More statistical properties of order books and price impact

doi:10.1016/S0378-4371(02)01896-4

Physica A: Statistical Mechanics and its Applications

Volume 324, Issues 1–2, 1 June 2003, Pages 133-140

https://doi.org/10.1016/S0378-4371(02)01896-4 Get rights and content

Abstract

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 power-law, dependence of the price response on the volume. The weak time dependence of the response function shows that the impact is, surprisingly, quasi-permanent, and suggests that trading itself is interpreted by the market as new information.

Introduction

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) (for a recent review see Ref. [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 from an empirical [4], [5], [6], [7] and theoretical [5], [8], [9], [10], [11], [12], [13], [14], [15] point of view.

The order book is the list of all buy and sell limit orders, with their corresponding price and volume, at a given instant of time. We will call a(t) the ask price (best sell price) at time t and b(t) the bid price (best buy price) at time t. The midpoint m(t) is the average between the bid and the ask: m(t)=[a(t)+b(t)]/2. When a new order appears (say a buy order), it either adds to the book if it is below the ask price, or generates a trade at the ask if it is above (or equal to) the ask price (we call these ‘market orders’). The price dynamics is therefore the result of the interplay between the order book and the order flow. The study of the order book is very interesting both for academic and practical reasons. It provides intimate information on the processes of trading and price formation, and reveals a nontrivial structure of the agents expectations: as such, it is of importance to test some basic notions of economics. The practical motivations are also obvious: issues such as the market impact or the relative merit of limit versus market orders are determined by the structure and dynamics of the order book.

The main results of our investigation of some major French stocks were as follows [8]: (a) the price at which new limit orders are placed is, somewhat surprisingly, very broadly (power-law) distributed around the current bid/ask; (b) the average order book has a maximum away from the current bid/ask, and a tail reflecting the statistics of the incoming orders. We studied numerically a ‘zero intelligence’ model of order book which reproduces most of the empirical results, and proposed a simple approximation to compute analytically the characteristic humped shape of the average order book (see also Ref. [13]).

In this paper, we give the results concerning some of the Nasdaq order books, as observed on the Island ECN (see www.island.com), and discuss the similarities and differences with the French data. Second, we give some results on the price impact function that quantifies how a transaction of a given volume affects the price (on average).

Section snippets

Results on Nasdaq stocks

We denote by $b(t) Δ$ the price of a new buy limit order, and a(t)+Δ the price of a new sell limit order. A first interesting question concerns the distribution density of Δ, i.e., the distance between the current price and the incoming limit order. We found that P(Δ) for French stocks was identical for buy and sell orders (up to statistical fluctuations); and very well fitted by a single power-law: $P(Δ)∝ Δ_{0}^{μ} (1+Δ)^{1+μ}$ with an exponent μ≃0.6. This power-law was confirmed in Ref. [7] for British stocks,

The price impact function

Recently, several studies have tried to determine quantitatively how a market order of a given volume affects the price. This information is extremely important for many purposes. First, for model building: many agent-based models of markets use as a starting point a phenomenological relation between price changes and order imbalance [16], [17], [18], [19]. Second, as far as trading is concerned, the control of market impact is crucial when one wants to manage large volumes.

The most naive idea,

Acknowledgements

We thank Jean-Pierre Aguilar, Jelle Boersma, Damien Challet, J. Doyne Farmer, Xavier Gabaix, Andrew Matacz, Rosario Mantegna, Marc Mézard and Matthieu Wyart for stimulating and useful discussions.

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More statistical properties of order books and price impact

Abstract

Introduction

Section snippets

Results on Nasdaq stocks

The price impact function

Acknowledgements

Physica A

Physica A

Physica A

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An Introduction to Econophysics

Théorie des Risques Financiers, Aléa-Saclay, 1997; Theory of Financial Risks

An empirical analysis of the limit order book and the order flow in the Paris Bourse

J. Finance

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Quantitative Finance

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Physica A