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Statistical properties of the stock order books: empirical results and models
 Quantitative Finance
"... We investigate several statistical properties of the order book of three liquid stocks of the Paris Bourse. The results are to a large degree independent of the stock studied. The most interesting features concern (i) the statistics of incoming limit order prices, which follows a powerlaw around th ..."
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Cited by 92 (5 self)
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We investigate several statistical properties of the order book of three liquid stocks of the Paris Bourse. The results are to a large degree independent of the stock studied. The most interesting features concern (i) the statistics of incoming limit order prices, which follows a powerlaw around the current price with a diverging mean; and (ii) the shape of the average order book, which can be quantitatively reproduced using a ‘zero intelligence ’ numerical model, and qualitatively predicted using a simple approximation. Financial markets offer an amazing source of detailed data on the collective behaviour of interacting agents. It is possible to find many reproducible patterns and even to perform experiments, which bring this atypical subject into the realm of experimental science. The situation is simple and well defined, since many agents, with all the same goal, trade the very same asset. As such, the statistical analysis of financial markets also offers an interesting testing ground not only for
How markets slowly digest changes in supply and demand
, 2008
"... In this article we revisit the classic problem of tatonnement in price formation from a microstructure point of view, reviewing a recent body of theoretical and empirical work explaining how fluctuations in supply and demand are slowly incorporated into prices. Because revealed market liquidity is ..."
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Cited by 81 (11 self)
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In this article we revisit the classic problem of tatonnement in price formation from a microstructure point of view, reviewing a recent body of theoretical and empirical work explaining how fluctuations in supply and demand are slowly incorporated into prices. Because revealed market liquidity is extremely low, large orders to buy or sell can only be traded incrementally, over periods of time as long as months. As a result order flow is a highly persistent longmemory process. Maintaining compatibility with market efficiency has profound consequences on price formation, on the dynamics of liquidity, and on the nature of impact. We review a body of theory that makes detailed quantitative predictions about the volume and time dependence of market impact, the bidask spread, order book dynamics, and volatility. Comparisons to data yield some encouraging successes. This framework suggests a novel interpretation of financial information, in which agents are at best only weakly informed and all have a similar and extremely noisy impact on prices. Most of the processed information appears to come from supply and demand itself, rather than from
Statistical theory of the continuous double auction. Quantitative Finance 3
"... Most modern financial markets use a continuous double auction mechanism to store and match orders and facilitate trading. In this paper we develop a microscopic dynamical statistical model for the continuous double auction under the assumption of IID random order flow, and analyse it using simulatio ..."
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Cited by 71 (5 self)
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Most modern financial markets use a continuous double auction mechanism to store and match orders and facilitate trading. In this paper we develop a microscopic dynamical statistical model for the continuous double auction under the assumption of IID random order flow, and analyse it using simulation, dimensional analysis, and theoretical tools based on mean field approximations. The model makes testable predictions for basic properties of markets, such as price volatility, the depth of stored supply and demand versus price, the bid–ask spread, the price impact function, and the time and probability of filling orders. These predictions are based on properties of order flow and the limit order book, such as share volume of market and limit orders, cancellations, typical order size, and tick size. Because these quantities can all be measured directly there are no free parameters. We show that the order size, which can be cast as a nondimensional granularity parameter, is in most cases a more significant determinant of market behaviour than tick size. We also provide an explanation for the observed highly concave nature of the price impact function. On a broader level, this work suggests how stochastic models based on zero intelligence agents may be useful to probe the structure of market institutions. Like the model of perfect rationality, a stochastic zero intelligence model can be used to make strong predictions based on a compact set of assumptions, even if these assumptions are not fully believable. 1.
More statistical properties of order books and price impact
 Physica A
"... We investigate present some new statistical properties of order books. We analyse data from the Nasdaq and investigate (a) the statistics of incoming limit order prices, (b) the shape of the average order book, and (c) the typical life time of a limit order as a function of the distance from the bes ..."
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Cited by 69 (2 self)
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We investigate present some new statistical properties of order books. We analyse data from the Nasdaq and investigate (a) the statistics of incoming limit order prices, (b) the shape of the average order book, and (c) the typical life time of a limit order as a function of the distance from the best price. We also determine the ‘price impact ’ function using French and British stocks, and find a logarithmic, rather than a powerlaw, dependence of the price response on the volume. The weak time dependence of the response function shows that the impact is, surprisingly, quasipermanent, and suggests that trading itself is interpreted by the market as new information. Many statistical properties of financial markets have already been explored, and have revealed striking similarities between very different markets (different traded assets, different geographical zones, different epochs) [1, 2, 3]. More recently, the statistics of the ‘order book’, which is the ultimate ‘microscopic ’ level of description of financial markets, has attracted considerable attention, both
Agentbased models of financial markets
 REPORTS ON PROGRESS IN PHYSICS 70
, 2007
"... This review deals with several microscopic (“agentbased”) models of financial markets which have been studied by economists and physicists over the last decade: KimMarkowitz, LevyLevySolomon, ContBouchaud, SolomonWeisbuch, LuxMarchesi, DonangeloSneppen and SolomonLevyHuang. After an overvi ..."
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Cited by 26 (0 self)
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This review deals with several microscopic (“agentbased”) models of financial markets which have been studied by economists and physicists over the last decade: KimMarkowitz, LevyLevySolomon, ContBouchaud, SolomonWeisbuch, LuxMarchesi, DonangeloSneppen and SolomonLevyHuang. After an overview of simulation approaches in financial economics, we first give a summary of the DonangeloSneppen model of monetary exchange and compare it with related models in economics literature. Our selective review then outlines the main ingredients of some influential early models of multiagent dynamics in financial markets (KimMarkowitz, LevyLevySolomon). As will be seen, these contributions draw their inspiration from the complex appearance of investors ’ interactions in reallife markets. Their main aim is to reproduce (and, thereby, provide possible explanations) for the spectacular bubbles and crashes seen in certain historical
The Virtues and Vices of Equilibrium and the future of financial economics
, 2009
"... The use of equilibrium models in economics springs from the desire for parsimonious models of economic phenomena that take human reasoning into account. This approach has been the cornerstone of modern economic theory. We explain why this is so, extolling the virtues of equilibrium theory; then we p ..."
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Cited by 24 (1 self)
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The use of equilibrium models in economics springs from the desire for parsimonious models of economic phenomena that take human reasoning into account. This approach has been the cornerstone of modern economic theory. We explain why this is so, extolling the virtues of equilibrium theory; then we present a critique and describe why this approach is inherently limited, and why economics needs to move in new directions if it is to continue to make progress. We stress that this shouldn’t be a question of dogma, and should be resolved empirically. There are situations where equilibrium models provide useful predictions and there are situations where they can never provide useful predictions. There are also many situations where the jury is still out,i.e.,where so far they fail to provide a good description of the world, but where proper extensions might change this. Our goal is to convince the skeptics that equilibrium models can be useful, but also to make traditional economists more aware of the limitations of equilibrium models.We sketch some alternative approaches and discuss why they should play an important role in
A mathematical approach to order book modeling
 International Journal of Theoretical and Applied Finance
, 2013
"... Abstract. Motivated by the desire to bridge the gap between the microscopic description of price formation (agentbased modeling) and the stochastic differential equations approach used classically to describe price evolution at macroscopic time scales, we present a mathematical study of the order b ..."
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Cited by 18 (2 self)
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Abstract. Motivated by the desire to bridge the gap between the microscopic description of price formation (agentbased modeling) and the stochastic differential equations approach used classically to describe price evolution at macroscopic time scales, we present a mathematical study of the order book as a multidimensional continuoustime Markov chain and derive several mathematical results in the case of independent Poissonian arrival times. In particular, we show that the cancellation structure is an important factor ensuring the existence of a stationary distribution and the exponential convergence towards it. We also prove, by means of the functional central limit theorem (FCLT), that the rescaledcentered price process converges to a Brownian motion. We illustrate the analysis with numerical simulation and comparison against market data.
Fundamentalists Clashing over the Book: A Study of OrderDriven Stock Markets ∗
"... Abstract. Agentbased models of market dynamics must strike a compromise between the structural assumptions that represent the trading mechanism and the behavioral assumptions that describe the rules by which traders take their decisions. We present a structurally detailed model of an orderdriven st ..."
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Cited by 11 (1 self)
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Abstract. Agentbased models of market dynamics must strike a compromise between the structural assumptions that represent the trading mechanism and the behavioral assumptions that describe the rules by which traders take their decisions. We present a structurally detailed model of an orderdriven stock market and show that a minimal set of behavioral assumptions suffices to generate a leptokurtic distribution of shortterm logreturns. This result backs up the conjecture that the emergence of some statistical properties of financial time series is due to the microstructure of stock markets.
A Random Order Placement Model of Price Formation
 in the Continuous Double Auction,” in The Economy as an Evolving Complex System, III
, 2005
"... Most modern financial markets use a continuous double auction mechanism to store and match orders and facilitate trading. In this chapter we use a microscopic dynamical statistical model for the continuous double auction under the assumption of IID random order flow. The analysis is based on simula ..."
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Cited by 4 (0 self)
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Most modern financial markets use a continuous double auction mechanism to store and match orders and facilitate trading. In this chapter we use a microscopic dynamical statistical model for the continuous double auction under the assumption of IID random order flow. The analysis is based on simulation, dimensional analysis, and theoretical tools based on meanfield approximations. The model makes testable predictions for all the basic properties of markets, including price volatility, the depth of stored supply and demand, the bidask spread, the price impact function, and the time and probability of filling orders. These predictions are based on properties of order flow such as share volume of market and limit orders, cancellations, typical order size, and tick size. Because these quantities can all be measured directly in real data sets there are no free parameters. We show that the order size, which can be cast as a nondimensional granularity parameter, is in most cases a more significant determinant of market behavior than tick size. We also provide an explanation for the observed highly concave nature of the price im