Now, after we talked about what the LOB is, let's talk about how we can model it. As a short summary of what we discussed so far, an LOB aggregates both sides of the market. It gives multi-level summary of the state of supply and demand sides of the market, and there is a substantial empirical research that shows that the state of the LOB is very informative of a short-time future. This means that the short-time price movements are quite predictable if we condition our predictions on the state of the LOB. In other words, we can say that features constructed of the LOB data should be useful for building short-term forecast models. Understanding of the mechanics of the LOB is quite important for a number of applications. One of them is optimization of trade execution and minimization of marketing, but we already mentioned both these topics in our previous course on reinforcement learning. Other applications deal with the design optimal in today trading strategies. Understanding of the LOB is important even for those players that operate that different frequencies, for example, daily frequency. The reason for this is a propagation of dynamics and signals between different timescales in the market. One such example of connections between high-frequency trading can traditional low-frequency daily trading is the Flash Crash of 2010. You can read about this event in the agent list for this week. Now, let's talk a bit about the differences between a portfolio allocation problem and an optimal placement problem for the LOB. First, for portfolio allocation, the event horizon is in weeks or months or even years. Traits for this setting usually extend from minutes to days, and portfolio managers solve their own problems of portfolio optimization. The cost of execution for them is a fixed cost that they pay to brokers. Now, for brokers, the task is to keep the largest part of the sphere as a profit. So, they have to solve their own optimization problem, namely the problem is now how to split the market order received from the trader optimally and this translates into a stream of orders that they start to place at very high frequency. Their decision space may be focused on only one stock but it's still pretty high dimensional space. The broker should decide on the time sequence of orders and for each order should be either a market or limit order. It should also specify the order size and it should specify the venue, namely to what exchange the order should be submitted. So, together, this produces a high-dimensional decision space even for a single stock. So, the decision space is high-dimensional and discrete. If we want to trade a few stocks simultaneously, the dimensionality will increase exponentially. The dynamics of the LOB is also quite complex as it has so many moving parts. So, a big question is, "What kind of models will work best for such settings?" There is one view that says that simple and analytically attractable models should be preferred. In particular, one reason for this is that those models should work fast. Another view is that the dynamics are so complex here that we better should rely on machine learning methods that are model free. A bit more general view of different approaches to modeling the LOB is to classify them into three categories. The first class is formed by economics-based equilibrium approaches. In this approach is a single agent is involved in the sequential game against the market represented by a state of the LOB. Other approaches focus directly on modal and observables of the LOB. For example, statistical and machine learning methods focus on statistical patterns of order flows, and they do not try to explain the mechanics of the LOB. Instead, they operate with exogenous random processes they describe order flows. Finally, there are physics motivated approaches that have you read them order flow says connected to stochastic dynamics of interactions within the LOB. We have already mentioned many potential applications of models of the LOB and here I list them again for you. They include market-making, optimal execution, risk management, daily strategies, option trading, and regulation. Finally, I want to talk about different types of data relevant for the LOB. So, depending on their level of aggregation, there are three levels of the LOB data which is something that we already mentioned before. So, to reiterate, Level-1 data, also called the quotes data, shows only the top of the LOB on both sides. In other words, it only shows the best bid and ask prices and the total liquidity, that is the number of trades available at this level, and this is the most basic information from the LOB that can be combined with the trading data as we'll discuss a bit later. The second level is called Level-2 data and it shows the aggregated state of the LOB for five levels into both the demand and supply sides. So, it gives you five bid quotes with quantities and five ask quotes with quantities. Finally, Level-3 data is the most defined data that provides information about each individual limit order. Out of these categories, Level-1 data is the most accessible and cheapest. There are a number of providers of such data, for example, tickdata.com. Level-2 data is considerably more expensive because it's the most interesting available data for market participants. It provides considerably more information than Level-1 data and many market players prefer to use this data. Finally, Level-3 data is of course the most valuable piece because it provides complete data about these data are generally available only to regulators.
