 in izgledaj se, da je to skupno vse jaz so prišli, da se je zelo tudi vsočen, kje je bomo grano, in ki se prišli, da je zelo možno nekaj zelo počunil. Zelo se naj način co je nebezpečno zelo, da se je srečen, da je nebezpečno zelo, da je bilo vse še vse. To je OK, naredaj sem tudi neko pogledo, zelo na veseli. srečnje pošli, če se pošli vstajilo, in pošli, da je šte in načinja norma za druga dela čas, načinja ni raz头ne, pa pošli do najšča maglouščne zespe. Kako so načinj, je naredil prije. Z Sabu če tečna je iz carefullyredna bilo, uvršalfully ležite, da je bolj sklepeč, ako je bilo zespečne prviče, ste obcleti nekaj prakčri, je pravčne vse zasedo, in to potem ležite. But its flat later is what one would expect, that correlations in price you don't have long range correlations. On short scale the fact you have that you have some structure. Well we will discuss it a bit later looking at models, what they say about this, but we have the guess that is related to the fact that order flow itself is what correlated. And I think what was more interesting, is the question of this meta order. je zelo vzelo. I tako je zelo jedno vzelo, da je tudi za vrčenje, vzelo v tudi vzelo. Zelo za ljude. Zelo se je... To je tudi vzelo. Zelo se je... To je tudi za vzelo. Vzelo je, da... ... ...zelo, da... ...zelo, da je... ...zelo, da je... in zelo tudi prikaj, da je tudi v čas, v čas, in in v čas, kaj je, tako, vse je, da je, kaj je, veči da je skupajštik tukov, način je, način ni se vse vse, da idem, da je, naprej z 0,5, je to nekaj, nekaj, nekaj, 0,5. In okej. Tudi je bilo vse prej, da bilo srednji svoj, da se pošli v sistem, in vse bo dobro vzupo. Spravim to, da je to nekaj vzupo. To je pa vse poslednja. Prej, da bilo na tjerek, pa pošli za kratilovate, da se pošli za vzupo se srednji vzup, ali bilo na različni kratil, da nezapravimo, da bo vse, da je ni zelo vzupo. Ne zelo pošli za vzupo. In In, OK, so, as we discussed, this is the volatility, so it's just to have a scale which is of prices. The important thing is here, OK, this is what we see on this curve, just to prove it, and we saw that actually it's for very different markets, it holds. And, OK, so this starts to be, I think, interesting because it seems that, it's all the things that we said yesterday, so the fact that you have a nonlinear response is unusual, it means that something strange is going on in the system. Obviously you see that there is some memory, so, OK, so what does this mean? So, if you trade at quantity Q and you look at how much the price moves, it will have some behavior like this, OK, this is a square root, just if I can draw well, it's something like that. What does it mean? Well, you have some, in principle there is a diverging susceptibility here, of course in practice you have discretization, whatever, but so for very small volumes you can push the price by a lot. So in some way to say this, that for very small volumes this has an anomalously large impact and of course there is a sort of flattening out here, so the more you trade it the less you are pushing the price, which means that there is a memory in the system, first half of your trade pushes much more than the second and in many ways to say it. I expect that these are understood, so this is more a language that you know. And we discussed, OK, just to make it clear, we discussed a bit also, OK, this is nice, it's square root, but what are the real orders of magnitude? So we discussed a bit at the very end, the idea of for some typical numbers, how big does this queue have to be for this thing to be comparable to fixed costs, like the half spread in the book, in the limit order book, right? And OK, I made a quick calculation and I just looked at Apple how much you have to trade on a given day in the day before and we came up with an umberizing like 50,000, so if queue is $50,000 then this already dominates over fixed costs, and just to keep in mind, OK, $50,000 can seem a large number or can seem a small number, it's still something that we can at least write down, it's not something enormous, but it was for the stock of Apple, which is probably the most liquid. So actually if you look for an average liquid stock in the US market, it will be more on the order of a few thousand euros that you have to trade in queue for this to have to be not at all negligible. So it's OK, it's clear, yeah. OK, so I just, we looked at it yesterday, but so in a crude approximation, what you would say here is that OK, so sigma t is something like square root of t, right? The more, the longer it is for not very short times. This is OK, and actually what you would ask, so what is VT, VT is the volume that is traded in a market while you are acting, that is again for not very short scales you expect it to be linear. So in two hours the volume traded by the entire market will be double of what is traded in one hour. So what we say is that if this holds, and if this is 0.5, then the t dependence is not there at all, in ideal case, this might not hold perfectly and this might be exactly 0.5, but even if you have a t dependence it will be something extremely weak. It becomes hard to measure, so if you have a t dependence here of power 0.1 it's, you cannot do much of it. And so one more thing that I mentioned yesterday but I wanted to just put up a figure about this is what happens, OK. We discussed what the price does while you are trading and what's the final impact and we said that it's the same since the market doesn't know when you are going to stop in the same functional form, but what is also important, OK, at a given point you traded all your quantity and you stop and just wait how the system relaxes. You kick the system, what happens and it relaxes or not, OK. It does relax. What we see here is actually what you would have, so how the price moves later. The problem is that I put Q here so if we said that, OK, instead of Q we are talking about the time that we are executing then this will be a correct picture to do like that. I'm being vague here. So one thing is that, yes, there is a clear relaxation in the market, yeah. There is a? How do you know? Because I drew it like this. It's not known. So in practice this is what you see, OK. Yes, it's not obvious. In practice it's very hard to measure because so this is on days like 50 days so what does it mean? On one given day you're trading and then for 50 days so you can add up what's the volatility of the process without anything so you really need to have a lot of data. Of course the further problem is that probably it's not that you trade once and then you wait 50 days. You will be doing something else in the future so in practice it's very hard to measure, it's hard to get good data for this. What is clear is that there is a decay. It seems to be a power-lowage behavior in the beginning. Actually it starts to decay fast here and then it's not very clear. So this figure would say in this paper which I took the figure from, actually it's not written here, but it's a paper from this January, claims that this goes to, that this flattens out. I mean visually it seems like after 50 days it seems to be flattening out. It's not at all clear. It depends a lot on how you condition your data so it's a very much ongoing research but what is important is that in all these questions of course there is a question of can you extract energy from the system and continuously. Can I buy something and resell the same thing and just buy my actions, push the price in the direction that I gained on all these. And so there are some theories how it should behave. Actually what it says, so if this goes like the square root increasing, the decrease should be also power-lowage so it should go to zero. I'm not sure if you can have a, I don't think if you have a permanent part you have some manipulation. I think it's an important question to study. So you cannot have trivial manipulations of the price, or that's what you expect, okay? Is this okay? Okay, so that's what we had yesterday and what we will do now is try to come up with some, okay this is what we see in the market. But we've seen, I think the main things in the past lecture is that there is a long-range correlation of the order flow and prices and trades push the price of how can we put this together in some at least phenomenological type of models. We will see more complicated as well probably on Monday. So what we will do is we will try, today we will try to look at models at the micro scale. So there are models which are more on the long scale but we will try to do now is how can each trade affect the price so each one of them and of course what would this add up to in total, okay? So we will look at a couple of models. So one is sort of simple that it's good to see it once. So if we are naive, if we have a naive model and we say that every trade pushes the price by the same quantity, okay? So it's a y9. So every trade impacts, I call it g1 by g1, okay? So every trade that happens does this. I don't know if everyone signed this paper. There is a paper going around. So then of course what does it mean that the price responds so the return will be if I define it like this. So just for the indices, I think every lecture I define slightly different indices. This is how I define it now. I'm just coming down the stairs. If I now really define it, I will mess up something. So then everything, it will be something like this, right? So this is, what am I writing? I'm simplifying life here but so this is some noise term. I say that this is zero and it has some variance. In the Gaussian noise, epsilon will be the sign as we had it yesterday and so I'm already simplifying things. I only care about the sign. We saw that dependence in volume is weak and I just forget it for now because life is much simpler, okay? I'm talking about one product here, yeah. So that's why there is no other index. I'm considering that, so actually one thing that I mentioned yesterday, that there is a type of cross-impact. It was just anecdotically, it's super interesting and a lot of work going on. We will forget it. So yeah, there is one product and all the things that happen on this product can impact it. Of course, what's in this noise could be effect of everything else. So okay, so in a model like this you can write up of course what this means is always right in this profound way. Okay, so I didn't do anything, yeah? Times are related or not? The what? The noise is uncorrelated. It's some idiosyncratic thing that doesn't, it's not, so we are in an ideal world. And so I didn't do anything. I just said, okay, the price at time p will be some reference price in the beginning of the history. Times, all the effects. Okay, that's clear. And okay, one can play in a model like this. It won't be surprising what we get from it. Is that, is that okay, you can have two situations here. Okay, g1 is constant, so if there is no correlation in epsilon, in epsilon in time is independent, which is okay, then things are okay. So if no in epsilon, the price response will be, so which we defined yesterday like this, this expectation will be equal to g, I don't write out, okay? Everyone can test it, so what happens if you write it out, there are no correlation terms, it will be only g1 that remains out of this. Okay? This here? In the indices you mean, the return is price change from t to t plus one? The mid price change. Yes, I'm talking about mid prices here. Yes, sorry. Most of the time I will talk about mid prices because, so what do you think, so there is a mid price and there are the two ask and bid prices. You hope that the mid is somehow more informative, it's more everything. So the bid and ask is, so already if you have a buy, it will happen here, if you have a sell here. The bounce in between the two, which you don't think contains real information for us on any timescale longer. So I'm talking about mid prices. You're right. So okay, this is the price response and actually one can write up, I'll write it here so that I have space, but you can also write up what we call this variogram for this mid price. This thing here is visible, which is, which will look something, it will be actually this, but what's important is that it will be linear in L. Things are okay, things are diffusive. So if there is no correlation, sure, we can have a model like this and no surprise here, I could have just omitted writing it. But the problem is that there are correlations in the price, we have seen it the other time and if you have correlations, so if you have correlations like we saw yesterday, then things do not work anymore, of course, here and you would have actually a correlation in between returns as well, which are of the same structure, essentially it would be like this. It's not surprising, I could have also said we don't even write this up, but since we all do models, you can see that this is called G1, so it is a hint that we will try to do something a bit more complicated, but in the same manner. So what it means that if there is an order flow correlation, that implies correlation in the returns, which is at odds with what we have seen before. So, okay, this won't work like that. And so the goal is, and actually I think what we will sort of look at today probably won't go further if you have also the exam later, it's okay, let's stay in this type of world, so we might want to have a linear world, but more complicated. There is what we actually call propagator models. It's clear what we did here. We came up with something, it doesn't work good. So propagator models in which the idea is that, impact is not just a direct kick, and then nothing happens, but somehow the effect of a trade propagates in time. So, which would be not very different in a first approximation so you say that price can be written up somehow like this. And so, what would be the limit? So what am I doing here? It's a linear model. So this is just a sum. What I say is that every trade that happened has an effect that propagates through time, which is this G tau. So what I do is sum up, to get the price of T, make some reference price far away, and I sum up all the trades that happened until now, and I say that the way they impact the price is their sign times how sometimes they are propagator. Sort of, I think it's clear. So how much did the effect of them fade away until now? So the trade at n will propagate to T via this propagator, the G, T minus n, and okay, there is a noise term. It's clear what we are trying to write up. So, okay, it's a relatively okay model. It's not overly complicated. We can see, okay, what predictions it gives on different time scales. Actually, just for language, so this we call propagator model, there are people who call this transient impact model for obvious reasons. And okay, so we made a huge amount of assumptions here. We said something about G, but we can always try to fit a linear model. But we also made assumptions that that volume of trade doesn't matter, the volume of the single trade, we only have epsilon here, which is okay, which we saw that it's, we might do this. We also say that time of trade doesn't matter, and we also say that market conditions, which is a bit similar to the second point, it's clear. There is no, in this G, depends only on T minus N, but doesn't depend on N in this language, and there is no nothing extra. So it's obvious that we did these assumptions, and I think it doesn't do not surprise you, but keep in mind that this is a huge simplification of the world. The world is more complicated than this. You know that thing, it matters exactly from the conditioning of trades. And not on what happens. This contradicts it, but what we hope is that on average things will work out. Is this okay? Well, the first one we saw and we see that okay, volume of trade it's not completely flat, but there was a very weak exponent in the impact depending on this. So we can say, yeah, sure, it's not exact, but we can live with it. Okay, but market conditions we do not know how to define them well, so we will have to think. Time of trade, if it matters, we believe that it should matter, but we try to live without and see how much this predicts. And then we compare the predictions of this model with the real world and we will think, okay, so what should we... The time of the moment of the trade, so was it at 10 am or 2 pm? Which is a bit the same as marketing, what was going on? So there is no N dependence here. This is a usual way of... Actually, I was here in Deepak's talk in the beginning yesterday, that's what he said. Okay, let's try to throw away information. I mean, some conditioning and we will see at the end. So what would transient impact mean? What is the... How would one read this type of model if you have a propagator on each... What does it mean? Each trade has an impact that propagates in time, which is g. Well, if g is transient it doesn't mean what it will be. Yeah, exactly. So indeed, just this model doesn't mean that it has to be transient. So here you guess that it should be transient because otherwise you can rewrite it in a language like this and you would get... But this we will try to see explicitly. And so what's the plan now is that, of course, you want to understand how g looks like, how g behaves empirically and try to... And analytically, if you can. And try to understand what predictions it gives for all the things that we have seen yesterday. And then... Actually, I write up another form of this as well. Should I do it now? Actually, I write up another form of this just because maybe it's easier to understand, but maybe you understand. Is that you can do... You can write up a differential form of this usually, and often... I will get back to this. So actually what you can do is also do this so where we defined... So what I'm doing is simple stuff. I'm taking the differential form of this instead of prices. I take differences of prices and one can write up in this language you will have another propagator which will be the differential of this. We will get back to this because actually for some numerical reasons it's better writing like this. But maybe this is easier to understand. So what does it mean that the return at time t will depend on some direct immediate impact of the trade that happens at t. Or I mean just before t. Plus the memory of all... So this is the memory of everything in the market. And this is the immediate... It might help in understanding but it's not much deeper. So return at time t depends on all the trades in the past but it's also seen from there. And so the question is how to... So how to calibrate... You have this type of equation. What can you do in these cases? If you have real date of course what can you do? This is a linear model. You can do some linear regression if you want. I will show how one solves a system like this. So the problem is that if you want to do it in an explicit way so not just via some regression but solve it by hand where this g you cannot measure so you want to connect it to some measurable some expectations. So actually the way to calibration or solving the problem which again is this thing that... This is not really finance related here. So the way you usually do is you can write up. So you have this equation here and you can in the same manner write up for another time and do anything. I just changed the index. So what you want to do it's clear what is summing up. What is right? For the limits for the time. And which means that the measurable thing usually is this that you like to look at which will be the following shape. Sorry, today we have a bit of this. Unfortunately once these things have to be written up so what you can do is take this and subtract this and so what's the message? It's easy, we can do it. What's the message of this? You have all the trades that happened between before T so before the first price for all these you have the differential of two propagator so you will have the propagator is it clear? So how much trade before T affected T and how much it affected T plus L it will be the difference of these two times the trade. And all the trades that happened after T, of course before T plus L will just have their propagator trade. This is trivial up to now. What is usually the trick to have to solve this type of problems is that you want to get you want to see expectations. You write in this time series that you cannot really calculate with so actually what you usually do is the following so I write it up and then we'll discuss. So you say the response to the price which is exactly this the response to the trade so from this what you want to do is to get to a correlation in practice you are multiplying by epsilon T and taking the time average. So this thing how will it look like it will look in the following way so if we just use what's here obvious, these are ugly these equations wrong to write. Yes so what we say we are neglecting there would be a noise term we forget about noise part of the noise of course cancels out since we are looking at difference but the remaining for simplicity I threw it away but you can always have a nice term so is it clear what I did here so I'm just multiplying by the sizes taking correlations that we've seen correlation between price change and the sign and so what we see here is the correlation so in practice this is C T minus N and ok so what one can do is the following but then it will be the following thing actually I do not human rights that properly one can write up what did we do here from there but instead of epsilon we put the correlation and we paid a bit with replacing variables so what we do is once we do here we will do N we will call so what is T minus N we will replace by N we will do a step and actually in the second term there is another replacement of variables just to simplify that after we do just for you if you want to check it I think we are doing this so once do this replacement write it up and then in the second term whatever so we get this here is it ok I don't have to write it up I mean you can play it and and so this is what you want to solve ok so no actually we are happy because but everything else is measurable C we can measure, R we can measure so we have a system of equations that we can solve matrix equation that we can solve so we related this G to two measurable and of course we get back the result that we had before if no auto correlation then what you would get is R being G which comes that only one of this term will stay where C has zero in its argument which of course the correlation at leg zero is one so no surprise ok what is the problem is that of course this is nice to write it up like that we are always summing since ever in the past but first of all our time series is finite so we will have problems if we have two long legs and we have seen that the correlation decays we know that the correlation will decay under the noise level after some time so in practice to solve this actually it's a bit more complicated for the following reason is that in practice there is a let's call it L max which is the maximum leg that we care about because otherwise we are talking only about noise and which ok can be related to the size of the system and so to solve it with a maximum leg it means that you have some boundary condition in this system of equations you say that after this nothing matters anymore it's ok, what I'm saying the problem what is the sharp condition which means that I say it and you say if it's ready so what's your problem here what's the boundary condition in this system if you say that after some L max everything decays it would be that that this G itself if you want to solve this you would have a condition that G goes to 0 as L goes to infinity let's say so it's the the propagator itself so you measure up to the time and then everything is 0 so it's a very dangerous thing so that's why actually I wrote up the derivative form so you're much happier to write things in derivative form and have a boundary condition on the slope of this thing that it will flatten out is it clear what I'm saying and so in practice I just write it up so that so that well it can happen that you solve stuff like this it's not a very much finance related type of problem so it's good to know how to solve this so actually you can you can define the derivative I will write it up properly and then I'll say what I'm saying so I define a new variable which will be this there should be some simple derivative of the response fine, we are in the discrete time so derivative and then you can write up an equation for this simpler I think it's like this so let's get to an equation of this type where k as we defined before will be the what do you call it discrete derivative the difference is of g and s is the difference of the response so solving this system and then integrating you can you can get g and what you're happy about is that here your boundary condition will be that this k goes to 0 for large legs which is less dangerous so it's in practice so that's that's the way to solve a problem like this now what we want to do is to look at how does this g look like if it is solved but actually I want to give a homework here because that's the way to do so which is I think it's useful to get your hands on this type of calculations you might not like it so let's have simplified version so assume that ok what did we say here, we omitted many things we had one propagator let's assume that you have several type of events in the road, so let's say you have types of trades you'll call it L and S we'll call the type of event so what we'll call this pi so you have some type of definition what you call small trades and large trades volumes below 100 and above 100 for example you can define it for yourself and so then you can of course write up similar type of model just it becomes a bit more complicated so let's write it up then I explain properly what I'm talking about so what are the summing going on I think it should be like this so what I'm doing here is I have two types of event large and small, they might have different propagators and I want to check this, is it true or not it's a relevant problem that one can check but I can always write up my price as summing over all events converted by these propagators we didn't say but obviously it's a convolution and this will be a chronicle delta the way you choose so we are in this quick time in every moment there is one type of event that can happen so you have to do the choice on this sorry, yes what is given so in every moment in practice imagine that you have your date and in any moment you know if it was an L or S type of trade and you just want to to see what you want your question is ok you can measure things in your data there are the measurable SVD here you can measure responses and correlations and you want to determine g how would you do it there is a question so do ok, so is it sort of clear so the question is write it up, obviously the question do a point define relevant ways of response functions and correlations I mean do averages that you can measure on a time series and solve the system Shahin is not happy it's a homework is the problem ok? or nobody cares because there is an exam afterwards no, no, no a general equation of course so this is the point and give a closed form solution ok, what did I give here? I gave a system that you can invert the system now to get k of course so a type of solution like this I mean the hint is that of course it won't be extremely different but one has to so the goal of the exercise is to be familiar with what we wrote up here to be forced to calculate so you have a data you have measurables in it in this one dimensional case what are measurable as well r is a measurable and c is a measurable so we are happy because we have an equation here that relates r and c which one can solve if I give you r I give you c you can solve it we discussed here a bit question of numerical issues but ok, let's forget it for now we don't mind about it here in this solution the point is do something similar in which you can correct measurables from which you can ok, so I mean I don't know about the tutorials on Monday there is at least one tutorial or more maybe more because we are a bit I threw a bit of exercises here and there and they are summing up so ok no, no, no let's also do something so what is ok, we have a type of model a linear type of model in which impact propagates ok, let's first of all see how these things look like and I have to explain there are too many curves on these grounds but I will explain you so these are two examples actually I can tell you in CISCO these are two existing stocks so what you have here so the flat curves are the response functions that we have seen before so you have the price change as a function of time scale and so these decaying curves which actually it's a bit ugly these figures just they don't find very good ones I might show others later so these are these G's so you can see the G's here so that's I see that indeed they are decaying they seem to be if you look at this, you can imagine that indeed it's transient you don't see where it really goes but it seems to be the case so it's decaying as expected which will bring us in the direction that you would expect prices to be diffusive and ok, so what you see here I think these are extremely ugly plots sorry for showing these so what you see here is the G in a log log scale so what you see is that in the tail there is maybe some some parlovish dependence ok so it's exactly the same stuff here just in a log log scale actually we see that it seems to be the same exponent in the two cases around one quarter so it's ok it's decaying very slowly with an exponent a little smaller than one I have to clean up here and ok so this is actually we spoke about long range persistence of the order flow usually this is called long range resilience of the order book and you have these two long memory processes playing one against the other so the question I think that we want to ask is ok so here there is an exponent 0,25 but what is this can we get a hint on what this exponent is and so that's what we will try to look at in the following so it's ok, we have seen that that this is what we expect from crisis of the main what we call stylized fact was this, we expect this to be somehow linear in L right this is what we call the variogram and so what one can do is what one wants to do is write up this thing in the previous model and see how it behaves and it will be a bit simplifying here and I will do the following I say I care about asymptotic behavior I care only about large L I don't care about, actually you see that here there is some difference up to let's say like 100 it's not really following this power law so I will do the following simplification so large legs I will only care about the following ok so I am contradicting what I did before what did we say before that the price difference between T plus L and T will be essentially an integral on G and an integral on the derivative of G is it ok what I am saying what we said is that the change between T and T plus L will depend on all trades that happened before T and these will have two propagators how much they affect T and how much they affect T plus L and the difference of these plus the propagator of all trades that happened between the two things what I say is ok let's say L is large and I only care about the first term everything that happened before T is already forgotten ok so we are in an asymptotic world nothing changes one can write up roughly the same equations take the square of this so let's light up the following things so what will you make sign of the fact that this L is large I will just go to T minus 0 matter that much just to have a bit less less variables in the system and that can be ok you can write it up in a simple way if you will check it at home if I don't make an error ok so I am not doing anything very complicated what I say is that ok I use this fact I reindex the bit just to have less letters in the system so it will be the change here will be sum up to T of this propagator times the sign and so I have to take the square of it I will write it up properly and and so for if one wants you can say that you can call this thing here I will call it q and this thing here I will call p for simplicity I just change variables to have life bit simpler and what I will have is the following so what did I do I took but I changed variables and I just noted epsilon epsilon average product of the average is the correlation nothing new here and so ok actually if you move a bit to continuous time maybe it's easier to see this but let's say that so this somehow I just give the hint of how you can see the asymptotic exponent so this ok you could say that you write it like this much here just move to continuous time because life is easier there so I want to make a change of variables there are a couple of equations but the final solution will be very simple so don't worry and and so what can I what can I do next ok I didn't write it explicitly but what you see is that the behavior of somehow this g tau is something like this I will call it minus beta just to have a letter so it is some power low exponent there that I am looking for that I want to find and so we are almost at the end of this so actually what you can do is ok you can rewrite this here in the following way so use the information that we have I am just being a bit vague what do I do from yesterday this I know from today I want to find beta so ok from this thing here let's write up the dependence of them ok on p and q is it ok what I am doing I don't know if it's too simple it's too complicated I hope it's too simple so actually if you do a replace if you want to do the following you can say q some t times u and p will be something like t times v so I am just taking u and v which will be in 0 and 1 change of variables then what I get is the following ok I write the final thing it will be the following so what will we have we will have a t to the minus beta here we will have a t to the minus beta here and we will have a t minus we will have a t to the minus gamma probably here from the difference yes and something else which will be which we will write up but it won't depend on t so we will be happy about so actually the following happens you will write you will get how much is this gamma but actually you are integrating so of course here also you have a t depends so you will have a plus 2 which I don't have color so that's because of the integration and something else I write it up but just to be clear so you will have something like this if I didn't make an error which is unprobable anyway, something here which is okay, this is a number we don't care about it because we care about the asymptotics are we okay here so this is the behavior in time that we expect and what do we want for things to be diffusive to have this to go as t square in theory sorry, to go this linear in t so what we have diffusivity would be ensured if minus 2 beta minus gamma plus 2 is equal to 1 hopefully the solution of this is 1 minus gamma over 2 for beta check it it can easily happen that I made an error somewhere here but I hope not but anyway so if you want diffusivity we get a condition of the exponent here depending on the exponent here so let's stop now and let's think about it if it's clear what I'm writing the equations are I think it's simple to rewrite so we are not exact here we just want in the for large L to describe so is it clear what we are saying the main idea is the following we have seen yesterday this that there is this long memory in the order flow it's clear what was our problem that if there is a long memory in the order flow and each of the orders push the price how can you have no memory at all in the price process we see that if it's a simple direct impact it won't be the case you cannot have a model like this what we said is let's write the let's write this type of model so that your impact propagates in time and what we see is that it indeed decreases in time but what you say is ok I'm solving this problem so we don't have the equations anymore but how do I get g, it's a relation between autocorrelation c which is here and price response r which contains the fact that prices are diffusive except for some very small scales so what we say is that so let's write up this thing what is the diffusivity condition of our mid price and say how should g behave to get back what we were looking for there is no explicit way because we are not generating the data what we can do is measure certain things and what we are looking for is the relation how it behaves and if things follow ok, I think what your point is that if we know that prices are diffusive and if we made a model for the price then if you are solving our system we should get back the proper things should work so indeed this propagator is not some underlying wisdom of the world what we say is we have a model for this and we want to verify how this propagator should behave it's our construction so we get just by some whole conclusion to have diffusivity we get this relation and ok we want to check empirically if things work out because that obvious when you fit a model if things work out but yes, why we want to care about this because of course this is for single traders we also want to see ok so what predictions do these give on other scales so for that we should know if we have a description we can calculate things we could also say we just measure things and do only numerically integrating summing so this is the relation that we get which actually is in line not very much surface so what we saw yesterday is gamma is something of the order zero five which should be the case that well in practice you get something order one quarter for this beta but this is of course on average there are products for which you will see gamma zero six and beta zero two and their behavior is something like this so it's very slowly decaying and it follows what we would expect here actually ok sorry I want to show more properly so here we see this zero 25 but actually you can look at it's an uglier figure a bit but you can look at several products what we are doing here is exactly the same so it's the g on the y g which is plotted as a function of time lag and I don't know if it's visible so red is the data and there is a line on it the formula and so the tail behavior is indeed what you would expect so ok no surprise so so this is how g behaves but also this teaches us something before getting to to further analyzing it is that ok so you indeed we call it transient impact indeed the impact of a single trait dissipates in time so somehow goes away but very slowly so you see that it's a very low exponent so it has an effect again just to get some concrete things here the time scale is 1000 at least on this data where it was done this is at least the order of a day so it's a 1000 traits but actually I think it's all data so it's really a short I mean on these scales very short very slow decay and ok so we said to solve this puzzle that how can it be that order flow is long range correlated but prices are diffused where you have some extremely slow adaptation of the market to what you're doing to get back to diffusivity yeah that's empirical data you can do two things one is fit the power law and compare to this or calculate this it becomes relevant for several other things but it's not really relevant on this scale because you average it out simply this noise there is a lot of noise but it averages out to if you look at the volatility of the system which we actually want I think have time to go deep into it there are a lot of effects that that noise give you so if you look at the ok but in the price they average out ok I mean it's not perfect of course you see that there is noise in this and sometimes figures are not it's always the figures that you show are the figures which are nice but typically it works well there is no cheating here because if this noise had a non zero mean if there was some average in it then it would be exploited so in practice you think that it would be exploited by people so people measure this and that's why we call it noise ok so I wanted to say two things about this one is that actually we postulated this type of model but I'm sorry I'm thinking if I should start with this or if I should start with another claim I will start with another thing and then I'll get back to this so the question that remains is that we can have some type of description phenomenological description of how the impact propagates in time and how to get back the deficit of the market given the long range correlation here the question is we have seen this other fact the impact of meta order so if you have several connected orders that you are doing in the market how do you impact what we have seen there is this square root low which has many of these many interesting things so what would be the prediction of a super simple model like this what you can do is that if you know the propagator of each trade you can integrate it and get the propagator of a meta order so that's what we'll write up now and so for that we have to do some assumptions we have to introduce some meta order type of thing in this model that that you have as we said here only signs of trades counted so there was everything was volume one so what you want to do is you want to have a meta order which is of size n n trades that you want to do horizon t in the next t time steps I want to do n trades with a constant trade so I will I will say that there is the rate of my trades will be I will call it 5 will be definition will be this and it's constant so I don't know over 10,000 trades I want to do 100 so I will do every 100 trades from now on and to see how I am pushing the price why are we looking at it we saw the square root impact we see that there is a nonlinear behavior in this model what does it give so what would you get and of course you assume that there is auto correlation in the system as we had it before you are adding yourself to this but you don't really change the correlation in the system so what would you get as a result assume that you are let's say buying let's say that you buy here it doesn't matter and so what you want to calculate is what will be this price this price change this is the definition thing and so you can do the following so what it will actually be you can see that you can write this up in the following way it should be what are you doing here we made assumption which was hidden you are just summing up your trades in time there is nothing extremely interesting to say and and what we have seen is is it okay we have seen that this g will behave as just to keep this g t oh it's there okay there is no need to really write it up so one can write up what will this thing be do the sum here normally what you should get is the following it will be some type of behavior like this there is a big there is a big t and so if you do this sum you can do it this should be the thing that we get so what do we get there will be one because we are integrating and this is the exponent of g and okay so what does this so what would this mean given the fact that we know here it should be the following so it should probably do something like this if I didn't make an error can do the calculation the final result is good because I hope I didn't make an error here so you get a dependence exactly what we are looking for you check how does the price move if I do n trades if I over a time horizon t what we looked at before in met orders where we are talking about quantities talking about number of trades here is the same because we say all trades are units so the ratio of my trades to the other trades is exactly my ratio of my quantity to the total quantity so this is what we get which will say that there is okay so this is one plus gamma over two so for realistic gamma this is one half phase this would be something like n to the 0.75 or something like this we sum with a phi is what is it 0.25 probably so what does this model tell us is that okay it's not what we are looking for we are looking for square root of n dependence and no dependence on phi ideally what we get is not exactly this but not extremely far so what we see is okay there is a weak dependence in phi but it's 0.25 it's already something that is hard to verify or not so okay it's at odds with what we thought and we find something which is concave right so this is at odds with what we said so we had this idea that the same thing should be square root of n in reality so it's not good we don't get it it's a way higher parameter but still it's concave so we are able to get this concave behavior of impact so we are not orders of magnitude of but it doesn't work and of course the problem is that you could say oh but what gamma would we need to get back the square root of here square root of here but we would need a gamma 0 so a correlation that doesn't decay at all so we cannot massage this type of model to get back what we got here we go to simplified model already the what does the propagator model say it's only the sign of your trade it's only your trading plus one or minus one and we are among those people it's just the number of trades all these things can be done you can move to a world where you do properly quantities it's uglier to write up but it doesn't change life of course you could also say but what if this is not constant this rate ah if you can do yourself yeah but we are not studying so what we wanted here the question was okay you have a decision on the direction if you're doing buy and sell buy and sell you can have some strange effects but it doesn't change the world I mean it's the same type of integration how could you get to a square root nothing really you don't get to square root with this type of model but especially I mean it's not what you don't want to trick the system to go to square root because you know what you're measuring empirically and you want for that type of behavior to get the square root so okay so you get something concase what I want to stress is that sure you didn't get the result that we wanted but it's not completely stupid so the fact that somehow this idea that there is this okay we have a long range correlation and this we have the market which adapts very slowly to what we want to do so we have in physics language to the medium we are moving in somehow adapts to us if this happens slowly so if this is a power lawish behavior you get some what we call anomalous impact so not in a response in any case in the system but not the right thing for the moment so I wanted to just mention one thing and then I think you will stop and you can study for your other stuff if you want to study is that okay we postulated this model we said let's put a propagator in the system let's say that every trait does the same so the effect of each trait is the same but it propagates in time actually one could come up with other models I just want to mention if we won't go into detail another approach to this could be which is maybe a closer to a finance language approach would be is you say okay it's not that every trait does the same impacts in the same manner and there is a transient impact whether you have some permanent effect every trait does something and then it stays there forever but what you do but this effect depends on the history itself so you could write something like this which would be somehow related to what we discussed so some type of surprise in the system what you do now depends on what was the past so what is expected from you so these are called somehow history dependent models impact models and they are somehow based on the surplus of excess so what we assume in this type of models is that let's call this epsilon hat the following thing so my expectation at t minus one of what epsilon will be at t it's my best predictor of the near future and so if you define it like this then this type of HD this is called actually HDIM often history dependent I might use it sometimes like that what it says is that it's very similar is the following you say this alternative type of modeling so what you say is that the change in the system the response in the system in one step will be how much I surprised the system somehow how much was expected how much the system people acting in the system expected this epsilon to be and how I was different from it times something that doesn't depend on anything so what does it mean if there is a prediction in the market and perfect predictability exactly what they predicted happens then this will be zero of course what those epsilon is plus minus one but its predictor need not be plus minus one there is a probability if there is a perfect predictability nothing will move sure it was known to everyone before what will happen and otherwise there is some type of response and so one can check that actually in this type of models prices will be by definition non predictable martingale type of condition check predictable its by definition in this type of models it has to come out and so the type of ok I don't want to very much go into detail of this type of models but what you get out of this is I will write one more thing so I think I don't want to go 5 minutes I don't want to go much into detail so anyway this is also a type of modeling that one could do one thing I want to say about what you can do is write up proper expectations what is the expectation of the you have some type of prediction and then you can what is the expectation of the price move if the prediction was right if the prediction was wrong and of course you want to weight this by the probability of one and the other so this can be written up and you can write up properly what is the condition for how should the price move what is in one case and the other I don't want to sorry go into details I know that I am vague here the main thing that I wanted to say that actually one can prove that if you only have one type of event so not unlike for example in the exercise that I said and if epsilon hat is some special behavior actually these two models match up on the other so what we called propagator will be equal to this type of model if in a given case so not in general so is this not in general it cannot be said but there is one case I just want to mention it because it can come up elsewhere so if this epsilon expectation t-1 epsilon t is a linear combination epsilon in the past then yes believe my claim actually in the slides I think there are some you might also not care about all these details we have too many models you can say anyway why I wanted to mention it it's one obvious way of coming up with a model so let's not imagine a propagator in general it's not the same except if this condition holds and there is a special name for this type of processes they are called discrete regressive processes so if one is interested there is a special type of stochastic processes which assures the two are the same and I think I'll leave you now I mean of course I'm happy to get questions these noises is there any evidence that the Gaussian evidence for the Gaussianity in practice I wouldn't be that I mean the fact that they have average 0 that's okay so in fact if you start to look at higher order moments of the price process of the price changes you start to get a lot of deviations so you have effects so I would say that not I don't have a very explicit answer I would say no but I don't have a good way to yes but what we say is that we can assume this and we can most of these are testable but what we did here is not like for example for the beta we can test it so indeed at this level so we are looking at the first but it can have deviations above