 Vzlušajte. Jih treba deta. Zdaj nekaj vizitavaj prejšelj, ne bo, da je, da ne vizitajte. Vzlušajte, da je bit... Vizitajte, da je tudi odrečen. Zdaj zelo vse modul je inšta. Vzlušajte, da je intrest inšta. Vzlušajte, da je inšta. Vzlušajte, da je inšta. In če sem vzlušaj. in izgledajte, da sem ti se očetil. Zato si ste kako se skupajte. Zato sem vse nekaj, da so včasni vse. Zato, da smo se skupali ještje, če lahko se stavili, da bomo vse preddjeli kripe in da bomo vse moj skupati! Takaj sem izgleda, da se glavimo. In je to, da je vse vse izpronovate, da je neko vse izpronovate. Zelo je to, da je izpronovate. Vse odličen je, da se skupaj vse nekaj, da je vse izpronovate. In tako, da smo zelo, da smo izpronovati, da se zelo, da smo izpronovati. ⏠ tako, da je mobility sem, vi seациj pomekno da grade in namoči in da so na nekaj pravde, kaj sem semuge. Kaj so vteke iz mnega pravda? To je zelo mjelosti, ki in volume, ki se sve ti semuse. Tako je sem vsovalo tukaj da se rečim vseklju, ma neka bi se na način, če je izgleda in blizda. Adi je naša gra, zato to neki del, vzvebt, da je in večor nečisto, več, kaj pa vzve istotneших. In to je ovo, da imamo. To je nekaj... zนakvorilčne zvečnost. So prvi izgleda, ki smo priberati, je ta, da se bodo komand in na skupin rada. V tom, da smo ne več kažujem, pos envelopečenom ekonomi. Draže pa izgleda na stato priysok. More people would be ready to trade there. It's very basic reasoning. If the price is very high, more people want to sell. OK, some basic economic reasoning said that volume should grow with the distance from... I think I put it something like this. I said P0 is now the price. The further you go away, the more volume there should be in both directions. And so there we came up with the idea. OK, so let's try to model something that we might not see. And then think about why we don't see it. But let's say that there can be latent variables in the world. And let's model a latent order book, which is not what we see, but something much deeper. So we had some idea that it should be somehow like this. Very close to the current price, actually. It should match what we do see, because it obviously has to match. But at further away, things can be different. So this was the idea, I just wanted to stress it. It's a simple idea, but what one has to come up with it. So that you say that there are these... OK, so somehow these are the intentions, right? That might not materialize. It might have already materialized. In that case, you see them in the visible book. They might materialize later, or never. I mean, this is sort of long-term intentions of people. And we came up with a simple model for this. So this deposition-like model. We said that, OK, so let's start with something putting zero thinking into it. I mean, thinking for the agents or particles. And so we said, OK, and see if that can help. If it doesn't work, of course, one wants to put complicated life. But we said, OK, let's do a deposition-type of model. We had a lambda and a nu. So a deposition and an evaporation. Evaporation defines a typical lifetime. And by hand, we added something to it, which was diffusivity of prices. Which is somehow the effect of market orders, which it's not very elegant, right? We had something by hand. It breaks a bit the microscopic description. But, OK, and we had some results from this, is that there is a linear profile close to the price. So indeed you find in a simple model something that behaves linear close to the current price. And another thing, as we'll get back to this later, and another, which sort of comes from this, that vanishingly, small volumes best. At the best quotes, so close to the price. Just for the language. Is it clear what I wrote here? So it's not at best vanishingly, but at the best. So, OK, there's two things that you get out from a very simple model, which are sort of non-trivial. And actually, OK, so we have a linear profile. But, of course, one can think about what's the width of this linear profile. And it will be somehow related to the volatility on timescale 1 over nu. So the only timescale we have in the system is 1 over nu, which is the typical lifetime of this, or the memory time of this book. And so the widths of this profile will be on, well, one can write up property equations, it will be the typical price move on this timescale. Now, of course, here we have to think, OK, there is a model, it gives something linear somewhere, and it is not what we see for the volume itself. So you'll have to think, OK, so what, for a model like this to work to match data, it means that the square root impact type of behavior, so some linearity in an invisible volume, is on relatively long timescales, you can measure it on short timescales as well, but also on daily timescales, or several days. So which suggests that, OK, if you want to map this model to a real world, somehow this timescale that you get has to be the timescale of slow players in the market. So people who are there and who really have an idea of what this price is going to do on one day, ten days, something like this, and not people who are actually, you could call them high-frequency traders, newspaper you can read, who are just there very close to the price and placing on a millisecond scale for minor gains. So it gives you an idea of how you can map this type of model to a real world, OK? So this was what we saw yesterday, and, OK, so, well, from the linear profile it comes, but, OK, let's be explicit, it suggests that you have the square root impact, so you have everything that you sort of want because you have a square root impact, you have very low liquidity close to the best price, meaning that there will be, in a practical sense, there will be octocorrelation of signs because if you want to trade, you have to trade in several steps, but what was, OK, but the thing that was surely a bit missing from this is that, OK, but there was this put in by hand, so there was nothing really to lead to it, and so we had to come up with a numerical model, OK? So I wanted to discuss another way to get to this, and, OK, so I didn't discuss now explicitly the fact that you have this linear profile actually depends on the fact that there is this diffusive motion that eats the volume, which is just sitting there, so there was this model, and what we can do is one, is that what we did yesterday, OK, come up with some numerical enrichment of the model and study that, OK? We can help, it's good to have a numerical model. The other thing is to change the model a bit and add some things that make it more consistent and make it analytically tractable. So actually I will show slides, I think, today for a while to be faster. There's a lot of stuff, so this is what we have seen yesterday, OK? And another comment, of course, that, but it's sort of obvious, time skill in this, I won over this new time skill in the book, but anything that happens on longer times will have a linear impact, for example. So all memory is killed after this time. OK, so I wanted to discuss a more consistent model. So these models, by the way, are very recent, so these are from between 2012 and 2018 in the different models that I'm discussing here. OK, so this is what I just said, so this latent order book type of model gives insights, so you get an idea of what you're looking for, but there is this strong assumption of diffusive prices and the rest we just discussed. So I want to show another type of family, which is extremely, so the idea behind it, so these first three points, actually the first almost four points, are the same, so you want to model this volume in the background, but we change a bit the system, and instead of just having a deposition lambda here of something, you try to model two types of particles. So we want to model a sort of reaction diffusion system, so where you have particles, which are to buy and to sell, which if they meet, they react, they kill each other, and we'll try to get to describe similar systems. So this is the actions that we are discussing, so it's very similar, but slightly more complicated. OK, we have a deposition of rain of new intentions, whatever, they are particles falling. With lambda, again, we simplify this lambda, it doesn't depend on distance. OK, by the way, I comment here that of course we discussed a bit of how lambda has to behave for this to hold, so this is a very wide range of behavior. So just you have rain of particles, you modulate it, actually in this case we'll just use a heavy side, meaning that one type of particles fall on one side, the other part of particles fall on the other side. You could have some other type of function, but of course you want to have some increasing function in one direction. You have a cancellation, so this can be some partial or complete with some lifetime exactly as before, but what we add is these two things. So first of all, you add a diffusion of the particles themselves, and you will add a type of drift, which is this guy. So this could be somehow reassessment of prices. So someone is sitting, a particle fell somewhere, then he can change his position or she can change her position afterwards. It's like, OK, you change the idea of where you want to be, and it will have two parts, so there will be some part which is agent-specific, so each of them diffuses, and this will contribute some diffusion coefficient, but also we will add a common component, so this one needs to do for things to work. So we also add some type of, OK, you could think about it as a collective information, but somehow everyone is shifting in the direction, so some process V, and OK, of course if this V, underlying V is a white noise, then I mean you can set this V whatever so to make the final price diffusive. So this is a bit still by hand, but OK, so you have this two, so you have a drift, which is from V, and of course if V is diffusive, then drift is diffusive, and the agent-specific diffusion, and OK, and you have transactions, so what we do not put by hand some market orders in, but we say OK, so there is some rate, we will consider a high rate, but with some probability when they meet they kill each other, they eat each other, OK, so it's traditional reaction diffusion type of model, and that's it, I think that's clear, yeah? Yes, but I will consider that actually, so this is a drift, I mean this is whatever, V, which drifts everyone, but it need not be a constant drift, so I will define Vt to be a white noise, yeah, so it's a noise behavior that pushes everyone to follow it, noise in time, not in space, exactly, it's different in different time steps, so it's not a traditional drift, but it's collective, exactly. So it's OK, it's not a, I mean it's hard to solve these models, or sometimes hard, but it's easy to understand. So OK, I don't want to go very much into all the details, but OK, so you can write up the equation of motion S we did yesterday, it's not very different. What are the differences? You have two equations, OK, it's different. So you have two equations, you will have an equation for B and for A, so this is again the density of the equation of motion for the density. So OK, let's go from here, so here there are things which are the same as before, these middle three terms. So there will be a diffusion term, OK, exactly as yesterday. There will be an evaporation term exactly as yesterday and the deposition term similar to yesterday, so we have a heavy side here, because, OK. And the two things that are new is one, is that you have this well, this drift component, which we just discussed, which will be the same on the two sides. And, OK, and you have this reaction component, which is exactly the same term, of course, in the two sides. So it's a great kappa. There is this annihilation of the two particles. So this is a hard thing that we have. Sorry? And we'll try to eliminate it in practice for decorations, because it's hard to handle reactions. We set the problem. We set the problem. We are writing the model. So we decide what we say is that it's a... OK. What we will in practice says that K is infinite, so it's probability one, if two particles A and B meet, they eat each other. And, actually, if we put a heavy side here, then it can only be at the front of the two... There is only one reaction in front where they can meet. But one could generalize and make life harder, but it won't change the many things. So, and, of course, there is one... Yeah, OK. So, here we have a Pt. In the heavy side, of course, the definition of this price is where the reaction front, where the density of the two type of particles is the same. This is... Well, in this system, yes, it's mid-price, but, actually, in a system like this, there is no reason to have any other price than one. So, see, if there are particles diffusing and they annihilate when they meet, then, OK, for well-behaved... So, unless new is some extremely... It should be only one point where they meet. So, it shouldn't be aligned. So, OK, this is a bit harder to... It's hard to handle this, because we have the reaction, but, of course, we don't care about the behavior of everything that we care is the price, essentially. So, what we will do is simply... We don't want to do look at the dynamics of rho a and rho b. Is it OK that I use slides and I don't write up here most of the things? So, we don't want them separately, but if you actually look, you say what you care about, for some reason, I wrote a phi here instead of rho, but whatever. So, phi, so you look at the difference between these two densities, you get this equation. You can... You don't care about the actual details of the reaction anymore, right? What this means is that, OK, you had... We always thought about the... Just to understand, what we had in mind, it was somehow like this type of volumes. Of course, however we define, we will have something. So, actually, this will redefine somehow the densities on one side, we'll call it positive the other. So, what do we have? Well, we have exactly what we had before here, here and here, but we only need the sign. So, I mean, it's trivial what we do here from the heavy side because we get the sign and we eliminated the reaction term, OK? And we can also make life easier that we change a bit the reference frame, so you want to move in the reference frame of this... This is what we call drift, so VT. So, what we do is simply we redefine a new price which is what we had before. So, which is X minus P hat, where P hat is the integral of overtime of this drift, OK? It's clear. Is it clear? P hat T is the integral of up to now of this V, which is so this, let's say, drift term. What we simply do is, OK, there is everyone shifting like this. We take off this collective, we put ourselves in the preference of this collective motion. To čuze... The boundary condition. But at some given point you're somewhere and then on you... There is a collective motion that you don't really care about. What you do is simply take this out of the system. OK. So, what you get is this. This has been removed. OK, now we are back to a system that actually we have seen sort of a system that we have seen before. So, indeed you get something, which is the same that we had yesterday. Right? So, we didn't have to put in by hand what we had and we got back what we... got back a local linear profile. So, I had in mind something. So, what do you have here? So, that the stationary shape of the book, OK, we have it written this way. So, for y being negative, it will be this and it will be this way the inverse on the other side. And so, what is this? It means that that the profile will be linear close to the best. I don't know if I discuss this and there will be... So, there will be a width of the linear region will be so, width linear, I think, I don't know if I use the same notation but it will be something like this. OK. So, again you have one over... It's the same stuff. One over new is the lifetime. This is the diffusion. So, diffusion is the same. And so, OK. So, this is not a... I mean, it's good to show here. This is a known result that the diffusion with some close to an absorbing boundary has this behavior. So, that is good to show, you know. And so, and so I just want to introduce something that is not on the slides but I think I will use the letter later. So, actually, if you zoom in the linear parts, so you say, OK, you will have some linear behavior here. So, just, right? And if you zoom in, you can... you can define somehow the volume, the transactive volume, midtime. So, so, so. You can define, OK, it will be something like this. Is it fine now? Right? It will be something like this. So, you look at the slope locally times the typical diffusion. So, it's OK. Actually, this you can call some type of current. And, and actually, one can... OK, this will be the slope. Should I write this all up? I won't write this all up. So, what you get is that, that actually, so the slope, if you write a bit pay, pay the thing, you will get a slope which is, which will be this, I call it L, which will have the following behavior. Behavior, it will be somehow like this. OK, it doesn't really matter. It's OK. The orders of magnitude are clear. So, this is the slope close to the mid normally. So, this should be coming from this. What is interesting is actually just to a bit, try to connect to what things that we discussed elsewhere before getting further into model. So, OK, you can define some type of slope here, but of course, if you get back, so for example, in a model like in the Kyle, there was a lambda which we defined. Well, it's somehow so, so I think the inverse of this should be something like the lambda there. Of course, we have a completely different model, but this idea how much, how much the price should react to a given volume, so how much the price should change, it's the same type of behavior. So, in a collective language, if you have all these different particles in this type of model, you have a mapping from what there was a lambda to the inverse of this. But, OK, so this is just to say that you can map a bit dismal so the main variables in one model in the other are similar. Yes. So, what you say is there are these stuff falling, they are diffusing in both directions, so there is no explicit current coming from elsewhere anywhere, so they are falling. So, I think the only thing you care about is that they in the middle they annihilate. I don't think you really care about boundaries. OK. I think so. I don't have to think about, because you could have versions of this type of models that you have two ends of the system and there are currents coming in. So, somehow there is a current, but then they are diffusing. I think you don't have to do what did I want to say. So, OK, we have the linearity in a similar model to before, but slightly different. And so what you can actually do in this type of thing, just to show what how this continues. So, of course, what you care about is now, OK, now we have something that we like. So, yeah. Let's say I wrote something. OK, let's talk about this. Here what you say is that I'm one, why is this negative? So, I'm defining one side. So, I'm on this side, let's say. If y is negative, then this is, of course, a decrease exponentially. So, that's OK. And so, what you have on the other side, what you should be having is, well, you flip y in the parenthesis. It should work, OK? And you flip this type of definition. No? Does it seem OK? Yeah? I think. To me, it seems good the way I wrote it, but I could... For the zero, you mean? Or that it's not larger or that it's equal? In zero, it's zero. Everything is zero. In zero, this is one. OK. So, which is the definition, the way you define the price is the point where the two densities are the same. OK. So, we have, again, we have this background system that behaves as we like, that diffuses, that has a local linear profile. And what we want to do is, OK, add a method. So, how would adding a method go to the system? How would that affect? In the same way as before. Exactly, because in this numerical model, we discussed that, OK, it's great to have some linear profile, but if things are fluctuating a lot, it's not that obvious that you can really measure this. So, numerically it works, but there was no analytical solution. So, what you can do is that, you add an extra current. I think it's written here. Yeah. So, you have an equilibrated market. So, you just decide, and you add an extra current, which arrives at the transaction price. So, it's people really know what they do. They are setting a method on the, arrived at the transaction price, and OK. So, we can model it. What we do is simply, OK, there is a delta function here that it has to fall on the current price. And it has an intensity m. OK. It's with another term. Yeah. I mean it could be, if you want to do calculations. Yeah. It's a statistic. We'll keep it general. Then sometimes you say that you want to have it to be constant in time. Yeah. So, OK, and we simplify life a bit. We say, OK, we are close to the best price. That's where we care about things for the moment. Then we will generalize this. What you say is that, OK, locally there on short times, on short physical scales, evaporation and then deposition locally won't change much. So, what we say is that, OK, let's keep only the first and the last term of this equation. Which is actually a nice thing. So, for which there are solutions. So, this is what's called heat equation. Yeah. Is, OK, sorry. Mt has nothing to do with any price. It's just by chance that I call it m. I should have called it, because it's intensity or something. Oh, no, it's m, because it's met order probably. That's why I did it. So, mt is an intensity of particles falling to the point defined by this Dirac. So, it's nothing. Sorry. So, it's not at all a price. And, OK, so we get this equation to which there are solutions. If you put zero here, so if there was no met order coming, then it's traditional heat equation. So, the temporal derivative and the secondary derivative in space. If there is an intensity, you can look up the solution to this. And, OK, so there is some solution. What do I show? Yeah, OK. So, OK, so one can look up the solution. So, what you get is that, actually I would like to simply get to the last line here. It comes from the other, is that the behavior of the price will be some integral on this intensity. So, this met order falling. Is it clear what I'm assuming? This is the Lax equation. You have this one over L here. We just defined on the blackboard. And, what you're doing is, you have an integral of this intensity. And, OK, and actually if you want to be for small impacts, when you're not impacting too much the price, the stuff which is in the exponential can be forgotten. And, you get a model so simple. So, where are we? We simplified life. So, we say that it's a small impact case and we are on small time. So, the deposition of operation locally do not count too much. Right? So, it's a simplified life. But, what we get back actually is some type of propagator model. Just to connect. So, what you have is that, yeah, so there is a linear effect. You're just integrating. And, you have a one over square root of t behavior here. T minus s behavior here. Which would mean that you have a propagator in the language before. This would mean that g is something like, so t minus s to the minus one-half. And, OK, there is four p and stuff in it. So, you get some propagator model. OK? So, when you're small, these impacts add up linearly, but you have a square root decay of impact. Which actually what we have seen is that, what this means is that, on the short terms that we are looking, so we can say that there is diffusivity due to the collective motion. But on the short terms, so this is what we call beta. In other lectures also, what you say it will be a sub diffusive behavior. So, this we discussed. Is it clear? Are people following me? And, OK, on the longer terms, you will have diffusion. So, OK, you have something which is a bit, it's not super good. We have to check, OK, in practice, on what time scales is there this sub diffusion. Actually, there is a solution to this type of problem, which we won't discuss at all here. But you could say that, OK, if nu is not one number, but it is very much dependent. You can get to a model where orders are not diffusing, really, themselves, so there is a fraction of diffusion, but you get back no sub diffusivity on short time scales. If anyone is interested, I can give you the reference to it. It's an interesting paper. And, of course, all this is valid for if you are doing very slowly, so you are not kicking too much the system. If you are more aggressive, then it is not. It is not why it implies sub diffusivity, because sub diffusivity of the price itself. Why it implies, because, well, we wrote it up for the actual market, this relation between the response, this stuff, so this is a convolution. We have this type of relation. Now, in general, here we don't know what C is, if there has to be C, but what you can say is that if G goes with, so there is the relation with, so if you say that this is, so this was the relation that we had, for the price to be diffusive, if beta is one-half, then gamma has to be zero, actually, in practice. So for no finite value of gamma, can it work? So it will always be sub diffusive on short scales. Is this okay? Yeah. The rate at which, probably yes, but I'm not sure to do, I tell you, and you tell me if it's the same that you think. So it's, yeah, the rate at which you make, so you want to do a matter, the rate at which you are making your particles fall in the center. So, so that's why we say that if your rate is extremely, so if you're extremely aggressive, then things break down, then you cannot, well, then surely this, you cannot use this, but you at least have the exponential term. There is also the question if you can eliminate, so the eliminated deposition and the operation terms. YT is, is, so YT is the is the current price, in the price at the moment, T. So we had this P minus P hat, so we are a bit playing going further, but it should be the price, so this stuff here, this equation is the dynamics of the price. So, but anyway, I want to give the main idea, I think these are the interesting things, but it's exactly the equations you don't have to check today for tomorrow. But, but okay, so check them. So what did we say, so this is what we have for when you're pushing the market slowly, and so this is the behavior in time, and actually I wanted to show, so how does the, of course this means that the book is distorted, so all this is understood somehow in my reasoning, but this is what one sees in a simulation, so of course, okay, so now we flip this back for visibility, but so you have a local linear profile in an equilibrium book, and then you start kicking it and you put an extra meta order, and you have this type of distortion that on one side you will have a higher slope, and not necessarily it starts to become nonlinear, so this is what one expects, but it's nice to see in simulation. Yes, sorry, meta order is pushing, sorry, I didn't define it anywhere. Okay, so the meta order is pushing in this direction. So, I mean, it's logical, you have a system like this and you start eating, because, okay, this part is clear that you're eating into it here in the meantime, of course, there is more deposition coming because you break the symmetry, you're pushing in one direction, and the next thing to discuss about this model is, okay, and then you can look at the actual impact of a meta order on it, so we saw the dynamics of the price, but how does it work, and so I won't go through the equations, but actually one can write out that if you say, okay, let's simplify life for calculability, someone was asking this exactly, so let's say that, okay, you have a constant M0, I mean constant M, and you're executing in this way, so just for simplicity, you can do the calculation, that you will come up with, and you find an impact, which is indeed, in all cases, here there is a problem, some other letter, J, this is a 2, no, actually, so this capital Q here is a 2, forget it. So anyway, you have a square root behavior in quantity in both cases, so this is what you were looking for, I just make the statement, we won't study it, and then what's interesting, that you will have two type of regimes, in one case, when you're very aggressive, so this M0 is much larger than the typical current that we discussed, then it's exactly square root, it's obvious, because it's easy to understand, you have a linear system, and if you're extremely aggressive, you will be, I mean, linear profile, you'll be really, nothing else matters, just that you push the price, it will be exactly square root, because it's for geometric reasons. So there are no fluctuations that play a role in the simultaneously. If you're below, so if you're slower, then well, the behavior is there, so you have a square root behavior, but you have some correction to it, which depends on your intensity. But the weak correction, so we can live with it, it might be there empirically. So that's where I, so just to summarize this stuff, is okay, so we recover the square root, so you recover the square root in this model, so it's more an actual okay, microscopic model, where you can do, where you can solve analytically, you don't need to do simulations, and there are a big couple of things in a very rich type of model that you can study, that I don't study here, is how, what does the price do after, so how does the things relax after you start to stop trading. There was all this question of absence of price manipulation that I discussed several times, but only tangentially, is that of course in an actual system you don't want to be able to pump out energy because of the way you push the price itself, so this would be price manipulation, which is illegal as well, but if it works like this, everyone would do this, so you can prove that there is absence of price manipulation, so what does it mean, let's say you buy a lot, and then you sell the same amount, you cannot gain on this consistently, and then you did it. You did it? Not consistently. And there is a last question, of course, which is all hidden here, is that we discuss again getting back to this, so what do we discuss, it's these intentions, so what we call latent volume, of course if you want to, there would be one step from this is get okay, what's the relation between latent volume that you do not really see, but affects the market, and the revealed volume. Latent volume is all the things that people want to do, but they don't do it in the given moment, they react a bit later, so this volume will be here, but only if the price comes closer that it will realize, but so what is the mapping from one to the other? It's an important question, and we won't discuss it here. Of course, if you have any questions, we answered that, and I wanted to summarize a bit. So I remove everything. So I'll try to write up what I have in mind about the relation of all these things. By the way, does it make sense where we went from and where we wanted to go, where we went from, where we got to? It's not an exam question. Is it? You don't have to come out here and do it, because you guys did you have the time to think about this? Is there someone who says yes? You don't have to say yes, it's really... No. Someone said explicitly no, but I think nothing is no. OK, so... OK, so I'll write it up, then you tell me if it's... OK, what did we do? We spent a lot of time, so we had all these empirical results, we spent, I don't know... There were several cases of empirical, but we spent the first two lectures on these. One has to understand what are the empirical things, and so what was it? We said that there is diffusivity. We said it many times, and probably for us this was z t, which is so, unpredictability. Unpred. This is how we think about it. And there were other things, which are important to know, but we didn't follow up too much, so there was... It's extremely important for any model in these fat tails, and we discussed the question of... OK, that... Even on relatively long time scales, the central limit theorem doesn't hold, because you have correlations in the system. Right? And what were these correlations? The size of moves. So there was this what we call the volatility clustering, meaning that volatile periods are... For sure, sometimes you have a volatile period, then you have a non-volatile, so it's correlated in time. These are clear. I mean, we can also go back to the figures if anyone has doubts about these things, but if you want to go back to the figures, you tell me. So there was this type of behavior. We had two types of figures, but this... So what is volatility? It's the size of the moves. Let's forget the sign, typical size, as we use it in English, in everyday language as well. And so that... OK, here you have a 100-year period. There are visibly periods when things are much more volatile and visibly periods when things are much more calm. This would be the case for a random walk. So this is what we call... So volatility is clustered in time. And we saw here that actually... OK, that's on 100 years. This is on 5 minutes, but you have... OK, it's less... Visually less obvious, but you have the same. You have a period here when it's very calm, a period when it's very... So on all scales you seem to have this. You can of course define, this is the autocorrelation of the volatility, which will be a very slow decay correlation. Visually. OK, we'll get to this. OK, we'll get to this in a second. So I just wanted to... So there were these. We discussed something, which sort of seemed... It was a bit side question, just I think it is. So the question of the correlation matrices. It was the only time when we discussed the non-single product behavior. I think it was OK in its own to understand, but... And OK, so essentially there were these things, and I guess the... OK, my reasoning is the following. OK, so there was this main question of... OK, so why are prices really unpredictable? Why are prices diffusive? And so we tried to answer this. So OK, why diffusive? Why prices are diffusive? And so this is where we go to a type of modeling which you hated, which is this type of economic type of models, so say. OK, there is this traditional picture which is fundamental efficiency. So what was this? It was this type of strange model. It's not super scientific, but that's the canonic way of looking at it, is that prices contain information. In a simplified way contain all info. And it's somehow this underlying... So there is an underlying process of information or fundamental price and so it says that OK, there is an underlying process which is unpredictable which some people know. So there are some chosen people who know this and they follow it. OK? The claim is OK. There are critiques of this already in the economics literature so... Well, not very scientific. I don't even... It's OK if I say, you know. I don't want to write too much. That it's not very scientific. There was this reasoning of OK, so let's go to some quantitative ideas. So there was this type of... What he said is, OK, what does efficiency mean? It might mean that the price and this really underlying true value are not more than 100% away. So it's not extremely scientific and we don't have it here. Maybe I will decide it. You can come up on what time scale should prices really follow this and you see that it doesn't work. And there were also... In the economics language there were these puzzles. OK, we actually discussed two of these puzzles. So one was... Because in economics you call these puzzles that one was excess volatility. So if... If prices can only move when there is this underlying... When there is a real news coming in, then why are prices so volatile? So real news comes much less often. Why do prices move so much? In the same language, why do people trade so much? If it's very rare to have real news coming. And something that we didn't discuss but I want to mention that of course we saw that prices behave in a diffusive manner on the time scales that we cared about. But of course on long times we also saw this in the beginning that there are trends in prices. You can see that that on several years typically are going up. But typically they are going up. This doesn't matter. So for diffusivity the fluctuations are much larger than the typical trend. So it's only on a decade or more that you can see this. But there is something that sort of contradicts this idea that it's only when there is a news. If it's only real news that govern the prices this type of behavior shouldn't be there. Also... There are other things which you can use to critique this. But... And so what we said is that we said one type of solution. So one solution at least to why people are trading so much. And this was these... So what we call... Models incorporate information. Ok, so there is this Kyle and lost a milligram type of model. So we can give intuition, but... Is it readable? But so one solution to at least why people are trading so much is that there are noise traders who trade so much. Ok, we found the culprit for one part of the thing. It's strange. We discussed the critiques of all these type of models. Why should be people either noise or informed. Ok, so that's one type of solution in a more economic way. But we went into another direction. Apart from these puzzles that were here we showed another I think quite explicit contradiction which was the... What we call the long memory. So this is what Gabya was asking. Right? So here we... Models incorporate. So it's not just a puzzle there is an explicit contradiction. If the decision of people are made on much longer time scales if people have a big decision but execute it on several days which causes this then if you cannot have all information in the prices. Ok? On micro scale. Yes, so what I say here is that what does it mean that all information are contained in the prices? Well, it's not a super clear claim. But what is sure that if, what you want to do now is buy 10,000 of something. But what you can every day you are able to buy only 30 because there is no one else to sell more than 30 days you will be buying. There is no way that the price incorporates the information 10,000 in the beginning. You could say that on some longer time scales it could be the case but on minute, hour, day time scale for sure not. It cannot be the case. They are different. I'll get to this at the very end. Not really. So when efficiency is different type of definitions of what there is but no, so statistical efficiency so I will get to. Statistically you measure this it will be always the case but it doesn't say the underlying reason. So here we had a clear competition and there, there so in my mind the direction where we went to is there was questions of orders of magnitude here. Why it has to be the case that you cannot have all information. What we did is another way that alternative view to this is the question of impact or the idea of impact which is a completely different idea so it's an alternative view I would say. It's much more physics approach if you want. So what we said is that it's not some of this underlying fundamental value that counts and that people are following but it's actually order flow in the system and regardless of information so this type of modeling says that if there is something you break the balance of supply and demand that you change the balance that pushes the price and it doesn't matter if someone is informed or not you don't have to have a model of noise traders and inform traders and this type of modeling but it also means what the others do what you want to predict is more what the others do than some fundamental idea I don't try this here but it sort of comes and it has a lot of further suggestions it means that things will be more and X so the dynamics will be and the genius if it's the actions of each person that each trader that drives the prices then you expect the system to be and the genius and or mostly, not all has to be and the genius and we will get some other points here and so what this suggests here is that order flow moves prices we didn't say why do we have diffusivity this is where it says this is just a statistical effect so this is what is called more statistical efficiency in economics that it's actually people finding inefficences and trading to exploit these which remove the inefficences you find that you can buy very cheap you will go and buy very cheap and you will be pushing the price up and after a while you cannot buy cheap anymore and that it's continuously people are continuously looking for this so it's a statistical effect it's not some effect like this and it's also important to say what were the facts that we have seen so this is the idea behind it what were the empirical facts that we have seen it was for this all empirically what we discussed is single trade so impact of single trades and trying to model how this can lead still to efficient prices so all these propagator models so this phenomenological models and we saw that there is an effect of conditioning ask me if there are things in the channel I am talking about things that we discussed so I am just saying keywords and then there was the other points the meta orders looking at meta orders it's not the conditioning that counts but it's real true response in the system and there was this square root impact so this is what we did so I think sort of the main claims here that I will get back in a second but this picture against this picture so to have two alternative ways of discussing and then ok so from here we discussed these two types of models so one was let's say phenomenological I don't know what's the good name but propagator and company and there were more microscopic for agent based for another community this agent based models which we discussed yesterday so this was the sort of the map for me I don't know if it makes sense now so I mean if there are logical connections for you yes well it's essentially because of this point that if you say that so what the prices the way prices move is that there is some fundamental news coming and all news is incorporated in the price that's what leads it immediate immediate or on short scales of course of course you can define this does the prices contain information and follow some fundamental value on the scale of 70 years but that's not really I mean it doesn't mean much sure you can always increase the time scale and say this is true on the history of human being but yeah so what it says is that there is this underlying process which is the fundamental value it follows something but that's what's unpredictable it only changes when there is a news coming but so these are weekly scientific claims that's why you have problems with it because these claims are like that that's why we are trying to do another approach so it's completely okay but you would be pushing the price down when you're selling so it's a question not related to this so you want to go and make money now this is what I call price manipulation okay so side remark to answer this what you would expect if some picture let's say you're pushing the price up when you're buying this should be a square root you stop here and when you say okay great I pushed up the price let's sell it now actually you can calculate the average of this somewhere here what would happen is of course sure it would be great to sell here but now you want to sell the same stuff so what will happen is this what you will do to the price in a simple picture this will be the final price in practice probably you would push the price even more in a model like this latent or the book because you it's not balanced in this moment but okay let's forget this it should be somehow the average it would be selling here so you're losing money and this is what you expect from a model like this if your impact were linear you did this at least you would be paying and it would be the same but okay so this is somehow the map that that I had in mind here so think about it a bit if it makes sense and so it's okay what is the final claim that we have two views one is this traditional picture one is a more physics like picture that it's an endogenous process and what the result is that you have actually two sort of self-consistent ideas one is the economics idea the one is the more that it's impact the price prices and the statistical efficiency comes from people using the arbitrage opportunity so you have these two things of course just understand so this means that there is an underlying price coming from somewhere else it's this fundamental information so actually in the language level here you would say that it's price discovery that happens exists but people have to discover it and so that's the way people talk in economics and here you would say that it's more like price formation this is only a question of language but it shades lights on what we are saying and which means that okay prices would exist without trading and prices only exist because you are trading and we didn't give a final answer we sort of showed two views on this and okay what I think is that it's easier, it's much more plausible to have a type of model like this which also matters, it's not just what your predictions do how they work but also it's more plausible but okay this is a one may argue against it and also we have so there are all these different puzzles that one can come out so empirical facts we seem to explain much more empirical facts in this picture than in this picture but okay it's an open discussion and I'm saying this but most of them would believe in this picture and what else did I want to say well okay so things just some keywords that if you come from physics can be important so as you said this is an endogenous process mostly which means that there will be of course feedbacks between the different actors and you can easily have for example model bubbles in a system like this if people actions change the price and people of course can look at each other what the others are doing and there can be hurting effects and feedback loops what we have also seen actually which is interesting but it comes from the model here is that well in these models to find the diffusivity it means that actually you are in a very strange point so you have some state variable that you can look at you seem to be what we call around a critical point for example in this latent order model we have seen that that to have diffusive prices with this autocorrelated order flow you have that there has to be a parameter we can define which parameter exactly if it's the decay of the impact or what which is at some critical value which means that of course you expect the system to be super fragile that the small changes the behavior of the system can change a lot so it is interesting for analyzing I think and it is also important for regulation so it means that understanding more this type of fragility in the system and these endogenous loops and feedback loops can help you devise better rules better fee system maybe so you want to avoid people trading too much without much information there are fees you can trade with me you can have to pay 50 cents so you will only do if you think that you can gain more than 50 cents so there is a strong effect on regulation of all this and so I would say that's it general conclusion from this is like the way I started is that if you have a system in which there are many things that you do not understand probably it's better not making big theories like this and try some simple microscopic models so all the microscopic models we had did not contain some specific intelligence we said what can we explain without intelligence of course you can look at what didn't I explain and think better things but it is a trivial claim but good to keep in mind that if you don't understand things then try to not come up with great big theories so that's what I wanted to say ok if you have 15 minutes I mean if you have questions then try to not come up with them so first about this but if not general then I wanted to say a few points but everyone start this so if you do a meta order or if you do a one shot it's very different it's very different from impact formula we cannot we cannot have a very clear relation between the two so there was this type of picture that we had so the volume is ok we said this later is a latent volume and there is an actual somehow visible volume so what does latent volume mean that there are these people and they might realize if you do not realize to realize themselves it's to come to the market so exactly so if you want to trade a quantity of this much ok say this much so the integral here if you do it in one shot you will push the price here ok you take the volume it's a trivial case you take the volume which is there no one has time to react while in this to get the same amount of volume if you're integrating on the blue you would be I don't know doing this so what you think is that if you do trading slowly you will impact less because people have the time to come there and provide liquidity to you there is no one-to-one correspondence but actually you would in practice never do the white solution you wanted something no? ok so I wanted to say two things one is a very general thing that I don't know how to do it but if you have feedback on any of this or on the lecture it would be good to have them you know my email you can send it I don't know there is no prescribed way doing it here but if you feel I would be happy to get any feedback now or after the exam of course if you like it doesn't matter of course but whenever really and I just wanted to list a few things that we didn't discuss which could be interesting for people is the effect of if exactly so it is this type of that if there is a there is a good word for this which doesn't come to my mind if there is an arbitrage opportunity so if there is an opportunity to gain make a gain in the market by the fact that your trading will move the prices exactly in the direction against your against the arbitrage opportunity so on some scale it will be eliminated of course it doesn't says that it will be exactly diffusive but still it gives a framework why these do not exist for long times and what you say is that so there are all these people in the markets looking for opportunities they can be machines or people so the simple opportunities will be gone and then don't misunderstand me diffusivity here doesn't mean that locally there are no predictable patterns on higher I mean with better statistical measures there are of course they are very hard to exactly and for example for that if you measure so it's a bit tautology because if you know that diffusivity is here and know that long memory is here there somewhere has to be something which counterbalances this long memory and in practice for example you can measure indeed the reacts also so the way the price and also others because the price reacts the way others react so the way the price react to a trade which is very predictable is very different from when it was unpredictable so if you have a strong predictability and the prediction realizes the response in the system is very different from when it doesn't realize it's sort of tautology it has to be the case but you can measure this and you can so the answer is empirical level it's simply this and why it is the case because people know in this case people know that that your actions will affect the market so they have an expectation of what will happen and they will change their behavior if their expectation doesn't realize or they can in practice in these two tendencies sorry, what do you mean by the two tendencies? by loops I mean it's that there is feedback in the system I'm not sure I get the question no, they do not drag you but understanding what's going on can help you come up with good regulation so if I am probably I want a very good example but you know that the market is very fragile so typically the response of the system is in a way but sometimes if there is a small fluctuation it can be completely different and there can be huge jumps in the prices you can come up with an idea in terms of regulation so of course these are particles falling in the model these are people we can tell them if you fell down you have to stay there for 10 minutes or some regulation rules to make the system more stable so it's not auto regulating this research can help you come up with ideas how to regulate the system because then we are back to the original question and of course we can model it in a reaction diffusion model but it's real money that can be lost for certain people and real economy so there is a lot of stuff that we didn't something that I wanted to discuss we didn't discuss at all so of course you can go in the same type of picture further you have an idea of how the price behaves given that you are trading and now let's say that you really have one day to buy a given quantity you can come up with optimizations there is a field which is called optimal execution how should you be trading if you want to minimize and you yourself are impacting the price so you want to put your trades the further apart so that the system forgets you maybe but at the same time you want to be able to sell or buy all you want it so there is a final condition and there are difficult optimization problems that one can come up I thought I would show some things on this but ok, you can live it out for the limit order book there are several queuing models in different orders until when do you want to stay in a queue when do you want to get out because it's too long and you want to do another action stuff like this and then there are some other completely different questions that we didn't discuss I just wanted to mention which I think for physicists can be mass physics can be interesting so there is all a field of derivative pricing that you can consistently do in the market so if if if you have three products A, B and C and then there is some linear combination between them so A plus B gives C or so in a very simple if there is any linear combination of products that gives an other product that gives you a condition on the fact that the price a condition on the pricing that you cannot have the basket same as another product must have the same price so there are some underlying interesting things we didn't discuss about risk measures at all so we discussed only volatility as some type of variance of price sometimes of measure of risk but of course if you have fat tails you can do much better we didn't discuss again so I mentioned here but all these herding models we didn't discuss all and there is a big literature we didn't discuss non-linear correlations actually there was a question once we discussed correlations but only at the linear level so you could, if you have fat tails you could do better to understand interdependences there is a very big many people working on getting more to economics or network systems a system in which well some network interactions and looking at face transitions on these systems etc what else did I write ok, one thing that we just mentioned but we didn't go into the discussion at all so at the correlations we mentioned the portfolio optimization problem it was in a tutorial which ok, it can be quite well mapped to spin glass problems so it's very much physics like approaches and so this is more the finance questions and then there is a lot of things more in economics so studying the distribution of wealth in a society and how you can get what you see empirically and getting to more these macroeconomic models so to model there are many physicists working on there is this general equilibrium models in economics which say that there is an equilibrium with very mild fluctuations around it try to make these models better and via this we have predictions on inflation and monetary policies and it's not finance on the small scales but economics on large scales there is also well, many things to be done so it's a bit similar approaches are a bit like this and the world is very different so ok, I just wanted to list these things if someone is interested to go into other directions yes, on what level do you ask because this question can be like who to work with or to so you're talking about economic research or moving into finance it's which part of it ok, so well, what to study I don't have a good answer to this because I just listed these things there are enormous amount of different things that can be studied so more at the end of macroeconomic systems, which actually I do not know much about which is I think it can be super interesting so to understand how an entire economy behaves and how to model it and as I said there is the other end on the micro scale and in between there are many things and my interest is more on the micro scales I'm more interested in these impact models but ok, that's a question of taste so to the question of what to work on I don't really have good suggestions I can say some groups where I think I don't know if that interests you ok so ok, there are two sides you can work in academia on more like what we call econophysics even if sometimes it's not econo but more the finance and you can go to work in a company doing finance if you work in a company doing finance I guess it's better having at least ok, the place I work at you should have a PhD often it's even good that if you have a postdoc in whatever physics and I think there are two approaches one can go to work in finance in a bank which might be much more mathematical than physical more math than physics it's more mathematical finance and ok, I have a biased view but my view of it is less of a creative work more there are existing models that have to be followed and then you can do calculations on those there are existing models because they are strongly regulated so in a bank you cannot come up with a completely you can come up with a completely new risk model but they don't really want you to because the regulation says that this type of model should be used and so they prefer that you use that while in a hedge fund for example what I work in it's much less regulated and it's much more free so if you come up with new ideas in any type of any type of new ideas if they work and if they make sense then they are good and usable so this is completely to quite different words for me but in any case, probably having a PhD is useful and ok, I don't have a very general remarks on this but ok, I don't know one thing is that if you want to go into so there is the other direction is that you stay in academia but work on these questions most of the things that we discuss here you can do perfectly research in an academic world that's one thing so very good if you move to finance it's hard to get back so that's also why I think after a PhD maybe to even do a postdoc so that unless you're completely sure that this is your direction because it's easy to move in one direction but harder to move back ok, so thank you very much for these two weeks