Okay, welcome. One question for you. I'm sending the slides. In the end of the day, you get those, right? Because I just have this super long list of emails. I'm sending it. So if you have questions about that, of course, it's not complete. But main things are in it for now. So, okay, so as usual, I think it's good to have a very quick recap of what we discussed yesterday or up to now. And then get to a bit continuing correlations, and then get to modeling. People are still... So, okay, so what did we discuss yesterday? I am writing this recap because, I mean, what is there to keep out of what we discussed? So there were the questions of, okay, we were finishing up a bit stylized facts. So what did we see? We saw that there are correlations in the... Well, let's write it explicitly. So we discussed this, the question of volatility, clustering, or in another way to say, oh, the correlation of volatility in time and actually also of activity, which we discussed for its own right, and we discussed to see the result that, well, okay, we saw the factors of price change distributions on a scale of a few minutes to one day, but we saw that even on much longer time scales, we do not have Gaussian distributions due to the fact that's why the central limit theorem doesn't work, that there are correlations in the size of steps in our time series. So non-Gaussianity up to long scales. There was actually actually something I didn't discuss yesterday and just in the slides, if someone's interested, and it's just to know that actually what we discussed here is typically for quite liquid markets. So we discussed it was S&P 500 index and related things. There is actually one slide, so you can look at markets which are extremely non-liquid. Very few people trade or have very strange structure where you can have this idea of fat tails can go to the extreme, so they can be even more fat without actually a second moment defined. But if you want to look at it, you can. So what else did we discuss? I think that was the main things that we had yesterday. Was there anything else? I mean, before the correlations. I don't remember what was the day before. Okay, I think these were the main talk. There was, okay, we discussed this leverage effect, but okay, it's good to know. And then we discussed, okay, correlations. It's okay what they are, a bit why they are important, but that's also for you to think about a bit. And so what we discussed is, okay, empirical questions. So just to see how this matrix looks like, what can we do with it in a trivial manner to clustering, try to get some information. We showed essentially just looking at the matrix. There are several statistical methods to get information out of a correlation matrix if one is really, I mean, wants to understand the structure, the underlying structure. We discussed the idea of principal component analysis very quickly and applying it to the correlation matrix to look at eigenvalues and eigenvectors. And okay, so from this we got to discussing, okay, so try to have some model of what part of this correlation, so the spectrum of this correlation seems to be very noisy. What is information in it? And we discussed, well, this very brief introduction to random matrix theory, which is just to know, I mean, if there was this Marchenko-Pastore formula, which is good to know, I think it's very interesting. We just stated it. And we discussed, okay, so what is the information? What's the part of the spectrum which is information? I don't try it because I think you understood. Okay? So it's, I mean, I hope it was clear or if it wasn't clear you asked. And so there was one more thing that I wanted to discuss about correlations that I mentioned in the beginning. So what I, so when I introduced all the things, I said, okay, so the correlation is something like this. So between products i and j, so i and j are products. Sometimes I'm not clear on this, but let's be explicit. So there was this delta t index, which actually yesterday we didn't really use. Actually I didn't really define yesterday what is delta t. Right? I didn't, so delta t is where you are calculating. So you can say that our i delta t is, if you are in log price, let's say it's the price at t. It's the price at, so it's the size of the window on which you calculate your returns. But we didn't discuss it. Actually it was a correlation I showed. It was sort of daily correlations. So I want to very quickly go into this. It's, from a theoretical point of view, it's not a very hard discussion. At least the way I will discuss it, but it's good to know. So actually this time scale has an importance. And what it means that actually the cij delta t depends on delta t. I mean I show a figure, maybe that's easier. So it's growing this set of slides. Get there. So it's a figure that I made a long time ago. So what we see here, for some reason it's a row. What is a c here? It doesn't really matter. So it's a correlation between two companies. Here it's Coca-Cola and Pepsi just for easy understanding. And so this effect, which is actually called Epps effect, after a guy who is called Epps, Thomas Epps, he's alive. So the effect says that for high resolution data, so when this delta t here is very low, you get correlations that are significantly lower than the asymptotic values of the correlations for some high level of delta t. So you measure your returns. You can measure it on different scales. You can do the cross correlation matrix or just look at one product. And as you increase delta t, your correlation increases. Is it OK? The claim is simple. Yeah. There was an error. It's different. So log A over B is log A minus log B. Actually, afterwards at the moment I had this in mind that I don't remember higher of it. So yeah. So I use log here, of course, one could. So this is log minus log. One could stay in non-log prices. It will be a bit vague on this. It doesn't really matter. So sorry for that error. The y-axis. Yeah. So what I call rho is what I call C there. So I could rename everything here, but then I would get mixed up and I'm writing from my notes. What this simply means is that, OK, you can measure your correlations on your returns on different time scales. Let's see the first point here. Actually, I know it's at two-minute scales. So this is not extremely recent data. The initial values of frequencies might change in time, of course, as I will show. So if you measure two-minute returns and you measure the cross-correlation of this, you get a number of zero-seven, let's say. And as you increase your window to four minutes, you measure this. OK, so this is up to nine. This is essentially a daily level. You see that it goes to an asymptotic level, which is zero-three-ish in this case for these two companies. OK. So this is an effect. Do I have to write it up, or it's... If I increase more, it will stay constant. Yes, so I don't show, but you can guess from this that actually there is an asymptotic value, which is sort of the daily correlations, which is very often, if you look at correlations, it's the daily correlation that you look at. So the question, of course, what is the cause of this? What is the importance, of course? Well, we might want to understand it in general. Of course, there is also the importance that the bigger your windows are, the less number of windows you can have, OK, trivially. So, of course, you want to have some type of trade between good statistics and good measurement of the asymptotic value. Typically, I would guess that you're much more interested in the asymptotic value of this correlation because many of your decisions are more on the daily scale. But, OK, it's not obvious on what scale. You might care about the end-to-end curve, but typically people would care about the asymptotic value. So it's OK. I think the claim is clear. And so the question is what causes this? That's what I want to discuss a bit. And so, OK, one guess that you can have, or I give you and you can think of this. So one guess is somehow the question of asynchronousity, which I will discuss in a second. So by asynchronousity, what I mean is that, OK, so what are we doing? We are looking in these windows at returns. But, of course, price might not change at all in this window if the window is very small. And then you will have a lot of zeros in this average here. So, OK, you can say that, yeah, sure, the correlation will go down because you're adding an enormous amount of zeros into your sum in practice. And which indeed is, you expect is the case. I mean, the shorter the window, the more probably that nothing happened there. So the usual, the original approach to this, and actually there are several results of, OK, how can you measure correlations in asynchronous time series? But actually, if you look at it, it's not really the case. I mean, it's not the main case in itself. So I show here two figures. One is actually the right-hand side that you should look at, which is, OK, it's a measure of somehow a measure of asynchronousity. It is the average inter-trade time changing throughout the years. So, of course, what you would think is that, OK, first of all, what do we see here is that, OK, it's an old data set from 93 to 2003, and you see that average inter-trade time in the beginning was like order of one minute to few minutes. On this, OK, I didn't say. So these are quite liquid US stocks. Coca-Cola and Pepsi is there, but, OK, I don't know the names of others. It's a set of liquid stocks. So from a few minutes, it went down already to 2003 to, let's say, 10 seconds. And today, for the same product, I guess it's more on the one-second scale. So, OK, this is somehow a measure of asynchronousity. If there are much more trades, then you expect to have higher costs or less probability of nothing happening in a window, OK, to say it in a trivial way. But if you look at the left, you see the same curves for four different years. So 93, 97, 2000, and 2003. I just scaled them to one for simplicity because correlations can vary in time. So what you see is that actually the effects, so there is quite good collapse of the curves. The effects of somehow the characteristic time of this effect didn't change, while there was a factor of 10 change in asynchronousity of the typical frequency of trades which should change the asynchronousity, the characteristic time of the effect didn't change. So I think that this is not in itself, it's not, I would say it's not enough. And it's not just this. So we need another explanation. Actually, the explanation that I will give is extremely simple. I mean, from the math point of view, but it has some strong implications. So actually, OK, what you can think of and what sort of trivial is you can also, you can of course say that you want to write up the, you can write up the return in a window delta t. So that's what we had before. Of course, you can write up, it seems there's a sum of the returns in the smaller windows, not very deep. So you can define some, OK, let's define a delta t zero timescale, which is somehow your shortest timescale. Let's say that you have it here in our case, actually it will be two minutes, so the left-most point. And so, OK, so you can write it up like this. Hopefully, I didn't do a mistake in the next. OK, and this is what you do. OK, I have on this delta t scale, I have the price change. It will be the sum of price changes on smaller scales summed up. And so from this one can write up. OK, so if this is the case, actually you can write up that d above. So what is the correlation? That this correlation actually, I'm messing up when it's delta t is up and down. Sorry, this is for anyone. You can write up the correlation the following way. And so this x should probably go from something like this. So what I say here, is it readable? It's readable. OK, the first line is clear. Or I know it's not clear. So what we, OK.  We need to have access. So it's an exercise to check at home. It's a simple stuff. One can write it up. But it's good to write it up once. Actually, maybe, I don't know if I didn't make it. Anyway, so what you can do is write up, what is the meaning of this correlation? Is that the correlation on a scale delta t? So OK, so what is this guy here? This is somehow correlations on. OK, here it's only visible. So this will be somehow correlations of ij on some other scale, but lacked correlations, right? So we have delta t0 scale between i and j. But the time at which we measure the return is not the same. So OK, so this is an exercise. So is it from this? From this here? It might not even be an exercise. It's like, OK. Anyway, so the equation is exercising the sense that do it. I think it's useful. I mean, I won't come and check your results. OK, so the main message from this is that correlation on a scale delta t, relative long scale delta t, is some weighted sum, or it could be weighted integral if you're going to use time, of lagged correlations on finer scales. Right? This is a t. This is a t. Sorry, it's a tau. It's a tau which we never defined here. So it's a non-zero. What's in the stomach of this and this is not the same. It's a lagged correlation, right? It will be x delta t0, just since you're summing on x, I cannot put just one term. It's a sum of, is it clear what I'm saying? So what I'm saying is that this thing here will be some type of sum of tau, and here there is some f of tau, which is a simple f. OK, so what it should explain is simply, if you're in this picture here, the finer delta t becomes, so the smaller your windows in which you're measuring your price change, the more probable that nothing happened. The more probable it is that nothing happened, right? So if you look every millisecond and very often the price will be exactly the same, and if you look every day, it's very probable that something did change in the meantime. And since you're looking at cross correlations, you look at, OK, in one window, so you have the same window of some size of stock i and stock j, and it is a non-zero term that comes into your correlation if both of them changed. So the more often, so essentially the finer the scale it is, the more probable it is that at least one of them changed, so in your actual correlation. So this, of course, will be empirically the sum of r i, sum of this times this, so 1 over t. So you will be adding a lot of zeros into your sum, which will, OK, automatically decrease the sum, so which is the correlation. I don't know if it's true. So this would be the effect of a synchronicity. Why it is not really the explanation that, OK, this should mean that the effect of the typical frequency of market changing should behave as if you changed your window size, so the probability of nothing happening there should change. And actually, what this means here is that on the right hand side, what we see is the average time between two trades, which went down, let's say, by a factor of 10 in this period that we study. So the probability that nothing happens in a window should have gone down roughly by a factor of 10, which would suggest that the effect, so this type of curve, the increase of this correlation, so it has some characteristic time on what time it gets to its asymptotic value or close to it, should change. And what we see is that it actually did not change. It's relatively stable. Is it OK? So OK, so this claim here is relatively trivial, of course, on a longer time scale. So on a bigger window, if you measure correlations, it will be somehow the sum of correlations of finer scales. So mathematically, it's not a big claim. But actually, how does this look like? OK, so the question is, OK, how do these things look like? Actually, I have the curve. So indeed, so OK, what do we show here? Well, OK, what I call F here will be these correlations. It will be these correlations of the finer scale scaled in a way. OK, let's forget. So these are essentially the lagged correlations on a scale delta t0, OK? Is it OK what I'm saying? So now what we say is, OK, let's move into this point. This is the finest scale that we have. Let's look there, but not just the equal time correlation, which, OK, now I scaled it up to 1. But the value of this point should be the value of this point. Just it's rescaled in practice. That's why it's called F. Anyway, so what we see is, OK, their correlation time 0 is understood. But there is some finite time decay of this correlation, right? Which is OK. Here we have its value, the negative lag correlation between A and B. Yeah, yeah. So if tau is positive, so I mean, in this definition, if tau is the difference between this and this. If it's positive, then j is leading i. If it's negative, it's i leading j, right? It's inverting the time. It's inverting the pair, the two, i and j. Is it clear? This graph here is this thing here up to normalization. It goes to 1, but yeah. So what I just simply want to say is that indeed at non-zero tau, you have some signal here. At least these, let's say, three points here seem to be non-zero. And then, of course, for larger lags, you don't measure any more correlation. So you can use this. OK, this is only for one given year. So I don't look at the dynamics of this quantity. But actually, what you can do is, of course, use this measure. So this F, what is going to be built in here, and recompose the correlation at longer times. Because you can say, OK, let's measure all the lag correlations on a fine scale and extrapolate to longer scales. And that's actually what we do here. So this is what we get. So the blue points should be the same in theory. They're the same that were on the first slide. I didn't check them, but I did them. And OK, so it seems to work roughly. So the measured correlations are what we have seen before. And so the computed, well, it's analytically here. It's not really analytically. What you do is measure things at the fine scale, delta t0, and do this sum up for all points and extrapolate to all other time scales. It seems to work OK. It seems to explain what we are looking for. It's OK. I think it works. But I think the message of this is OK. This is the effect. So it's not simply an asynchronously, but there are these lag correlations. But actually, one message that I want to convey with this is that the fact that the characteristic time of this guy didn't change. The fact that these curves here for different years seem to behave in the same way means that this seems to behave in the same way for different periods, which means that, OK, even if you change the frequency a lot, somehow this typical time scale of left correlations do not change, which means that there is some type of, OK, there is or claim. OK, here it's not an exact claim. So or claim about this that the scale of these do not change, that there is something else going on than just the simple typical frequency of markets. I want to complete. So there is something else going on. Namely, what we call is we think that there is some type of human time scale in the system. So the fact that even if the market got 10 times faster, somehow these left correlations didn't really change a lot. It means that this typical few-minute scale here has some physical meaning. Well, our interpretation of this physical meaning is that it's somehow the time scale on which people react to news or other people acting somewhere. So somehow even if things became much faster, there is some type of time scale that remains there even if there are computers trading, which might be coded in many programs. Maybe there are programs can be updated every few minutes, which can keep a constant time scale in the system. Yes? And then you have to tell me if it's clear or not. Yes? Yes. So what I do here is delta t0 is my shortest time scale. So everything is multiple of this. So yes. Yeah, but you say that if I go to extremely, if delta t goes to zero, you expect everything to go to zero. Yeah, but it's hard. Well, delta t, obviously delta t, I don't go to zero. But actually if you can see that the first point here is at two minutes. So the shortest window that you consider, the top one is at zero. The top one is at zero. So it's equal time. But sorry, that doesn't change. So the length of your window is delta t0. That doesn't change. The fact that leg is zero, it means that you look at the same window for the two products. But it's a two minute window. So there is a, of course, if you go down to extremely short times, you will have an effect on this. So essentially what you can say is that you can define a shortest time scale that you're interested in the system that's a bit up to you knowing the system what it should be. It could be a minute. It could be a second. Probably it's not a nanosecond that you want and not a day. And from measuring everything on this scale, you can extrapolate to longer scales. Well, delta t0 is set by me.  It can depend on my data. It can depend on my, I mean, so. Of course you can change delta t0. And things will change. But the idea that from measuring everything on delta t0, you can extrapolate to delta t0 won't change. And the other thing, so this is OK. This is the equation part. And the other message is that I think the fact that it doesn't change, so things remain constant, is that it's not just an automatic time scale. It's not just related to the, so the autocorrelation here is not just related to the typical activity of the market, but somehow related to the time scale on which probably humans can digest information or that they pay attention. So it's an ill-defined concept. But the fact is that indeed the time scale of things do not change. So this is one explanation. So OK, so why is this important? Is that, of course, what it should help you do? I'm guessing to this, which we won't discuss here, but actually it comes up in, also, not in finance. So what it helps you do is, OK, what do we have? You can measure everything on this scale. On the shortest scale, you have a lot of points. Or you can measure everything on the long scale and you will have the asymptotic value immediately. And you have to have a trade-off between these. You want to have good statistics, but you want to have the proper value. This comes up, actually, not only finance. These type of problems can come up in many fields. OK, there is a method that you can connect these correlations and make a good trade-off between the two things. Is it OK? So OK, so that was it for correlations for me for now. Maybe we'll get back to it a bit later, but it's always comes up. And so what I wanted to discuss is OK. We spent essentially three lectures discussing some empirical facts. We had a bit of modeling, but so what I wanted to go into is to discuss, OK, so why do we think prices behave like this? I mean, it won't be in one lecture that we discuss all this, but of course the goal is going to this direction. And so there is a notion that we discussed, and we will mention it, but we won't be extremely clear on its meaning. So there is what we discussed is somehow market efficiency, which we in fact didn't very well define. So one definition, OK, it has actually several definitions, one that we can think of after all this, so is statistical efficiency, which is simply to say, OK, all predictable patterns in prices are inexistent. So no predictable patterns. So which is somewhat, OK, it's somewhat what we saw, right? We saw that there are no simple correlations, that prices seem to behave in a diffusive manner after extremely short time scales, so on short time scales there might be something different. So it's OK, we might say this, prices do not contain predictable patterns, we don't say why it is the case. And so there is a typical economics approach to this, which is actually called fundamental efficiency. So no predictable patterns. Of course what we discussed, actually this I wanted to mention, some are lack of time, so what we discussed were linear methods. So there is this linear simple method you cannot predict. One could already, but correlation, we could look at nonlinear correlations, we won't do this, so there are several other methods to go further in data, but we won't discuss it here. Someone wanted something, yeah? OK, so it's an observation, we saw it in the previous courses that it's another way that prices seem to be diffusive. But it's not a deep thing, it doesn't state why it is the case. Of course it's just another word for the same thing, but we'll get to the discussion. So there is this fundamental efficiency, which is a very, very much economics concept, which claims that there is a fundamental value, which is the fundamental value of a product, and that this is what an asset is really worth. So prices will eventually go to this. How will they go to this? It's the usual, it's called this type of arbitrage concept, is that there are some people who know the fundamental value, so this is known to some people, how we do not know. And so what they will do is that, OK, if they see that the price is below the fundamental value, then it's a good moment to buy it, by which they will be pushing the price up. And if it's above, they will sell it, by which they will push the price down. So those people who know the fundamental value will trade in a way that the price indeed goes to this fundamental value. I think it makes sense as a claim, but it's not very clear how the mechanism really works. But OK, we can imagine this, that's sure. If something is very cheap and people start buying it, then the price of it will be increased for some reason. We will exactly discuss what is the reason, if it's a mechanical reason or it's a more complicated reason. So this way, do I have to write it up or it's clear what I'm saying? I never know how much I should write. Is it clear what I'm saying? It's clear. Clear? OK, so fundamental value, some know this, trade accordingly, let's say. So to push the price to this value. And OK, so this stuff in itself, OK, what is efficiency in this? And then they say, OK, so this fundamental value, so FV stands for fundamental value, can only change if there is some real news in the market, some non-anticipated news in the market. Right? And OK, so you can say, sure. There won't be predictable patterns if the only way prices can change is via an unpredictable underlying process. OK, so it's not obvious how to falsify this in a first step approximation. So this says, OK, so this justifies unpredictable prices in its own world. OK? Again, there might be new concepts, but simple claims. Something that you cannot predict. Simply news that comes out, I mean news. So I'm really new, exactly, really new. So I don't know, if you know that tomorrow there will be elections in, or yesterday there were elections in Estonia, and if you really know what will happen, all polls tell you that one side will win and it's not really news when it comes out because you anticipated it. Non-anticipated is really some news. But the idea here is, OK, so this fundamental value will depend on something which is non-predictable, so the fundamental value, it won't be predictable. So somehow it justifies in a hand-waving manner. But there are some issues with this. Indeed, I think you can falsify it. So actually what you can falsify is that, OK, but real news doesn't happen that often. Right? Or I mean, that's what I claimed, and that's the main critique of this, is that it's actually prices. So this is actually, so this is a critique of the fundamental efficiency. Prices are much more volatile than news. So it's actually this question, OK, what is unanticipated news? How often does it happen that there is really a big news? Well, maybe once a day, or I don't know. But if you look at volatility of prices, they are on super-short scales, they are moving. You feel that it's not a very good explanation. And news is, no news, OK, so I didn't define it here. I say some know this. So it might be a news that only some know. So maybe it's only you have insider information on a company, and you get some information. Of course, you have to be big enough that your trading accordingly pushes the price there. So maybe there have to be more. So it's ill-defined, but it need not be all. So it can be some other type of information. But still, it doesn't change the case that, OK, it's a strange explanation. And also, another thing is that, OK, which we'll discuss a bit more, but OK. So if it's news that arrives, which is built into the prices, you would expect that there is some finite scale for doing this, actually. We saw some correlations on fluctuations. Why would it be an immediate? So from the diffusivity of prices, we see that after a few seconds, things are diffusive. So would it take so fast to interpret news? It's the question of interpreting news versus short-scale diffusivity. Right? It's clear what I'm claiming here? So of course, I mean, it's a trivial claim that interpreting news is hard. First of all, the news can have errors in it. I mean, you don't know where you get it from. Maybe it's not true. You have to understand it. Sometimes it's not obvious if a news is positive or negative and all this type of thing. So, OK, so there are, I think, some issues with this, and we will see further. The alternative view, so what is this really statistical efficiency? Well, the idea somehow is this point here to trade accordingly is there. So the statistical efficiency, well, it's not a nice theory. It's not like fundamental efficiency would be a nice theory if it were true. Statistical efficiency simply said that there are a huge number of algorithms and, well, it could be people trading, but algorithms trading in the market looking for correlations, trying to exploit trends or any predictable patterns. And trade accordingly, and exactly this way they exclude the patterns. So this statistical efficiency is it's essentially algorithms that are trading. So it could be people, but today it's more algorithms. Algorithms look for patterns, and actually they trade accordingly and remove these patterns. So it's a less nice theory, but maybe it's more true. So, OK, but let's try to be a bit more clear about this view here. It says, OK, we are hand waving. We say that it's hard to believe, but we can be a bit more quantitative. So, OK, what this says is that, OK, there is a fundamental value, price can go away from it, but it will come back to this level soon. So the question is, OK, what is the error around this fundamental value? So how far can price go from its value? So, OK, but what I say is that the fact that there are no patterns is a lot of algorithms that are looking for patterns. So even if there is a pattern locally coming up, it is removed. So any patterns are essentially one by one removed by people understanding that they exist. So there can be patterns that come up, maybe more complicated, maybe not simply in this linear, predictable fashion. But on average, you do not, you measure diffusivity because they are all exploited. There is a huge money you can make by exploiting them, so there is a lot of people trying to look for it. So if locally you find out that there is a, right if there is a clear mispricing between, I don't know, oranges in the US and Europe, I mean, taken into account the money to travel between or to import, if there is a huge mispricing, people will be trading in a way to eliminate this. So as a result, things will be eliminated. It's not that God eliminates them. So, okay, so still I'm on trivial claim, so fundamental value, if it's, okay, if it's zero one percent against, price can go zero one percent away from fundamental value, good, then it's a meaningful thing, and if it can go a hundred percent away, then it's hard to handle it in any way. So actually, I put a quotation, I like to put quotations, from Fisher Black, who is a super important economist. I think he was trained in physics. Actually, so there is this Black and Scholl's option pricing, he is the Black from it, and this is a paper called Noise, and I leave you read it, I read it for you. You can read it. So he wasn't the person to come up with fundamental efficiency, of course, as you can guess from this. So is it okay? Okay, so the claim is that, okay, we might define an efficient market where the price is within a factor of two of its value, which is fundamental value, so between half of it and twice, and double of it, two is, of course, arbitrary, in extra, but it's not just big, but it's arbitrary. Actually, the title of paper is Noise, and he discusses how much noise affects all these, and of course, I think almost all markets are efficient almost all of the time, almost all means at least 90 percent. So what does it mean? To me, it's a super non-scientific statement to say this fundamental efficiency. Of course, if anything is fundamentally efficient, and actually you can come up with, so if we base on this, if we want to believe black, then you can make a small calculation. You could say that, okay, so the difference, let's say delta is the difference between the fundamental value and the price now, and let's say, okay, so that it's 50 percent. Then you can write then, actually, you can estimate how much time it will take for price to go back to this. And so the time estimated, so is it clear? So time to get back to the fundamental value, if you are 50 percent away, you can be written as, so if you know the standard deviation of the process, it will be something like, you can write up something like this. So what you want is that, via the typical fluctuations, you can get back to the original price. And so actually, if you come up, so this was 50 percent. For typical products, this is, let's say, 20 percent per year. So the typical volatility, the typical standard deviation of prices, let's say it's 20 percent a year, which would give, I wrote six years, I think. So it means that, okay, so for a typical fluctuation of 50 percent away from the fundamental value, it will take six years to get back to it. So it's meaningless, right? You cannot do much with this. So we won't believe in this. It might be a useful concept, and statistical efficiency seems to be even true. But what we will try to see in the following is how the price is changing. This seems to, so the traditional claim in economics seems to be thrown away. Actually, I think it's fun. I put up this figure here from the New York Times. So in 2013, there were three people who got Nobel prizes in economics, which is this guy is Fama, and this guy is Schiller. The middle guy is written here. Actually, I don't know what he did. But so what Fama did, he is the father of this fundamental efficiency, fundamental value theory. And he still believes in it. If you ask him about how do you think that prices always make sense, he said, yeah, I do not believe in bubbles. I don't know what they are. It is indeed true. This guy here, instead, is the person I think who first showed this. Actually, I will write up the name of this. So the fact that prices are much more volatile than news would suggest is called volatility puzzle in the literature. So it first comes from... I don't think the name comes from him, but it comes from Schiller. It's pretty funny, I think, for this idea in economics that two people who say exactly the opposite got Nobel prize at the same time. So it's a bit of a question of related to yesterday's movie about this underlying big truth, if it exists or not. And okay, so this was more on the theoretical side. So what do we want to do in the next? We want to understand a bit, okay. Somehow information does exist in the market. Why do prices change? So essentially, how do these information usually get built into the market, to the prices, and try to become more... doing some models about it. But to me, it's... maybe it's true, but it's meaningless, right? If there is a fundamental value from which you can deviate so much that it will be in six years that you get to it, then for any practical reasons I don't care about this fundamental value. So it doesn't disprove in a sense. There might be some fundamental value somewhere, but nobody knows it. Of course, okay, I didn't mention something. So what is this fundamental value? Okay, so one would guess that fundamental value is the real value of the company, right? Apple might have a price now, but then you can write up or look at all the balance sheets and come up with the real valuation. And you say that, okay, it's overvalued because... So, okay, so I want to discuss how, well... how information gets built into the market, actually. I will call it like a short discussion before getting to the model, impact and info. So there is a notion that we discussed a lot, and actually that's the answer, eventually, to your discussion, so what is called market impact, which is okay, it's a trivial claim that I already made, so market impact or actually it's called price impact often. It's simply the claim that if you buy on average, okay, it's important, on average, buy trades, and I will define it in a second, we'll make the price go up and sell trades, we'll make the price go down. This is market impact, of course, it's easy to claim, but it's hard to really measure. I want to make just a short thing that we didn't discuss before, so something to be always kept in mind is this, is all figures, but this as well, which was okay the way the market functions. Of course, it's trivial to say that at any trade, there is a buyer and a seller involved in it, so you might ask, what does this mean, this claim? So the way we define the sign of a trade, so if it's a buy trade or a sell trade, is the aggressive party, so the initiator. In this case, so the person in this sense would put a market order, so there are all those limit orders sitting, and then someone decides, okay, boom, I want to trade, that's the only way to have a proper trade, so to have buyers and sellers meet, so it's the sign of the initiator, which is the sign of the trade, it's a definition, okay? Actually okay, sometimes if you look at data, sometimes it's not that obvious to decide the sign of a trade, which should be the simplest thing, but so this is the claim, okay? And so there are three usual, maybe I will just discuss, okay, three usual explanations for this market impact, so one is that it's somehow, that trades convey info to somehow private information, so the dynamics here would be that, okay, I have some information, so I trade, so others learn from me, so others update their expectations, so price moves, right? It's a simple, it's hard to write it up, but it's okay, so that means I have information that price will go up from somewhere, I buy, people see that I buy, they say, wow, this guy has information, maybe, but if he trades, if he doesn't have information, let's all think, let's follow him, or they might think that I'm an idiot, and let's do the opposite, but somehow they get the information from this, okay? So this is one way of thinking about it, one which is a more, actually, more physics approach, which says, it's okay, of course here what information is is an important question, and how do others know that I'm someone who has information and I'm not just a monkey? It's not obvious, and so the other is that you can, another theory can be that it's essentially any type of fluctuations of supply and demand, which can be somehow random, any random fluctuations have some mechanical, mechanically impact the price, okay? So it can be, I mean, you see already here, maybe people are looking at what's the number of sellers and buyers and if anything happens, they try to update the data to, sorry, so it's not really updating, it's a mechanical manner that if someone here trades, he will take away some part of the, so if he's buying, he will take away some part of the supply here, the red one, so the prices will mechanically move. Impact, so this would be somehow, this would say that impact is a statistical phenomenon, you can model it in simple ways, it's not some real information, so there are not trades containing information and trades not containing information, and also there is, okay, there's a third way of looking at it, which I won't even write up, which can be, okay, actually, okay, I write it up, which is that it's just a question of forecasting, of course you can say that, yeah, sure, buy goes up, if you buy, price goes up, but it's not this direction, it's more that you just forecast the price will go up and you're buying, it can make a correlation, so even if you weren't there, the price would do the same, so this is not really impact, so I won't discuss here. So the usual way to handle information in finance is the following, so we will get to doing a bit of models here to understand it better, but so the usual way is the following, you usually differentiate between, you say that in models, so usual models in finance, you say that there are informed traders, right, so you solve your problem about all this question of what is information, you say, yeah, okay, there is a subset of people who are informed and are rational, so okay, they are trading in a way to exploit their information that came from somewhere and not everyone has them, and so actually this is a bit like behavioral, finance issues, and the other is that you have noise traders, who are idiots, who are trading just, who don't have information, they are trading to make things work nice, so you can, okay, so noise traders do not have information or do not have the ability to process the information, whatever, it's the same, and okay, you can come up with explanations why they trade, so they might trade for other reasons, they might trade because, because from some liquid, for some other reason they have to buy, not because the price will go up, because somehow they really have to own this product or they have to trade for some risk reasons or they are simply irrational, they think they have information but they are doing random stuff, so okay, it's a way to model this, and so I want to discuss a few months just to get the taste of this, so what I want to start with is a model which is called Kyle model, because of a guy called Kyle from 85, which is actually a super nice paper, so of course, okay, when I'm talking about all these ideas in, typically in economics, I'm quite negative about it, but there are very deep ideas as well, so there are very clever people who may be following another approach that what you would do but come up with very good things, maybe I'm too negative sometimes. So anyway, there is Kyle, a paper whose title is continuous auctions and insider trading, if you want to read it, does anyone want to read it? Should I write up the title? Continuous auctions, insider trading. The title doesn't seem to be extremely sexy, but so the idea is that it's one type of model how information can get built into prices. Okay, so how info builds into prices, and actually somehow, so I said that we will discuss microstructure, this is somehow the main paper, how information builds, gets built into prices. So it's somehow the foundation of the field of microstructure. So it might be long to discuss it or not. It's a simple model, but it might be a new approach. So you have a setup, which is that you have one asset. Okay, there is only one product in the market. Let's forget all these correlations that we discussed, cross correlations. You have one informed trader, so that's also simple. Actually, she will have a name, she's called Alice. There is one market maker, who is called Bob. I'll discuss in a second, and there are noise traders. Okay, let me turn around. So we discussed one asset trivial, one informed trader, and noise traders, this we discussed. So what is a market maker? Again, we discussed it a lot of times. This is the person who somehow puts quotes in the market. So Alice and the noise traders have to trade against someone. It's this type of auction, so that they can decide the price, and they will trade at the end. So this was an auction that we discussed in the beginning. So they will be able to trade at the price that the market maker offers them. It's okay. So what do they do? What is their behavior, and what is their knowledge? The knowledge is the following. There is Alice who gets some information. So we are at T0. It's a one-step game. He knows at T0 she gets info on the price of this asset that we are having at T1, which we will call actually PF for fundamental. So we are at time zero. She gets some information about where the price will go in the next step. That's it. She's only her, only one to know, which is okay. So the market maker and the noise traders do not know, and so what does she want to do? Okay, she has this information. She has to decide what to do, how to trade. So this sides trade Q based on this information. Q is of course a signed quantity. She can buy or sell depending on what she thinks. Of course, okay. Of course, the price now is P0. And of course, she decides to trade in a way to maximize expected gain. It's trivial. And okay, there are some, in practice she can trade whatever quantity she wants. Okay, no risk constraints. I put it in parenthesis because you would have, not ask me if there is a risk constraint at all, but okay, it's important to know that okay, in real world, of course, the quantity that you want to trade depends on something. So this is okay. This is Alice. It's simple. There is the noise trader guys who are noise trading. So essentially they do a random order flow. There are several noise traders, but since they are random, we can bunch them together. We say, so there is a random order flow of buying and selling. So the quantity they want, which we will call the noise. Okay, so it's a signed quantity again, the sum of all these people there. We will discuss how they said. And so I'm discussing this in detail because it's okay. It will give some results, but also to get the taste of this type of model, which is different from typical physics. And so okay, there is the market maker, which is called Bob. And so what does he have to do? He has to clear the market. He has to trade against the people. So he has to set a price at which he matches volumes. So clear the market, which means okay, match volume of delta V, which essentially does the sum of these two. He will have to trade against these, so Q plus V noise at some price he had. And okay, again, we have an unrealistic assumption that he has no what we call inventory constraints, meaning that okay, whatever quantity Alice wants to trade, he will be on the other side. He will be available to trade. So inventory means the amount of things you have in your pocket. So he's not constrained by this. If you are Alice and you want to trade, and you say you want to sell me one million, I say okay, I'll buy one million, I don't have any problems. I'm not constrained by a maximum size of my inventory of my... Okay, it's not a limit. It's important in a real world and it's important in an economics language for us. I mean, you wouldn't have asked if it exists, but okay, keep in mind. To make realistic, you can eliminate these and you will have another model which is a bit more rich. So this is the setup and then, of course, the question is okay. So it has a taste of this game theory type of thing. So what do we want to do? Yes. Yes, which one? Bob. Bob. And? And noise. And, okay. So N is noise traders. Okay? I have to explain what they do or just the question of N was... So no noise traders generate... They do noise, they are... They have no information for some reason they trade. Don't ask me. And they generate a random order flow. Order flow means buying and selling. So each of them, we will discuss the distribution of this order flow, but each of them does some random stuff and then it's the sum of these random stuff, which we put into one variable because they are uncorrelated. And Bob. So what Bob does is... Bob is available. Bob sets the prices in the market. So he has to trade against the noise. So make the Alice and the noise traders happy. Set the price. This is the setup. Let's go deeper. And so he decides, okay, this is the total volume he will have to trade, the sum of these two. And he will have to decide a p hat where to trade. So his duty... We'll see what the dynamics are and then it will be clear what we are trying to solve here. So the dynamics are the following. It's a bit this theoretical finance taste. So Alice tries to maximize gain. Well, we already said it here. Okay, maximize gain. But what is his gain? Her gain will be... She knows that the final price will be PF. This information she has. And she will be able to trade at p hat, which is set by the market maker. And she's trading a quantity Q. So her gain will be this quantity here. Right? If I know that the final price will be 150, now it's 100, and I'm trading 100, then my gain will be... How 100 times the difference between these two is 50, which is 5,000. Okay? So Q is a quantity. It's a quantity that she decides to trade. And p hat is the price which is set by the market maker to trade, which we'll see in a second haul. So I'll write it up and then... So this is the first step of the dynamics. And what does Bob do? Okay, he doesn't have information. So what he wants to do is a usual condition in this type of models is what we call break even condition, which the idea here is he wants to set the price in a way that it will be this quantity. Okay, so what he does is he wants to set the price to break even, which means no systematic loss or gain. This is the most he can do because he doesn't have information, but he doesn't want to have systematic gain or loss. So what does he do? He observes delta V, that's the sum of all the trades, right? And condition on this, he wants to calculate the conditional expectation of the final price, so the real information that Alice has, condition on the only information that he sees, right? And sets the price there. That's the best he can do. Okay, so it's the information. Where did we put it? Here. Again, it's an F. The idea is okay. So what does this mean? Okay, in a practical sense. I think it's sort of a deep understanding or a deep idea of the market. So there are people who have information and there are other people who are trading there. Okay, you can ask why he's trading. We will have more complicated model where we understand this, but he's afraid, okay, his job is to trade, but he doesn't have the information that Alice has. He's afraid of losing on this. So he tries to set the price. If he understands that Alice has a lot of information that the price will go up, he will also increase his price because he doesn't want to sell her very cheap. Because of course, I mean, there is the next step. So of course, then he will have sold something that he would have sold for much more money later. Okay? Yeah. Okay, in practice, this is what you expect, right? Okay. It's a question or it's a claim. It's a question. Yeah, so okay, in a real world, just I have to clean up here a bit. In a real world, what you would expect is that somehow the market maker is there. He gets to understand something. It's not obvious. It's a super complicated process. But yeah, here we will have a simpler thing. So we need to give some more information to this. But okay, so this is the dynamics. And okay, so what are the further information? It's a further info. Okay, it's a bit this game games here. Because okay, Alice knows Bob's rule. Okay? Alice knows that Bob is going to do this. And Bob knows, okay, these are the usual trivial. Bob knows Alice maximizes. So this is, of course, a rule here. He will have some proper definition of this rule. And then, okay, there are two things. So the third is that Bob knows that Alice knows that... So I mean, everyone knows that everyone knows. I won't write it out. But okay, it's the usual. Of course you know. And there are two things which is very important to this. So actually Bob knows two things. He knows that this guy, so V noise, is a normal distribution with zero mean and something. Okay? And he also knows that this mispricing, so that Pf minus P0, so the information that Alice has, is also a normal. There's another. So this is a capital Sigma V, which V stands for this V, and Sigma F stands for Pf. So okay, it's essentially... He learned this somehow, or somebody taught him. I didn't understand. Why is it normal? Yeah, so I mean, okay, for V noise, you can imagine that it's not normal. If it's... I mean, we defined it as noise. Okay, why is it normal? Of course, why is it normal? Because you can solve it. But it's not that bad. I mean, sure. So what is this Pf minus P0? It's the amount of information that this girl has. You don't expect it to be some... A normal distribution is an okay approximation for this. It's not that she will learn that prices will go 10 times up tomorrow. Well, we know. They know, but they don't know what the other is doing. So it's as if it wasn't known. So what is important is that Bob doesn't know Q, right? He doesn't know the quantity that she is trading. If she knew Q, okay. She knows the total amount coming in. So sure, it's not... In a real market, you could assume that there is only one market maker. Sure, why do you assume that there is only one in-front trader? Actually, we'll get to a further model. I don't think today, maybe it will be tomorrow to make a bit more reasonable the model. But okay, it's a start. I mean, in a model, you want to start simple and see what that gives overall. Yes. I'm listening, just I lost my notes. P zero is the price at time zero. The price now. So what the dynamics is, Alice sees the price now and somebody comes to whisper in her ear that the price at one will be this value. It's an assumption of the model to be able to... It's stated. Bob knows that this information on average is normally distributed. Why? Well, because it will be easier, but also because, okay, I found my notes. Because it's an okay assumption, I think. But there is no underlying deep concept here. Alice knows the value of it. So okay, so is the main question clear? What the game is that Bob only sees this quantity and knows that Alice will be maximizing those distributions? What will they do? So, okay, let's make an assumption which is a reasonable assumption, but we won't go its meaning. Actually, okay, this will be a big critique of this type of model, is that, okay, let's assume the following relation that what Bob does, Bob's pricing rule is the following. The price he sets will be the price now plus a linear dependence on it. So, plus some lambda times delta V, right? So, he sees the total flow coming in and according to this in a linear manner, he changes his price. There is a factor here, lambda. Okay? We'll have to define what this guy... And what is important, so what does it mean that in this impact language that we saw before, so that price is changed, that price changes are correlated to trade flow, it's a linear impact model. The change in the price is linearly dependent on the volume, which actually we will see tomorrow that it's not a good approximation for real data. Real data is more complicated, but okay, it's actually, many models are based on this. So, what will happen? So, what should we discuss here? Okay, so Alice will maximize... So, what Alice is going to do in this case, she will want to trade a Q hat, okay? Which is, as we said, it's maximization. So, okay, in the math language, so he's maximizing his Q, her Q in a way at the expectation of her gain is... He's choosing Q in a way to maximize her gain, where, as we said, gain is here. So, this is the gain, but so then gain actually becomes more... If we know this rule and she knows this rule, then her gain will be this, because she knows the rule. And, okay, so we want to write up the expectation of this. I mean, we want to make the derivative of this and write up the expectation. So, who's losing this stuff? So, okay, so what we know, actually, is that the expectation of delta V is the expectation of... So, delta V is simply Q plus this normal variable. So, the expectation of delta V actually will be expectation of Q, will be Q. Simple. So, with this, what she can write up is that her choice of Q hat, you have to just do the derivation, will be the following... We are solving this maximization. So, it's a simple stuff. So, what does it mean? That, okay, at least we'll increase the size of her trade in a linear fashion based on her information, right? So, this is her information on the price change. So, she will increase his proportion to the information, the amount she wants to trade. And so, but the more complicated thing is what Bob tries to do. So, what does Bob... Where should I write? So, what Bob wants is estimate Q to be able to determine the value of... to guess the value of PF, right? This is his goal. This is his goal. So, what he actually wants to do, what he needs is the distribution, the conditional distribution. So, he needs... It's this what he wants to solve. Okay? Okay, I'll write Bob here because it's his information. Okay, and... Yeah, so here I wanted to give a bit of homework. So, actually... So, this is what he wants to solve. Okay? And so, how should I state it in a homework manner? So, do this calculation. So, what is this? Of course, here you can write up a base rule. Base theorem, right? It's own. So, write up... So, determine the probability. So, determine this guy here. This, of course, what he actually wants is the expectation. So, try to determine this. And give an expression for p hat. Okay, so it's... I think it's useful because you get a taste of what's going on. But it's not too complicated. Actually, it's quite simple calculation. So, this is what he will do. And there is the solution from this. Okay? How should p hat behave? Well, I'm giving away some hints, but okay. So, if you know p hat, you know how he's going to change what has to be his function. So, okay, that I will give you. So, the final solution to this is the way he will behave is lambda being the following one has to come up with it. So, this will be the final solution. So, what is this value? This is the value to ensure, just to make clear. I never know. I'm saying several times many things, but I don't know if I'm becoming boring or it's needed. So, this is the value to ensure what we said in the beginning that... Okay? So, this is to break even. Now, what does this mean? So, this is the solution to the model. It's a bit of dry solution. If you go through the calculations, it's more fun. So, what does this really mean? That it means the following. Okay? I'm using too much space here. So, that is the result there, but it's okay. What are the results or insights of this type of model? It is the following. So, there is something which is okay. It's sort of... The impact will be linear in q hat. It's sort of obvious if it's linear in delta v and delta v is a zero mean plus q hat, then it should be the case that, okay. So, there is a linear impact which is predicted in this type of model, which is sort of put in. But what is more interesting is that what Bob will do, so what we see there, is that lambda... Okay, so actually maybe it's good to... So, just this keep in mind. So, this was, right? This was the beginning. So, what is lambda? So, what does Bob do? There will be this lambda which grows with information. Probably the amount of information of the informed trader, right? So, the amount of information of the noise trader is sigma f. So, the typical size of information. The more traders can have information, the more he has to protect himself against them. It makes sense. At the end, what this type of model gives is things that make sense, but that's why you want to have a model. So, here needs to protect himself more, right? The more information. And it also goes down another thing which might be... So, this is more... The other is that delta decreases with the amount, the volume of noise traders, which is sigma, which is measured by this number. Yes, so it can have larger values. So, if sigma f is larger, but the difference between... So, the amount of information can be larger in absolute value, right? Because it means that the difference between price now and the price in the future that I was told is larger. Of course, the average remains zero, but it's not the average you care about, but it's deviation, because of course if... If Pf minus P0, so this is the amount of information, is negative, then I will sell, or Alice will sell. If it's positive, she will buy. So, it's the size of this information that matters because she can decide the direction in which to go to be in line with the... Information? Yes. Yes, so... Yeah, so Alice is the only person who has information, so that's why I didn't specify it here. So, the second is that the lambda decreases with the volume of the noise traders, meaning that the more there are these...  So, needs to protect himself less. So, this sort of main makes sense. They are in line with intuition, with this first intuition. What happens for Alice? Okay, for Alice, I think we're all... So, in line with what one should have been intuition. So, the expectation of her gain will be... Will be the following. Okay, I don't write it out. Again, it's an exercise, actually, if you want. It will be this. So, this will be an exercise here. And, okay, let's say that this is also an exercise. It's simple. We have all information. By now, we have all information. So, what does it mean? So, her expected gain will go this way, so that she gains, of course, it's increasing. So, increases with amount of information, trivial. Okay, so it's linear, increasing sigma f. But what is also important is that it also increases with some of this overall... Okay, I'll call it liquidity. Just in a second, I'll define it. So, okay, discussing about the amount of volume of the noise trading. Okay, it's some type of noise for Bob, but it's also... It behaves in a positive manner for Alice. She can hide herself in this noise, right? If there is a lot of liquidity, she can trade more. She's less visible. Again, it's in line with intuition. So, there is nothing extremely new in this, but I think it's a simple model that gives the results which are in line with what one expects in a market. So, to conclude on this, it's a first type of model. So, I mean, it's the way going through it, actually.  It might seem very trivial. It's an obvious thing, obvious schemes and obvious results, but they are not that obvious when you have to build it up. So, it's the first model that I think was in line with these basic intuitions that there is a market maker. He has to protect himself from those who have information. How can he do this? Well, we gave the solution and all that comes out might be in line with intuition, but it's good to think about. So, the more the inforter there are, the more he is in danger and the more he has to change the price and the more people on which he can make money. So, the noise traders, who don't have information, the better for him. And instead, for Alice, her expected gain grows with both sigma, capital sigma, and small sigma. So, we started at nine. Okay, so my idea was to discuss another model, but I think maybe it's the right moment to stop and not start there. No? It's too early to stop? I mean, for ten minutes to start something and restart. Okay, so that's what we'll do. If you have questions, of course. But I think, I hope it was clear.