades in the market, you can see. So it could be a strategy that you try to do what others do. I don't think probably it's not a good way, it's not so that others know very well, but ok, it's trivial. So another correlation could be that it's between different people and ok, so I said an example for this, Herding, of course, I won't go into details of this, there are enormous amount of models here. It could be some direct Herding, so you see others do something and you do that, but actually it could be also a common field that people are following. It could be a common news, everyone is reading the same newspaper, they are following that, so it could be an indirect Herding. It has a long... Or some people have. I mean for the correlation it's enough, I mean it's all that everyone has to be like this, but if there is a non negligible percentage of the market who are behaving like this, then that's the case. And, ok, so I'll try not to go to extreme amount of details here. Of course this can be... This can be... If you have information of who is doing what, this can be decided. So to do this, actually you need information on the people, info on who is trading when. Ok. And if you have this information you can decide on this. I don't want to go extremely deep into this because we are a bit behind schedule, I think. But so what you can say is that this type of information can be obtained, so you can get a type of information in the following way. You can say that I'll write up and then I'll explain. So you can define it for environment levels, so you can define some type of time series in which you have an extra index, which I call I here, which is the person who is trading or the entity who is trading in the market. It's hard to obtain this data, but it's actually for academic research you can get this type of data. And so you redefine, so we have this epsilon here, which is the sign of the trade, and I say, ok, let's say epsilon t of t i is non zero if it actually was trader i if trade, ok, so if trader, sorry, trader i. So if it was this person, I'm looking at a given person, so this will be plus or minus one if it was him acting in this moment at time t. And it will be zero otherwise if it's someone else, ok? And what of course in this revolution you want this to be something which we call event time, so from a continuous time to go to a time that only in the moment that something happens that silver and your clock moves. He didn't do anything, but since we are in event time, someone else will be doing something because otherwise the time wouldn't tick. So it's, of course, the sign of this here is if he bought or sold and at any given moment only one epsilon, so one actor can be active in any given moment. So it's a simple writing up with which, ok, if you have this definition you can write up in a trivial way the previous correlation, the following way. So the empirical version of that with this new information can be written like this. So I really, I hope it's ok, but I did it simply, ok, I have a new variable in the system, which is I, so the correlation will be, can be rewritten as a double sum, so still we are summing on T, ok. This is the number of trades. Actually I could put T here, sorry. So it's the number of trades all over and you're summing on each actor, simple. And actually, ok, for simplicity I will remove this here. Let's forget it on average, this is very small, close to zero. So if you have a long data, there is no one who is, for years, buying. He would own too much. So sometimes he's selling on average. It becomes zero. So this is the type of correlation that we have so you can already see in a trivial manner that, ok, what do you care here about splitting and herding? Well splitting will be the cases when I equals J, right? So it's the same actor at T and T plus tau and herding will be when I is different from J. So T is the number, it's the normalization of this also. Number of trades overall, so it's your normalizing. It's simply, before the sum is on the INJ, it's simply this can be written up as your empirical data. Sorry, it's N is T. It's the same normalization we are doing. And so, ok, your goal here is to write up somehow this C, T, the following way. So what you want to do is come here because it's going to be visible there. So actually what you can write up is, well you can, what your goal is that you have total correlation that you had before. What you would like is to somehow decompose into a part coming from splitting and herding. So let's say, some definitely we will try to define these things. And so the way that one can define is the following. You can say, for this you can try to look at, define it in the following manner. C split will be at the top and then I will define it, ok? Before this, so it's one thing that one needs to define just for simplicity. You can then, in a similar manner, define the following correlation which is between INJ. It's conditioned on a given INJ. You define it like this. One over minus the average of them. This is just the, we'll probably see it in the slides but you can write it up. So what I'm simply doing here is, ok, taking this, so I have this epsilon i epsilon j, I can define the correlation, the sample out correlation for people i and j in a similar manner. It's, I mean, I did nothing here. What I have to define, of course, is this, will be the number of times that j, so these equations are ugly but they are really super simple. So what this means is that, how many times does it happen that these two people act all time apart? It's just the number of non-zero elements that you will have here. So you have to normalize with that. Ok? Are you following me? Sort of? Ok, we'll try to finish up fast. So if you have this definition here, I write up the claim and then I show the results so you can have the following. You can define this thing here in the following manner. Let's simplify life. It will be just like this. Ok? So what I do, why I, ok, there are things that I have to define here. What is P i i will be, ok, P i j will be T i j over T. So this thing, ok, it's the number of cases that they are active if tau times apart, but the probability, so number divided by number of points in total, ok, so that's simple. And so you can write up the, so what we said, the total correlation you want to write up as something coming from, splitting something coming from herding. And you can do this, why it's not an exact equality because there is all these averages that remains. So I assume that the average of the processes is not relevant and it's really just the correlation term, ok, sort of. So what is actually important from this, we'll see what these results give, but actually what you get is that, ok, so the terms in the correlation, I mean that which gives an insight, you will have a term which is not related to your sign, it's just the number of times that you act together, so somehow correlation of activity. Is it ok what I'm saying? Ok, you ask after, if you don't ask, I don't know if it's ok. I assume it's ok. And so there will be a Cij, which is the correlation of signs. Ok, so this is just an insight that actually a correlation will be, will be the, will be somehow the product of being active at the same time and doing really the same, going in the same direction. So there it will be somehow the product of these and ok, so I won't go much into it, I mean these are, it's just algebra writing up, it's nothing, it's sums and products, so, but if you have question you can come and of course I don't know, I mean there is a tutorial, so maybe if things are, I'm not sure if this is needed, if things are needed to write up these calculations properly, it's going to be done at the tutorial. I, you have to tell me if you need it or not. But ok, let's get the result what they give. So actually what I plot here is exactly this splitting and herding component of the total autocorrelation. Sorry. So the red and the blue curve here sums to the black curve that we had before. Believe me, just here we are on a log log, here we are on a linear log scale. And so what do you see is that, well it seems to be a clear result that the red one is way above the blue one for the time scales that we care about here, so for up to a day, then you get in 10 noisy periods. So what you see is the end and ok, we have some definitions of error bars and we can see that the difference is outside the error bar. So we can see that for short times you have some herding between different members of the market, let's say up to 10 trades in this case. Then actually it seems to be somehow negatively correlated, it might be obvious or not, I won't discuss it here. And but the term which is about splitting is way stronger for all scales. So indeed what governs this long memory of autocorrelation is people doing the same for an enormous amount of times. So it was not just an order of magnitude estimation that I gave, ok, if you want to trade a realistic quantity, how much time you have to spend trading this, but it can be quantitatively seen that indeed it's a splitting part that dominates. So, ok, so I think we should stop. I wanted to give one, yeah? Yes, so yeah, ok, so the correlation between us on average, I mean, you know, I don't know, 100 hectares, I will give a sort of exercise still, so, or something to read, something people are leaving. So yeah, so what it means is that yeah, ok, for very short scales they are comparable, even if it's much lower. But overall what you can say, yes, so the correlation between different people is much weaker than the autocorrelation. Ok, so the autocorrelation coming, sorry, from the same people, yeah? Yeah, so the longer memory behavior of this is due to this. It's people, yeah? That this extremely long range persistent of the order flow is because people are doing things for long, the same thing for long times. So, which is interesting for its own sake and it will be interesting for the next thing. For example, if everyone is super autocorrelated, you're doing all the same, the same thing at the same time, how can price be non predictable at the end? If I know what you guys will be doing tomorrow, how can it be non predictable? This will be the main question of course. And also, ok, so in the more philosophical level, ok, so if people are doing the same for a long scale, it means that at any given moment the price cannot contain the information that you already know, that you will be trading all the time, the price cannot know this. So, ok, I wanted to give a paper to read and to write it up and then, because I didn't give much exercise, I think compared to others, no? So, no, I'll give it. So, no, I wanted to give a paper which I think gives a quite good insight in this. It's a nice paper, I just write it up here. So, the idea there is to connect this long memory in the order flow. What we said that, ok, most probably it's due to people wanting to trade a lot. So, the idea is to connect the auto correlation of the order flow to the distribution of the size of trades. So, if people have enormous trades that they want to do on long scales, that will have an effect on the correlation and what's the relation between the distribution of the two. So, there is a nice paper on this which is the following. So, it's by Lillo, Mica and Farmer. Is there access to vis-à-vis here at ICTP? How does it look? Ok, I can send actually to you. So, vis-à-vis sits from 2005 and I don't have the title here. So, it's on the archive as well, one can find. I can send you the title, I can send you the PDF if you want. So, the idea here is that if I give a new notion and I stop. So, normally single orders, one calls orders or trades and you call meta orders, all these decisions that last for a long time. So, the real thing that you want to do, sorry, we'll get back to this. You don't have to read it for tomorrow. So, it says that it's the distribution sizes of decisions to trade or to buy or sell, which we call meta order in relation to the auto correlation that we just discussed. So, my goal here, ok, there is no real exercise to do here. I think it's useful to read the paper. No, maybe not all of them. So, actually it contains two models and there is a simpler model which is I think they call fixed n model. But also the other one is interesting. So, it just goes through the calculations if things are clear and maybe not today's tutorial but the other one you can discuss about this. It's a very simple model. It's quite simple calculations but I think it gives some good insights too. And it's a good way to understand someone else explaining things that I explained. It's always good to have others as well. Ok, so that's it.