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on Market Microstructure |
| By: | Antje Fruth; Torsten Schoeneborn; Mikhail Urusov |
| Abstract: | In financial markets, liquidity is not constant over time but exhibits strong seasonal patterns. In this article we consider a limit order book model that allows for time-dependent, deterministic depth and resilience of the book and determine optimal portfolio liquidation strategies. In a first model variant, we propose a trading dependent spread that increases when market orders are matched against the order book. In this model no price manipulation occurs and the optimal strategy is of the wait region - buy region type often encountered in singular control problems. In a second model, we assume that there is no spread in the order book. Under this assumption we find that price manipulation can occur, depending on the model parameters. Even in the absence of classical price manipulation there may be transaction triggered price manipulation. In specific cases, we can state the optimal strategy in closed form. |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:1109.2631&r=mst |
| By: | Nicolas Huth (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris); Frédéric Abergel (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris) |
| Abstract: | 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. We develop a theoretical framework based on correlated point processes in order to capture the Epps effect in section 1. We show in section 2 that this model converges to correlated Brownian motions when moving to large time scales. A way of introducing non-Gaussian correlations is also discussed in section 2. We conclude by addressing the limits of this model and further research on high frequency correlation. |
| Date: | 2011 |
| URL: | http://d.repec.org/n?u=RePEc:hal:journl:hal-00621244&r=mst |
| By: | Maria Elvira Mancino (Dipartimento di Matematica per le Decisioni, Universita' degli Studi di Firenze); Simona Sanfelici (Dipartimento di Economia, Universita' di Parma) |
| Abstract: | We propose a new methodology based on Fourier analysis to estimate the fourth power of volatility function (spot quarticity) and, as a byproduct, the integrated function. We prove consistency of the proposed estimator of integrated quarticity. Further we analyze its efficiency in the presence of microstructure noise, both from a theoretical and empirical viewpoint. Extensions to higher powers of volatility and to the multivariate case are also discussed. |
| Keywords: | volatility, covariance, quarticity, microstructure, Fourier analysis |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:flo:wpaper:2011-06&r=mst |
| By: | Frédéric Abergel (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris); Aymen Jedidi (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris) |
| Abstract: | We present a mathematical study of the order book as a multidimensional continuous- time Markov chain where the order ow is modeled by independent Poisson processes. Our aim is to bridge the gap between the microscopic description of price formation (agent- based modeling), and the Stochastic Di erential Equations approach used classically to describe price evolution at macroscopic time scales. To do this, we rely on the theory of in nitesimal generators and Foster-Lyapunov stability criteria for Markov chains. We motivate our approach using an elementary example where the spread is kept constant (\perfect market making"). Then we compute the in nitesimal 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 that the order book is ergodic| in particular it has a stationary distribution|that it converges to its stationary state exponentially fast, and that the large-scale limit of the price process is a Brownian motion. |
| Date: | 2011 |
| URL: | http://d.repec.org/n?u=RePEc:hal:wpaper:hal-00621253&r=mst |
| By: | Anirban Chakraborti (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris); Ioane Muni Toke (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris); Marco Patriarca (IFISC - Instituto de Fisica Interdisciplinaire y Sistemas Complejos - Instituto de Fisica Interdisciplinaire y Sistemas Complejos); Frédéric Abergel (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris) |
| Abstract: | This article aims at reviewing recent empirical and theoretical developments usually grouped under the term Econophysics. Since its name was coined in 1995 by merging the words "Economics" and "Physics", this new interdisciplinary field has grown in various directions: theoretical macroeconomics (wealth distributions), microstructure of financial markets (order book modeling), econometrics of financial bubbles and crashes. We give a brief introduction in the first part and begin with discussing interactions between Physics, Mathematics, Economics and Finance that led to the emergence of Econophysics in the second part. Then the third part is dedicated to empirical studies revealing statistical properties of financial time series. We begin the presentation with the widely acknowledged "stylized facts" describing the distribution of the returns of financial assets: fat-tails, volatility clustering, etc. Then we show that some of these properties are directly linked to the way "time" is taken into account, and present some new remarks on this account. We continue with the statistical properties observed on order books in financial markets. For the sake of illustrations in this review, (nearly) all the stated facts are reproduced using our own high-frequency financial database. Contributions to the study of correlations of assets such as random matrix theory and graph theory are finally presented in this part. The fourth part of our review deals with models in Econophysics through the point of view of agent-based modeling. Using previous work originally presented in the fields of behavioural finance and market microstructure theory, econophysicists have developed agent-based models of orderdriven markets that are extensively reviewed here. We then turn to models of wealth distribution where an agent-based approach also prevails: kinetic theory models, and continue with game theory models and review the now classic minority games. We end this review by providing an outlook on possible directions of research. |
| Date: | 2011–06–24 |
| URL: | http://d.repec.org/n?u=RePEc:hal:journl:hal-00621058&r=mst |