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619 results in Econophysics and Financial Physics

Page 8 of 31

Quantum Field Theory for Economics and Finance
  • Belal Ehsan Baaquie
  • Published online:
    22 August 2018
    Print publication:
    23 August 2018
  • An introduction to how the mathematical tools from quantum field theory can be applied to economics and finance, providing a wide range of quantum mathematical techniques for designing financial instruments. The ideas of Lagrangians, Hamiltonians, state spaces, operators and Feynman path integrals are demonstrated to be the mathematical underpinning of quantum field theory, and which are employed to formulate a comprehensive mathematical theory of asset pricing as well as of interest rates, which are validated by empirical evidence. Numerical algorithms and simulations are applied to the study of asset pricing models as well as of nonlinear interest rates. A range of economic and financial topics are shown to have quantum mechanical formulations, including options, coupon bonds, nonlinear interest rates, risky bonds and the microeconomic action functional. This is an invaluable resource for experts in quantitative finance and in mathematics who have no specialist knowledge of quantum field theory.

  • 6 - Interaction

  • Edited by Domenico Delli Gatti, Università Cattolica del Sacro Cuore, Milano, Giorgio Fagiolo, Mauro Gallegati, Matteo Richiardi, Università degli Studi di Torino, Italy, Alberto Russo
  • Book:
    Agent-Based Models in Economics</I>
    Published online:
    02 March 2018
    Print publication:
    22 March 2018, pp 109-142

  • Summary

    Introduction

    It is a fact of life that the preferences, information, and choices – hence the behavior – of an agent affect and are affected by the behavior of the agents she interacts with. In fact, there is a two-way feedback between individual and aggregate behavior: agents’ interaction affects the evolution of the system as a whole; at the same time, the collective dynamics that shape the social structure of the economy affect individual behavior.

    For example, consider the case of the adoption of a new technology by initially uninformed consumers. Each agent, based on her preferences, may have some ex-ante evaluation about the quality of new products introduced in the market. However, by interacting with their peers, agents may gather fresh information about the new product and, eventually, they may decide whether to buy it or not. This influences the adoption rate of the product, which can be in turn exploited by other consumers as a parameter to be employed when subsequently considering whether to buy the product or not. Therefore, individual decisions may be affected by agent interactions, then impact on the aggregate state of the system, which can in turn feed back to individual behaviors.

    Traditionally, economic theory has largely overlooked the importance of interactions among economic agents. In standard economic theory, interactions are treated as externalities or spillovers. In general equilibrium theory (GET), the presence of externalities is often treated as a pathology of the model, leading to possible nonexistence of equilibria. Therefore, in the model it is often assumed that externalities do not simply exist – i.e., that agents only interact indirectly through a nonagent – that is prices, whose role is to aggregate individual information. Hence, in GET, agents are totally disconnected, dots living in a vacuum without any connections (links) between them.

    To appreciate the importance of externalities in mainstream economics, one has to resort to game theoretic models. In this setup, agents interact directly with all the other agents in the game. Interactions are captured via strategic complementarities: the payoff of any single agent depends directly on the choices made in the game by all the N − 1 other agents. This configures a scenario completely at odds with the one portrayed in GET, namely one where agents live in a fully connected world, where they are linked with anyone else.

  • PART V - PRICE IMPACT

  • Jean-Philippe Bouchaud, Julius Bonart, University College London, Jonathan Donier, Martin Gould
  • Book:
    Trades, Quotes and Prices
    Published online:
    26 February 2018
    Print publication:
    22 March 2018, pp 205-207

  • Summary

    Introduction

    In this part, we jump into the core topic of this book: understanding price dynamics. More specifically, we will consider the important question of how order flows interact to form a price. In particular, how can the predictable order flow elicited in the previous chapter produce unpredictable prices? This question will be our core motivation throughout the remainder of the book.

    As ever, we start with two fundamentals: concepts and data. We begin by zooming out and asking questions about how market participants’ actions impact prices, and, more broadly, the state of the market.We are immediately faced with a difficulty: since history cannot be replayed, how can we know how an action really affected the market? Put another way, how can we know that what we observed was not going to happen anyway? This conundrum leads us to the necessity of decomposing impact into two components: the reaction impact, which describes how an action directly affects future prices, and the prediction impact, which describes what would have happened anyway, even in the absence of the action. The sum of reaction impact and prediction impact will correspond to the observed impact, as measurable by an outside observer.

    Because it does not require any information about the circumstances in which trades occurred, observed impact often seems to be a natural quantity of study for an outside observer who only has access to public data. However, upon deeper reflection, observed impact reveals itself to be rather non-trivial. On the one hand, some aspects behave as we would expect. For example, buy market orders push the price up on average, sell market orders push the price down on average, and observed impact first increases rapidly before reaching a stable plateau. On the other hand, other aspects are much more surprising. One prominent example is that impact scales as a concave function of order size, such that the seemingly natural idea that this relationship should be linear actually breaks down.

    In the quest for reliable measurements of impact, private data is a prerequisite. By allowing a deconvolution of the prediction impact at the scale of the metaorder – and thus getting rid of most spurious conditioning effects – private data offers a reliable way to estimate how order flow dynamically affects the market.

  • 14 - Generalised Propagator Models

  • from PART VI - MARKET DYNAMICS AT THE MICRO-SCALE
  • Jean-Philippe Bouchaud, Julius Bonart, University College London, Jonathan Donier, Martin Gould
  • Book:
    Trades, Quotes and Prices
    Published online:
    26 February 2018
    Print publication:
    22 March 2018, pp 270-286

  • Summary

    When I want to understand what is happening today or try to decide what will happen tomorrow, I look back.

    (Omar Khayyám)

    Price Micro-Mechanics

    In the previous chapter, we studied a class of propagator models that consider the change in mid-price that occurs between subsequent market order arrivals. The models that we studied in that chapter consider all market order arrivals on an equal footing, irrespective of their size and how aggressive they are.

    In the present chapter, we extend those models in two different ways. First, we consider a propagator model that partitions market order arrivals according to whether or not they consume the entire opposite-side best queue upon arrival. We introduce a mathematical framework that enables us to distinguish the impact of orders partitioned in this way, and we show that this extension helps to solve some of the problems with the one-event-type propagator models from the previous chapter.

    Second, we introduce a generalised propagator model that considers not only market order arrivals, but also some limit order arrivals and cancellations. In this framework, we are able to track all events that cause price changes, and we are therefore able to monitor the evolution of impact on a more microscopic scale. However, we also argue that performing such a granular analysis of order flow requires working with rather complex models, which can be difficult to calibrate. We then turn to a more intuitive formulation of these multi-event propagator models that naturally encompasses the idea of history-dependent liquidity.

    Limitations of the Propagator Model

    The propagator model that we introduced in Chapter 13 is a reduced-form description of LOB dynamics. The model assumes that all market orders lead to the same impact dynamics, characterised by the propagator G(l), and does not explicitly track other LOB events that occur between market order arrivals. Clearly, this approach neglects many effects that could be useful for understanding or modelling price changes in real LOBs. For example, partitioning market orders according to whether or not they consume all the available liquidity at the opposite-side best quote is very useful, not only for assessing whether such market orders will cause a change in mid-price immediately upon arrival, but also for predicting the string of other LOB events that are likely to follow.

  • 22 - Market Fairness and Stability

  • from PART IX - PRACTICAL CONSEQUENCES
  • Jean-Philippe Bouchaud, Julius Bonart, University College London, Jonathan Donier, Martin Gould
  • Book:
    Trades, Quotes and Prices
    Published online:
    26 February 2018
    Print publication:
    22 March 2018, pp 406-421

  • Summary

    When men are in close touch with each other, they no longer decide randomly and independently of each other, they each react to the others. Multiple causes come into play which trouble them and pull them from side to side, but there is one thing that these influences cannot destroy and that is their tendency to behave like Panurges sheep.

    (Henri Poincaré, Comments on Bachelier's thesis)

    As is stated in any book on market microstructure (including the present one!), markets must be organised such that trading is fair and orderly. This means that markets should make the best efforts to be an even playing field for all market participants, and should operate such that prices are as stable as possible. As we emphasised in Chapter 1, market stability relies heavily on the existence of liquidity providers, who efficiently buffer instantaneous fluctuations in supply and demand, and smooth out the trading process. It seems reasonable that these liquidity providers should receive some reward for stabilising markets, since by doing so they expose themselves to the risk of extreme adverse price moves. However, rewarding these liquidity providers too heavily is in direct conflict with the requirement that markets are fair. In summary, if bid–ask spreads are too small, then liquidity providers are not sufficiently incentivised, so liquidity becomes fragile; if bid–ask spreads are too wide, then the costs of trading become unacceptable for other investors.

    The rise of electronic-markets with competing venues and platforms is an elegant way to solve this dual problem, through the usual argument of competition. In a situation where all market participants can act either as liquidity providers or as liquidity takers (depending on their preferences and on market conditions), the burden of providing liquidity is shared, and bid–ask spreads should settle around fair values, as dictated by the market. As we saw in Chapter 17, this indeed seems to be the case in most modern liquid markets, such as major stock markets, futures and FX markets, in which the average bid–ask spread and the costs associated with adverse selection offset each other almost exactly.

  • 3 - Agent-Based Models as Recursive Systems

  • Edited by Domenico Delli Gatti, Università Cattolica del Sacro Cuore, Milano, Giorgio Fagiolo, Mauro Gallegati, Matteo Richiardi, Università degli Studi di Torino, Italy, Alberto Russo
  • Book:
    Agent-Based Models in Economics</I>
    Published online:
    02 March 2018
    Print publication:
    22 March 2018, pp 33-42

  • Summary

    Introduction

    A rather common misunderstanding about simulations is that they are not as sound as mathematical models. Computer simulations are, according to a popular view, characterised by an intermediate level of abstraction: they are more abstract than verbal descriptions but less abstract than ‘pure’ mathematics. This is nonsense. Simulations do consist of a well-defined (although not concise) set of functions, which relate inputs to outputs. These functions describe a fully recursive system and unambiguously define its macro dynamics. In this respect, AB models are no different from any other model: they are logical theorems saying that, given the environment and the rules described by the model, outputs necessarily follow from inputs. As in any other model, they provide sufficiency theorems: the environment and the rules are sufficient conditions to obtain the results, given the inputs. The resort to computer simulations is only an efficient way – given some conditions – to obtain the results.

    In this chapter we offer a characterisation of AB models as recursive models. The chapter has a simple structure: Section 3.2 places AB modelling in the wider context of simulation models; Section 3.3 introduces the notation and the key concepts; finally, Section 3.4 concludes elaborating on what constitutes a proof in an AB setting.

    Discrete-Event vs. Continuous Simulations and the Management of Time

    Computer-based simulations face the problem of reproducing real-life phenomena, many of which are temporally continuous processes, using discrete microprocessors. The abstract representation of a continuous phenomenon in a simulation model requires that all events be presented in discrete terms. However, there are different ways of simulating a discrete system.

    In Discrete Event Simulations (DES) entities are thought of as moving between different states as time passes. The entities enter the system and visit some of the states (not necessarily only once) before leaving the system. This can be contrasted with System Dynamics (SD), or continuous simulation modelling, a technique created during the mid-1950s by Jay Forrester at the Massachusetts Institute of Technology (Forrester, 1971), which characterises a system in terms of ordinary differential equations (ODEs). SD takes a slightly different approach to DES, focusing more on flows around networks than on the individual behaviour of entities. In SD, three main objects are considered: stocks, flows and delays. Stocks are basic stores of objects, as the number of unemployed workers.

  • Appendix

  • Jean-Philippe Bouchaud, Julius Bonart, University College London, Jonathan Donier, Martin Gould
  • Book:
    Trades, Quotes and Prices
    Published online:
    26 February 2018
    Print publication:
    22 March 2018, pp 422-440

  • Summary

    Description of the NASDAQ Data

    Throughout the book, our empirical calculations are based on historical data that describes the LOB dynamics during the whole year 2015 for 120 liquid stocks traded on NASDAQ. On the NASDAQ platform, each stock is traded in a separate LOB with a tick size of ϑ = $0.01. The platform enables traders to submit both visible and hidden limit orders. Visible limit orders obey standard price–time priority, while hidden limit orders have lower priority than all visible limit orders at the same price. The platform also allows traders to submit mid-price-pegged limit orders. These orders are hidden, but when executed, they appear with a price equal to the national mid-price (i.e. the mid-price calculated from the national best bid and offer) at their time of execution. Therefore, although the tick size is $0.01, some orders are executed at a price ending with $0.005.

    The data that we study originates from the LOBSTER database (see below), which lists every market order arrival, limit order arrival, and cancellation that occurs on the NASDAQ platform during 09:30–16:00 each trading day. Trading does not occur on weekends or public holidays, so we exclude these days from our analysis.When we calculate daily traded volumes or intra-day patterns, we include all activity during the full trading day. In all other cases, we exclude market activity during the first and last hour of each trading day, to remove any abnormal trading behaviour that can occur shortly after the opening auction or shortly before the closing auction.

    For each stock and each trading day, the LOBSTER data consists of two different files:

  • • The message file lists every market order arrival, limit order arrival and cancellation that occurs. The LOBSTER data does not describe hidden limit order arrivals, but it does provide some details whenever a market order matches to a hidden limit order (see discussion below).

  • • The orderbook file lists the state of the LOB (i.e. the total volume of buy or sell orders at each price). The file contains a new entry for each line in the message file, to describe the market state immediately after the corresponding event occurs.

  • 4 - Rationality, Behavior, and Expectations

  • Edited by Domenico Delli Gatti, Università Cattolica del Sacro Cuore, Milano, Giorgio Fagiolo, Mauro Gallegati, Matteo Richiardi, Università degli Studi di Torino, Italy, Alberto Russo
  • Book:
    Agent-Based Models in Economics</I>
    Published online:
    02 March 2018
    Print publication:
    22 March 2018, pp 43-80

  • Summary

    Introduction

    In order to achieve her goals, an agent must decide a line of action (a behavioral rule). Mental representations of the environment and of the behavior of other agents are key in taking this decision. The availability of an adequate and appropriate information set and of cognitive capabilities to process information, in turn, are key in forming these mental models. In a context characterized by uncertainty, one of the most important cognitive process is expectation formation. In this chapter we overview the way in which rationality, behavioral rules and expectation formation are connected in modern macroeconomics.

    In Section 4.2 we set the stage by discussing (optimal) decision-making in an environment of full rationality and certainty. From Section 4.3 on, we discuss the consequences of uncertainty – in its wide range of specifications – and expectation on individual decision making and on macroeconomic performance.

    Section 4.3 is devoted to the theory of choice in the presence of measurable uncertainty (risk). Uncertainty is measurable when agents are able to attach probabilities to uncertain events. In this setting the probability distribution of the variable of interest replaces the true value of the variable (which is available only in the case of certainty) in the information set of the agent. We will provide simple examples of choice in the case of risk neutrality (Subsection 4.3.1) and risk aversion (Subsection 4.3.2). Moreover, we will discuss choice in a multi-period setting (Subsection 4.3.3).

    We will show that it is straightforward, and extremely useful, to extend the notion of measurable uncertainty discussed in Subsections 4.3.1 and 4.3.2 to the multi-period setting. Also in a multi-period context, in fact, the true values of the variables of interest are replaced by probability distributions.

    The Rational approach to Expectation formation (RE) is the natural candidate to model expectations in such a setting. In fact we introduce a Linear Stochastic Difference Equation at this early stage of the analysis.

    We illustrate its solution by means of a graphical tool which exploits the two-way relationship between current and expected value of a variable of interest. The true (or actual or current) value of variable x is a function of the expectation of the same variable xe, in symbols x = f (xe) (represented by the True Value, or TV, schedule).

  • 21 - Optimal Execution

  • from PART IX - PRACTICAL CONSEQUENCES
  • Jean-Philippe Bouchaud, Julius Bonart, University College London, Jonathan Donier, Martin Gould
  • Book:
    Trades, Quotes and Prices
    Published online:
    26 February 2018
    Print publication:
    22 March 2018, pp 384-405

  • Summary

    I will remember that I didn't make the world, and it doesn't satisfy my equations.

    (Emanuel Derman and Paul Wilmott “The Modeler's Oath”)

    Optimal execution is a major issue for financial institutions, in which asset managers, hedge funds, derivative desks, and many more seek to minimise the costs of trading. Such costs can consume a substantial fraction of the expected profit of a trading idea, which perhaps explains why most active managers fail to beat passive index investing in the long run. Market friction is especially relevant for high-turnover strategies. Given the importance of these considerations, it is unsurprising that a whole new branch of the financial industry has emerged to address the notion of best execution. Nowadays, many brokerage firms propose trading costs analysis (TCA) and optimised execution solutions for their buy-side clients.

    Trading costs are usually classified into two groups:

  • • Direct trading costs are fees that must be paid to access a given market. These include brokerage fees, direct-access fees, transaction taxes, regulatory fees and liquidity fees. These fees are all relatively straightforward to understand and to measure. As a general rule, direct trading costs are of the order of 0.1–1 basis points and SEC regulatory fees are 0.01 basis points.

  • • Indirect trading costs arise due to market microstructure effects and due to the dynamics of the supply and demand. In contrast to direct trading costs, indirect trading costs are quite subtle. These costs include the bid–ask spread and impact costs. As we discussed in Chapter 16, the bid–ask spread is a consequence of the information asymmetry between different market participants and is determined endogenously by market dynamics. Spread costs are typically a few basis points in liquid markets, but can vary substantially over time, according to market conditions. Note that the tick size can have a considerable effect on the value of the spread, especially for large-tick stocks.

    • Impact costs are also a consequence of the bounded availability of liquidity. These costs also arise endogenously, but are deceptive because they are of a statistical nature. Impact is essentially invisible to the naked eye (after a buy trade, the price actually goes down about half the time, and vice-versa), and only appears clearly after careful averaging (see Chapter 12).

  • Preface

  • Jean-Philippe Bouchaud, Julius Bonart, University College London, Jonathan Donier, Martin Gould
  • Book:
    Trades, Quotes and Prices
    Published online:
    26 February 2018
    Print publication:
    22 March 2018, pp xiii-xvi

  • Summary

    The Microstructure Age

    In recent years, the availability of high-quality, high-frequency data has revolutionised the study of financial markets. By providing a detailed description not only of asset prices, but also of market participants’ actions and interactions, this wealth of information offers a new window into the inner workings of the financial ecosystem. As access to such data has become increasingly widespread, study of this field – which is widely known as market microstructure – has blossomed, resulting in a growing research community and growing literature on the topic.

    Accompanying these research efforts has been an explosion of interest from market practitioners, for whom understanding market microstructure offers many practical benefits, including managing the execution costs of trades, monitoring market impact and deriving optimal trading strategies. Similar questions are of vital importance for regulators, who seek to ensure that financial markets fulfil their core purpose of facilitating fair and orderly trade. The work of regulators has come under increasing scrutiny since the rapid uptake of high-frequency trading, which popular media outlets seem to fear and revere in ever-changing proportions. Only with a detailed knowledge of the intricate workings of financial markets can regulators tackle these challenges in a scientifically rigorous manner.

    Compared to economics and mathematical finance, the study of market microstructure is still in its infancy. Indeed, during the early stages of this project, all four authors shared concerns that the field might be too young for us to be able to produce a self-contained manuscript on the topic. To assess the lay of the land, we decided to sketch out some ideas and draft a preliminary outline. In doing so, it quickly became apparent that our concerns were ill-founded, and that although far from complete, the story of market microstructure is already extremely compelling.We hope that the present book does justice both to the main developments in the field and to our strong belief that they represent a truly new era in the understanding of financial markets.

    What is Market Microstructure?

    Before we embark on our journey, we pause for a moment to ask the question: What is market microstructure? As the name suggests, market microstructure is certainly concerned with the details of trading at the micro-scale. However, this definition does little justice to the breadth of issues and themes that this field seeks to address.

  • 10 - Epilogue

  • Edited by Domenico Delli Gatti, Università Cattolica del Sacro Cuore, Milano, Giorgio Fagiolo, Mauro Gallegati, Matteo Richiardi, Università degli Studi di Torino, Italy, Alberto Russo
  • Book:
    Agent-Based Models in Economics</I>
    Published online:
    02 March 2018
    Print publication:
    22 March 2018, pp 222-223

  • Summary

    The economic crisis the world has experienced in 2007, and is still ongoing in some parts of the globe, has been also a crisis of the economic profession. “Over the past three decades, economists have largely developed and come to rely on models that disregard key factors – including heterogeneity of decision rules, revisions of forecasting strategies, and changes in the social context – that drive outcomes in asset and other markets. It is obvious, even to the casual observer that these models fail to account for the actual evolution of the realworld economy… In our hour of greatest need, societies around the world are left to grope in the dark without a theory” (Colander et al., 2009).

    This predicament was not new. Back in 1995, Frank Hahn and Robert Solow fiercefully argued against the new classical basic methodological principle according to which “the only appropriate micro model is Walrasian … based exclusively on inter-temporal utility maximization subject to budget and technological constraints … [This model] proposes that the actual economy can be read as it is … approximating the infinite time discounted utility maximizing program of a single immortal representative agent … There is simply no possibility of coordination failures … Of course that is the economy of Dr. Pangloss and it bears little relation to the world” (Hahn, 1995, p.2). Since then, some developments of economic thought have gone in the right direction, but overall their criticisms have gone largely unnoticed. The straight jacket of axiomatic Walrasian micro-foundations has limited the scope of the research for alternatives.

    Walrasian micro-foundations should be considered the wrong answer to a right research question, the most stimulating question since the very beginning of economic thought: How does a completely decentralized economy composed of myriads of self-interested agents manages to coordinate individual actions?

    Agent-based models provide a promising tentative answer to this question. There is still a long way to go, but the path has been traced. These elements present and discuss the basic toolkit for researchers interested in building ABMs.

    If the reader arrives so far in this book, we will be happy. If, from now on, it is we who will follow the reader's progress, we will be blissfully happy.

  • 20 - The Information Content of Prices

  • from PART VIII - MARKET DYNAMICS AT THE MESO-SCALE
  • Jean-Philippe Bouchaud, Julius Bonart, University College London, Jonathan Donier, Martin Gould
  • Book:
    Trades, Quotes and Prices
    Published online:
    26 February 2018
    Print publication:
    22 March 2018, pp 366-380

  • Summary

    Most, probably, of our decisions to do something positive, the full consequences of which will be drawn out over many days to come, can only be taken as the result of animal spirits – a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities.

    (John Maynard Keynes)

    After our deep-dive into the microstructural foundations of price dynamics, the time is ripe to return to one of the most important (and one of the most contentious!) questions in financial economics: what information is contained in prices and price moves? This question has surfaced in various shapes and forms throughout the book, and we feel that it is important to devote a full chapter to summarise and clarify the issues at stake. We briefly touched on some of these points in Section 2.3. Now that we have a better handle on how markets really work at the micro-scale, we return to address this topic in detail.

    The Efficient-Market View

    Traditionally, market prices are regarded to reflect the fundamental value (of a stock, currency, commodity, etc.), up to small and short-lived mispricings. In this way, a financial market is regarded as a measurement apparatus that aggregates all private estimates of an asset's true (but hidden) value and, after a quick and efficient digestion process, provides an output price. In this view, private estimates should only evolve because of the release of a new piece of information that objectively changes the value of the asset. Prices are then martingales because (by definition) new information cannot be anticipated or predicted. In this context, neither microstructural effects nor the process of trading itself can affect prices, except perhaps on very short time scales, due to discretisation effects like the tick size.

    Major Puzzles

    This Platonian view of markets is fraught with a wide range of difficulties that have been the subject of thousands of academic papers in the last 30 years (including those with renewed insights from the perspective of market microstructure). The most well known of these puzzles are:

  • • The excess-trading puzzle: If prices really reflect value and are unpredictable, why are there still so many people obstinately trying to eke out profits from trading? […]

  • 9 - Estimation of Agent-Based Models

  • Edited by Domenico Delli Gatti, Università Cattolica del Sacro Cuore, Milano, Giorgio Fagiolo, Mauro Gallegati, Matteo Richiardi, Università degli Studi di Torino, Italy, Alberto Russo
  • Book:
    Agent-Based Models in Economics</I>
    Published online:
    02 March 2018
    Print publication:
    22 March 2018, pp 183-221

  • Summary

    ‘How absurdly simple!’ I cried.

    ‘Quite so!’ said he, a little nettled.

    ‘Every problem becomes very childish when once it is explained to you.’

    Arthur Conan Doyle. The Adventure of the Dancing Men.

    Introduction

    The ultimate test of a theory is its empirical validity, so the question whether a model ‘fits the data well’ is crucial. In the last chapter, we introduced some of the many issues involved in model evaluation. Here, we dig into the problem of tuning the values of the parameters. Moreover, we are also interested in the values of the estimated parameters for interpreting the model behaviour and to perform what-if type counterfactual (e.g., policy) evaluation exercises. Agent-based (AB) models are in general complex non-linear models, and can therefore display many different behaviours depending on the region of the parameter space being sampled. Assessing the performances of the model in the right region of the parameter space is therefore important for model evaluation. Once this region has been identified and the model deemed appropriate for its scopes, lessons can be learned about what might happen in the real world if some of the parameters changed, either as a consequence of some unforeseen developments (scenario analysis) or due to some specific actions purposefully implemented (policy analysis).

    Our goal, broadly speaking, is comparing (possibly an infinite number of) instances of the model with different parameter values and select those that fits the data better.

    Before going on, a first remark is necessary. Generally, we do not aim at calibrating or estimating a model by getting to a single optimal choice for all the parameters. In a frequentist approach, we rather look at confidence intervals – that is, ranges where the ‘true’ value of the parameters, assuming the model is correctly specified, is likely to lie – while in a Bayesian approach we focus on the posterior probability distributions for the parameters – reflecting our uncertainty about the parameters values given our prior knowledge and the information contained in the data. In this chapter we will provide examples of both approaches.

  • PART III - LIMIT ORDER BOOKS: MODELS

  • Jean-Philippe Bouchaud, Julius Bonart, University College London, Jonathan Donier, Martin Gould
  • Book:
    Trades, Quotes and Prices
    Published online:
    26 February 2018
    Print publication:
    22 March 2018, pp 75-77

  • Summary

    Introduction

    In this part, we will start our deep-dive into the mechanisms of price formation at the most microscopic scale. In the coming chapters, we will zoom in – in both space and time – to consider how the interactions between single orders contribute to the price-formation process in an LOB.

    In line with our overall effort to start with elementary models before adding any layers of complexity, we will initially focus on purely stochastic, rather than strategic, behaviours. In short, we will assume that agents’ actions are governed by simple rules that can be summarised by stochastic processes with rate parameters that depend only on the current state of the world – or, more precisely, on the current state of the LOB.

    We will start with the study of the volume dynamics of a single queue of orders at a given price. Such a queue grows due to the arrival of new limit orders, or shrinks due to limit orders being cancelled or executed against incoming market orders. We will investigate the behaviour of the average volume of a queue, its stationary distribution and its time to depletion when starting from a given size. We will repeat each of these analyses with different assumptions regarding the behaviour of the order flows, to illustrate how these quantities depend on the specific details of the modelling framework.

    From there, extending the model to account for the joint behaviour of the bid and ask queues will be a natural next step. We will introduce the “race to the bottom” between the best bid and ask queues. This race dictates whether the next price move will be upwards (if the ask depletes first) or downwards (if the bid depletes first). We will then fit these order flows directly to empirical data, in a model-free attempt to analyse the interactions between order flow and LOB state.

    We will end this modelling effort by introducing a stochastic model for the whole LOB. This approach, which is often called the Santa Fe model, will allow us to make some predictions concerning the LOB behaviour within and beyond the bid–ask spread. Comparison with real-world data will evidence that such a “zero-intelligence” approach of a purely stochastic order flow succeeds at explaining some market variables, but fails at capturing others.

Page 8 of 31