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Über dieses Buch

The primary goal of the book is to present the ideas and research findings of active researchers from various communities (physicists, economists, mathematicians, financial engineers) working in the field of "Econophysics", who have undertaken the task of modelling and analyzing order-driven markets. Of primary interest in these studies are the mechanisms leading to the statistical regularities ("stylized facts") of price statistics. Results pertaining to other important issues such as market impact, the profitability of trading strategies, or mathematical models for microstructure effects, are also presented. Several leading researchers in these fields report on their recent work and also review the contemporary literature. Some historical perspectives, comments and debates on recent issues in Econophysics research are also included.



Order Book Data and Modelling


Trade-throughs: Empirical Facts and Application to Lead-lag Measures

Order splitting is a well-known behavior in trading: traders constantly scan the limit order book and choose to limit the size of their orders to the quantity available at the best limit. Order splitting allows traders not to reveal their intention to the market so as not to move too much the price against them. In this note, we focus on the other trades, called trade-throughs, which are trades that go through the best available price in the order book. We provide various statistics on trade-throughs: their liquidity, their intraday distribution and the spread relaxation that follows them. We also present a new method to get empirical distributions of lead-lag parameters between assets, sectors or even markets. This empirical study is based on tick-by-tick data of major EU and US equity futures from TRTH (Thomson Reuters Tick History) database.
Fabrizio Pomponio, Frédéric Abergel

Are the Trading Volume and the Number of Trades Distributions Universal?

Analysis of dynamical phenomena in financial markets have revealed the existence of several features that appear to be invariant with respect to details of the specific markets being considered. While some of these “stylized facts”, such as the inverse cubic law distribution of price returns indeed seem to be universal, there is less consensus about other phenomena. In particular, there has been a long-running debate in the literature about whether the distributions of trading volume V Δt and the number of trades N Δt occurring in a given time interval Δt, are universal, and whether the volume distribution is Levy-stable. In this article, we analyse data from the National Stock Exchange of India, both daily and high frequency tick-by-tick, to answer the above questions. We observe that it is difficult to fit the V Δt and N Δt distributions for all stocks using the same theoretical curve, e.g., one having a power-law form. Instead, we use the concept of the stability of a distribution under temporal aggregation of data to show that both these distributions converge towards a Gaussian when considered at a time-scale of Δt = 10 days. This appears to rule out the possibility that either of these distributions could be Levy-stable and at least for the Indian market, the claim for universality of the volume distribution does not hold.
Vikram S. Vijayaraghavan, Sitabhra Sinha

Subpenny Trading in US Equity Markets

We study sub-penny trading in the US equity markets.
Romain Delassus, Stéphane Tyč

“Market Making” in an Order Book Model and Its Impact on the Spread

It has been suggested that marked point processes might be good candidates for the modelling of financial high-frequency data. A special class of point processes, Hawkes processes, has been the subject of various investigations in the financial community. In this paper, we propose to enhance a basic zero-intelligence order book simulator with arrival times of limit and market orders following mutually (asymmetrically) exciting Hawkes processes. Modelling is based on empirical observations on time intervals between orders that we verify on several markets (equity, bond futures, index futures). We show that this simple feature enables a much more realistic treatment of the bid-ask spread of the simulated order book.
Ioane Muni Toke

Price-Time Priority and Pro Rata Matching in an Order Book Model of Financial Markets

Using our recently introduced order book model of financial markets we analyzed two different matching principles for order allocation — price-time priority and pro rata matching. Price-time priority uses the submission timestamp which prioritizes orders in the book with the same price. The order which was entered earliest at a given price limit gets executed first. Pro rata matching is used for products with low intraday volatility of best bid and best ask price. Pro rata matching ensures constant access for orders of all sizes. We demonstrate how a multiagent-based model of financial market can be used to study microscopic aspects of order books.
Tobias Preis

High-Frequency Simulations of an Order Book: a Two-scale Approach

Models of market microstructure at the order book scale can be split into two families:
  • First, the agent-based models [5] aiming at simulating a large number of agents, each of them having its utility function or feedback rule. The philosophy of this kind of modelling is similar to Minsky’s paradigm in artificial intelligence in the eighties: build each agent so that if you stealthily replace, one by one, each real person interacting in the market with such a virtual ersatz, you will finally obtain a full synthetic replica of a real market. The actual limits faced by this research programme are: first, the difficulty to rationalise and quantify the utility function of real persons, and then the computational capabilities of today’s computers. Last but not least, the lack of analytical results of this fully non-parametric approach is also a problem for a lot of applications. It is, for instance, usually difficult to know how to choose the parameters of such models to reach a given intra-day volatility, given sizes of jumps, average bid-ask spread, etc.
  • Second, the “zero intelligence”; models [9] aiming at reproducing stylised facts (Epps effect on correlations, signature plot of volatility, order book shapes, etc.) using random number generators for time between orders, order sizes, prices, etc. This approach is more oriented to “knowledge extraction” from existing recordings than the agent-based one. Its focus on stylized facts and our capability to emulate them using as simple as possible generators is close to the usual definition of “knowledgerd (following for instance Kolmogorov or Shannon in terms of complexity reduction). It succeeds in identifying features like short-term memory, Epps effect on correlations, signature plots for high-frequency volatility estimates, dominance of power laws [25], and the general profile of market impact [11], among others, that are now part of the usual benchmarks to validate any microscopic market model. The limits of this approach are: first, the usual stationarity assumptions that are made, and the difficulty of linking the microscopic characteristics with macroscopic ones, for instance linking characteristics of the underlying probability distributions to market volatility (even if recent advances have been made in this direction using Hawkes models [2] or usual distributions [7]). The search for such links is motivated by the fact that as they are probability-based, their diffusive limits (or equivalent) should behave similarly to usual quantitative models on a large scale (for instance Levy processes [24]).
Charles-Albert Lehalle, Olivier Guéant, Julien Razafinimanana

A Mathematical Approach to Order Book Modelling

We present a mathematical study of the order book as a multidimensional continuous-time Markov chain where the order flow is modelled by independent Poisson processes. Our aim is to bridge the gap between the microscopic description of price formation (agent-based modelling), and the Stochastic Differential Equations approach used classically to describe price evolution in macroscopic time scales. To do this we rely on the theory of infinitesimal generators. We motivate our approach using an elementary example where the spread is kept constant (“perfect market making”). Then we compute the infinitesimal generator associated with the order book in a general setting, and link the price dynamics to the instantaneous state of the order book. In the last section, we prove the stationarity of the order book and give some hints about the behaviour of the price process in long time scales.
Frédéric Abergel, Aymen Jedidi

Reconstructing Agents’ Strategies from Price Behavior

In the past years several Agents Based Models (ABMs) have been introduced to reproduce and interpret the main features of financial markets [7,14]. The ABMs go beyond simple differential equations with the aim of being able to address the complex phenomenology of a dynamics. This phenomenology is usually interpreted in terms of the Stylized Facts (SF) which correspond to complex correlations beyond the simple Random Walk (RW). The ABMs give the possibility to describe the intrinsic heterogeneity of the market which seems to be responsible for many of these SF [6, 12]. The main SF are the fat tails for the fluctuations of price-returns, the arbitrage condition, which implies no correlations in the price returns, and the volatility clustering which implies long memory correlations for volatility.
Valentina Alfi, Matthieu Cristelli, Luciano Pietronero, Andrea Zaccaria

Market Influence and Order Book Strategies

I review in this paper1 my findings on order driven market modeling. Following my previous works on robust agents based modeling in finance [13,5], I study specific characteristics of order book markets. By controlling the descriptive time scale of the dynamics involved, I show how market impact, linear by definition, and trading strategies lead to precise pictures for clarifying order book dynamics, consistent with what is observed empirically. I then discuss more specifically the role of market impact in the created dynamics and structure of the book and the economic implications of my studies.
The article is organized as follows. In Sect. 1, I describe financial market dynamics in an agent-based market model that clarifies the role of volatility in characteristics observed on a wide range of descriptive time scales. I define in Sect. 2 the limit order book model, agents’ strategies, and link liquidity provision to volatility estimates. I focus the analysis on the dynamics and structures of the book in Sect. 3. I discuss the economic implications of the results and draw conclusions in the last sections.
François Ghoulmié

Multi-Agent Order Book Simulation: Mono- and Multi-Asset High-Frequency Market Making Strategies

We present some simulation results on various mono- and multi-asset market making strategies. Starting with a zero-intelligence market, we gradually enhance the model by taking into account such properties as the autocorrelation of trade signs, or the existence of informed traders. We then use Monte Carlo simulations to study the effects of those properties on some elementary market making strategies. Finally, we present some possible improvements of the strategies.
Laurent Foata, Michael Vidhamali, Frédéric Abergel

High-Frequency Data and Modelling


The Nature of Price Returns During Periods of High Market Activity

By studying all the trades and best bids/asks of ultra high frequency snapshots recorded from the order books of a basket of 10 futures assets, we bring qualitative empirical evidence that the impact of a single trade depends on the intertrade time lags. We And that when the trading rate becomes faster, the return variance per trade or the impact, as measured by the price variation in the direction of the trade, strongly increases. We provide evidence that these properties persist at coarser time scales. We also show that the spread value is an increasing function of the activity. This suggests that order books are more likely empty when the trading rate is high.
Khalil Al Dayri, Emmanuel Bacry, Jean-François Muzy

Tick Size and Price Diffusion

A tick size is the smallest increment of a security price. Tick size is typically regulated by the exchange where the security is traded and it may be modified either because the exchange enforces an overall tick size change or because the price of the security is too low or too high. There is an extensive literature, partially reviewed in Sect. 2 of the present paper, on the role of tick size in the price formation process. However, the role and the importance of tick size has not been yet fully understood, as testified, for example, by a recent document of the Committee of European Securities Regulators (CESR) [1].
Gabriele La Spada, J. Doyne Farmer, Fabrizio Lillo

High Frequency Correlation Modelling

Many statistical arbitrage strategies, such as pair trading or basket trading, are based on several assets. Optimal execution routines should also take into account correlation between stocks when proceeding clients orders. However, not so much effort has been devoted to correlation modelling and only few empirical results are known about high frequency correlation. Depending on the time scale under consideration, a plausible candidate for modelling correlation should:
  • at high frequency: reproduce the Epps effect [1], take into account lead-lag relationships between assets [2]
  • at the daily scale: avoid purely Gaussian correlations [3].
Nicolas Huth, Frédéric Abergel

The Model with Uncertainty Zones for Ultra High Frequency Prices and Durations: Applications to Statistical Estimation and Mathematical Finance

The goal of this note is to describe a model for ultra high frequency prices and durations, the model with uncertainty zones developed in [27]. We also give some results from [28] and [29] which show how it can be used in practice for statistical estimation or in order to hedge derivatives. Before introducing this model, we briefly recall the classical approaches of price modelling in the so-called microstructure noise literature.
Christian Y. Robert, Mathieu Rosenbaum

Exponential Resilience and Decay of Market Impact

Assuming a particular price process, it was shown by Gatheral in [6], that a model that combines nonlinear price impact with exponential decay of market impact admits price manipulation, an undesirable feature that should lead to rejection of the model. Subsequently, Alfonsi and Schied proved in [2] that their model of the order book which has nonlinear market impact and exponential resilience, is free of price manipulation. In this paper, we show how these at-first-sight incompatible results are in reality perfectly compatible.
Jim Gatheral, Alexander Schied, Alla Slynko



Modeling the Non-Markovian, Non-stationary Scaling Dynamics of Financial Markets

A central problem of Quantitative Finance is that of formulating a probabilistic model of the time evolution of asset prices allowing reliable predictions on their future volatility. As in several natural phenomena, the predictions of such a model must be compared with the data of a single process realization in our records. In order to give statistical significance to such a comparison, assumptions of stationarity for some quantities extracted from the single historical time series, like the distribution of the returns over a given time interval, cannot be avoided. Such assumptions entail the risk of masking or misrepresenting non-stationarities of the underlying process, and of giving an incorrect account of its correlations. Here we overcome this difficulty by showing that five years of daily Euro/US-Dollar trading records in the about three hours following the New York market opening, provide a rich enough ensemble of histories. The statistics of this ensemble allows to propose and test an adequate model of the stochastic process driving the exchange rate. This turns out to be a non-Markovian, self-similar process with non-stationary returns. The empirical ensemble correlators are in agreement with the predictions of this model, which is constructed on the basis of the time-inhomogeneous, anomalous scaling obeyed by the return distribution.
Fulvio Baldovin, Dario Bovina, Francesco Camana, Attilio L. Stella

The von Neumann-Morgenstern Utility Functions with Constant Risk Aversions

Two Arrow-Pratt measures of risk aversion indicate attitudes of the individuals towards risk. In the theory of finance often these measures are assumed to be constant. Using certain intuitively reasonable conditions, this paper develops axiomatic characterizations of the utility functions for which the Arrow-Pratt measures are constant.
Satya R. Chakravarty, Debkumar Chakrabarti

Income and Expenditure Distribution. A Comparative Analysis

There are empirical evidences regrading the Pareto tail of the income distribution and the expenditure distribution. We formulate a simple economic framework to study the relation between them. We explain the Pareto tails in both the distributions with a Cobb-Douglas felicity function to describe the preferences of agents. Moreover, the Indian data suggest a thicker Pareto tail for the expenditure distribution in comparison to the income distribution. With a uniform distribution of taste parameters for various goods, we identify a process that can give rise to this empirical phenomenon. We also verify our observation with appropriate simulation results.
Kausik Gangopadhyay, Banasri Basu

Two Agent Allocation Problems and the First Best

We consider a general class of two agent allocation problems and identify the complete class of first best rules. By first best rules we mean allocation rules for which we can find efficient, strategyproof and budget balanced mechanisms. We show that the only first best rules are the fixed share allocation rules.
Manipushpak Mitra

Opinion Formation in a Heterogenous Society

Opinion formation and opinion leadership has attracted a lot of research among sociologists and physicists in the last decades. The first concept of opinion leadership goes back to Lazarsfeld et al. [8] in 1944. Larzarsfeld et al. found out that during the presidential elections in 1940 interpersonal communication showed greater influence than direct media effects. In their theory of two-step flow communication opinion leaders, who are activemedia users, select, modify and transmit information from the media to the less active part of the community. In later models sociologists gained a different view of opinion leadership by introducing the notion of public individuation. Public individuation describes how people want to differentiate and act differently from other people, see [9]. This attitude is a necessary prerequisite for an opinion leader, since she or he has to stand out against the masses. Characteristic features of opinion leaders are their high self esteem and confidence as well as their ability to withstand criticism. Although new technologies like the internet, blogs or instant messaging changed the way of communication and information dissemination globally, opinion leadership still plays a critical role in opinion formation processes.
Marie-Therese Wolfram

Opinion Formation in the Kinetic Exchange Models

We review the minimal multi-agent model (LCCC) for the collective dynamics of opinion formation in the society, which was based on the kinetic exchange dynamics studied in the context of income, money or wealth distributions in a society. This model has an intriguing spontaneous symmetry breaking transition to polarized opinion state starting from non-polarized opinion state. We also briefly review the simple variants and extensions of this model that have been proposed recently.
Anirban Chakraborti, Bikas K. Chakrabarti

Panel Discussion

In this brief section, we compile some of the remarks and comments made during the panel discussion session that took place during the conference.
Frédéric Abergel, Bikas K. Chakrabarti, Anirban Chakraborti, Manipushpak Mitra


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