Feedback and efficiency in limit order markets

A consistency criterion for price impact functions in limit order markets is proposed that prohibits chain arbitrage exploitation. Both the bid-ask spread and the feedback of sequential market orders of the same kind onto both sides of the order book are essential to ensure consistency at the smallest time scale. All the stocks investigated in Paris Stock Exchange have consistent price impact functions.     

Empirical regularities of opening call auction in Chinese stock market

We study the statistical regularities of an opening call auction using the ultra-high-frequency data of 22 liquid stocks traded on the Shenzhen Stock Exchange in 2003. The distribution of the relative price, defined as the relative difference between the order price in the opening call auction and the closing price on the last trading day, is asymmetric and that the distribution displays a sharp peak at the zero relative price and a relatively wide peak at the negative relative price. The detrended fluctuation analysis (DFA) method is adopted to investigate the long-term memory of relative order prices. We further study the statistical regularities of order sizes in the opening call auction, and observe a phenomenon of number preference, known as order size clustering. The probability density function (PDF) of order sizes could be well fitted by a q-Gamma function, and the long-term memory also exists in order sizes. In addition, both the average volume and the average number of orders decrease exponentially with the price level away from the best bid or ask price level in the limit-order book (LOB) established immediately after the opening call auction, and a price clustering phenomenon is observed.     

Empirical shape function of limit-order books in the Chinese stock market

We have analyzed the statistical probabilities of limit-order book (LOB) shape through building the book using the ultra-high-frequency data from 23 liquid stocks traded on the Shenzhen Stock Exchange in 2003. We find that the averaged LOB shape has a maximum away from the same best price for both buy and sell sides of the LOB. The LOB shape function has nice exponential form in the right tail. The buy side of the LOB is found to be abnormally thicker for the price levels close to the same best although there are much more sell orders on the book. We also find that the LOB shape functions for both buy and sell sides have periodic peaks with a period of five. The 1-min averaged volumes at fixed tick level follow log-normal distributions except for the left tails which display power-law behaviors, exhibit abnormal intraday patterns with increasing trend, and possess long memory that cannot be explained by the intraday patterns. Academic implications of our empirical results are also briefly discussed.     

Scaling and memory in the non-Poisson process of limit order cancelation

The order submission and cancelation processes are two crucial aspects in the price formation of stocks traded in order-driven markets. We investigate the dynamics of order cancelation by studying the statistical properties of inter-cancelation durations, defined as the waiting times between consecutive order cancelations of 22 liquid stocks traded on the Shenzhen Stock Exchange of China in year 2003. Three types of cancelations are considered, including cancelation of any limit orders, of buy limit orders and of sell limit orders. We find that the distributions of the inter-cancelation durations of individual stocks can be well modeled by Weibulls for each type of cancelation, and the distributions of rescaled durations of each type of cancelations exhibit a scaling behavior for different stocks. Complex intra-day patterns are also unveiled in the inter-cancelation durations. The detrended fluctuation analysis (DFA) and the multifractal DFA show that the inter-cancelation durations possess long-term memory and multifractal nature, which are not influenced by the intra-day patterns. No clear crossover phenomenon is observed in the detrended fluctuation functions with respect to the time scale. These findings indicate that the cancelation of limit orders is a non-Poisson process, which has potential worth in the construction of order-driven market models.     

Regularities and irregularities in order flow data

We identify and analyze statistical regularities and irregularities in the recent order flow of different NASDAQ stocks, focusing on the positions where orders are placed in the order book. This includes limit orders being placed outside of the spread, inside the spread and (effective) market orders. Based on the pairwise comparison of the order flow of different stocks, we perform a clustering of stocks into groups with similar behavior. This is useful to assess systemic aspects of stock price dynamics. We find that limit order placement inside the spread is strongly determined by the dynamics of the spread size. Most orders, however, arrive outside of the spread. While for some stocks order placement on or next to the quotes is dominating, deeper price levels are more important for other stocks. As market orders are usually adjusted to the quote volume, the impact of market orders depends on the order book structure, which we find to be quite diverse among the analyzed stocks as a result of the way limit order placement takes place.     

Mechanical vs. informational components of price impact

We study the problem of what causes prices to change. It is well known that trading impacts prices — orders to buy drive the price up, and orders to sell drive it down. We introduce a means of decomposing the total impact of trading into two components, defining the mechanical impact of a trading order as the change in future prices in the absence of any future changes in decision making, and the informational impact as the remainder of the total impact once mechanical impact is removed. This decomposition is performed using order book data from the London Stock Exchange. The average mechanical impact of a market order decays to zero as a function of time, at an asymptotic rate that is consistent with a power law with an exponent of roughly 1.7. In contrast the average informational impact builds to approach a constant value. Initially the impact is entirely mechanical, and is about half as big as the asymptotic informational impact. The size of the informational impact is positively correlated to mechanical impact. For cases where the mechanical impact is zero for all times, we find that the informational impact is negative, i.e. buy market orders that have no mechanical impact at all generate strong negative price responses.     

Preferred numbers and the distributions of trade sizes and trading volumes in the Chinese stock market

The distributions of trade sizes and trading volumes are investigated based on the limit order book data of 22 liquid Chinese stocks listed on the Shenzhen Stock Exchange in the whole year 2003. We observe that the size distribution of trades for individualstocks exhibits jumps, which is caused by the number preference of traders when placing orders. We analyze the applicability of the “q-Gamma” function for fitting the distribution by the Cramér-von Mises criterion. The empirical PDFs of tradingvolumes at different timescales Δt ranging from 1 min to 240 min can be well modeled. The applicability of the q-Gamma functions for multiple trades is restricted to the transaction numbers Δn≤ 8. We find that all the PDFs have power-law tails for large volumes. Using careful estimation of the average tail exponents α of the distributions of trade sizes and trading volumes, we get α> 2, well outside the Lévy regime.     

Complex stock trading network among investors

We provide an empirical investigation aimed at uncovering the statistical properties of intricate stock trading networks based on the order flow data of a highly liquid stock (Shenzhen Development Bank) listed on Shenzhen Stock Exchange during the whole year of 2003. By reconstructing the limit order book, we can extract detailed information of each executed order for each trading day and demonstrate that the trade size distributions for different trading days exhibit power-law tails and that most of the estimated power-law exponents are well within the Lévy stable regime. Based on the records of order matching among investors, we can construct a stock trading network for each trading day, in which the investors are mapped into nodes and each transaction is translated as a direct edge from the seller to the buyer with the trade size as its weight. We find that all the trading networks comprise a giant component and have power-law degree distributions and disassortative architectures. In particular, the degrees are correlated with order sizes by a power-law function. By regarding the size of executed order as its fitness, the fitness model can reproduce the empirical power-law degree distribution.     

Detrended fluctuation analysis of intertrade durations

The intraday pattern, long memory, and multifractal nature of the intertrade durations, which are defined as the waiting times between two consecutive transactions, are investigated based upon the limit order book data and order flows of 23 liquid Chinese stocks listed on the Shenzhen Stock Exchange in 2003. An inverse U-shaped intraday pattern in the intertrade durations with an abrupt drop in the first minute of the afternoon trading is observed. Based on a detrended fluctuation analysis, we find a crossover of power-law scaling behaviors for small box sizes (trade numbers in boxes) and large box sizes and strong evidence in favor of long memory in both regimes. In addition, the multifractal nature of intertrade durations in both regimes is confirmed by a multifractal detrended fluctuation analysis for individual stocks with a few exceptions in the small-duration regime. The intraday pattern has little influence on the long memory and multifractality.     

Modeling and simulation of a double auction artificial financial market

We present a double-auction artificial financial market populated by heterogeneous agents who trade one risky asset in exchange for cash. Agents issue random orders subject to budget constraints. The limit prices of orders may depend on past market volatility. Limit orders are stored in the book whereas market orders give immediate birth to transactions. We show that fat tails and volatility clustering are recovered by means of very simple assumptions. We also investigate two important stylized facts of the limit order book, i.e., the distribution of waiting times between two consecutive transactions and the instantaneous price impact function. We show both theoretically and through simulations that if the order waiting times are exponentially distributed, even trading waiting times are also exponentially distributed.     

Empirical regularities of order placement in the Chinese stock market

Using ultra-high-frequency data extracted from the order flows of 23 stocks traded on the Shenzhen Stock Exchange, we study the empirical regularities of order placement in the opening call auction, cool period and continuous auction. The distributions of relative logarithmic prices against reference prices in the three time periods are qualitatively the same with quantitative discrepancies. The order placement behavior is asymmetric between buyers and sellers and between the inside-the-book orders and outside-the-book orders. In addition, the conditional distributions of relative prices in the continuous auction are independent of the bid-ask spread and volatility. These findings are crucial to build an empirical behavioral microscopic model based on order flows for Chinese stocks.     

The structural role of weak and strong links in a financial market network

We investigate the properties of correlation based networks originating from economic complex systems, such as the network of stocks traded at the New York Stock Exchange (NYSE). The weaker links (low correlation) of the system are found to contribute to the overall connectivity of the network significantly more than the strong links (high correlation). We find that nodes connected through strong links form well defined communities. These communities are clustered together in more complex ways compared to the widely used classification according to the economic activity. We find that some companies, such as General Electric (GE), Coca Cola (KO), and others, can be involved in different communities. The communities are found to be quite stable over time. Similar results were obtained by investigating markets completely different in size and properties, such as the Athens Stock Exchange (ASE). The present method may be also useful for other networks generated through correlations.     

A precursor of market crashes: Empirical laws of Japan's internet bubble

In this paper, we quantitatively investigate the properties of a statistical ensemble of stock prices. We focus attention on the relative price defined as X(t) = S(t)/S(0), where S(0), is the stock price for an onset time of the bubble. We selected approximately 3200 stocks traded on the Japanese Stock Exchange, and formed a statistical ensemble of daily relative prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corresponding to the period in which internet Bubble formed and crashed in the Japanese stock market. We found that the upper tail of the complementary cumulative distribution function of the ensemble of the relative prices in the high value of the price is well described by a power-law distribution, P(S>x) ∼x^-α , with an exponent that moves over time. Furthermore we found that as the power-law exponents α approached two, the bubble burst. It is reasonable to suppose that it indicates that internet bubble is about to burst.     

Scaling in the distribution of intertrade durations of Chinese stocks

The distribution of intertrade durations, defined as the waiting times between two consecutive transactions, is investigated based upon the limit order book data of 23 liquid Chinese stocks listed on the Shenzhen Stock Exchange in the whole year 2003. A scaling pattern is observed in the distributions of intertrade durations, where the empirical density functions of the normalized intertrade durations of all 23 stocks collapse onto a single curve. The scaling pattern is also observed in the intertrade duration distributions for filled and partially filled trades and in the conditional distributions. The ensemble distributions for all stocks are modeled by the Weibull and the Tsallis q-exponential distributions. Maximum likelihood estimation shows that the Weibull distribution outperforms the q-exponential for not-too-large intertrade durations which account for more than 98.5% of the data. Alternatively, nonlinear least-squares estimation selects the q-exponential as a better model, in which the optimization is conducted on the distance between empirical and theoretical values of the logarithmic probability densities. The distribution of intertrade durations is Weibull followed by a power-law tail with an asymptotic tail exponent close to 3.     

Characterizing limit order prices

A computational model of a limit order book is used to study the effect of different limit order distribution offsets. Reference prices such as same side/contra side best market prices and last traded price are considered in combination with different price offset distributions. We show that when characterizing limit order prices, varying the offset distribution only produces different behavior when the reference price is the contra side best price. Irrespective of the underlying mechanisms used in computing the limit order prices, the shape of the price graph and the behavior of the average order book profile distribution are strikingly similar in all the considered reference prices/offset distributions. This implies that existing averaging methods can cancel variabilities in limit order book shape/attributes and may be misleading.     

Roughness and finite size effect in the NYSE stock-price fluctuations

We consider the roughness properties of NYSE (New York Stock Exchange) stock-price fluctuations. The statistical properties of the data are relatively homogeneous within the same day but the large jumps between different days prevent the extension of the analysis to large times. This leads to intrinsic finite size effects which alter the apparent Hurst (H) exponent. We show, by analytical methods, that finite size effects always lead to an enhancement of H. We then consider the effect of fat tails on the analysis of the roughness and show that the finite size effects are strongly enhanced by the fat tails. The non stationarity of the stock price dynamics also enhances the finite size effects which, in principle, can become important even in the asymptotic regime. We then compute the Hurst exponent for a set of stocks of the NYSE and argue that the interpretation of the value of H is highly ambiguous in view of the above results. Finally we propose an alternative determination of the roughness in terms of the fluctuations from moving averages with variable characteristic times. This permits to eliminate most of the previous problems and to characterize the roughness in useful way. In particular this approach corresponds to the automatic elimination of trends at any scale.     

Herding and zero-intelligence agents in the order book dynamics of an artificial double auction market

Effects of herding on the order book dynamics of a double auction market is studied by an agent-based model. This is done by comparing results from a zero-intelligence model and a model in which herding effect is implemented by aggregation of agents who take market orders into opinion groups. The number of opinion groups in a simulation step is determined from previous volatilities of the market as different agents compare the price change over different time intervals. Besides confirming that when herding is included the tail of the distribution of volatility is enhanced, we found several new results. First, the autocorrelation time of volatility is much shorter than the memory of most of the agents because limit orders have strong influence on the location of best bid and best ask. Second, from the relation between bid-ask imbalance and price return we find that herding reduces the chance for a small imbalance to produce a large price change. Furthermore, herding tends to decrease spread. This is because herding decreases the chance that a market order changes the size of the spread. Finally, we find that the relation between spread and volatility in our models does not agree with empirical data, this indicates a difference between agents with no strategies and agents in real financial markets.     

Mutual Information between Order Book Layers

The order book is a list of all current buy or sell orders for a given financial security. The rise of electronic stock exchanges introduced a debate about the relevance of the information it encapsulates of the activity of traders. Here, we approach this topic from a theoretical perspective, estimating the amount of mutual information between order book layers, i.e., different buy/sell layers, which are aggregated by buy/sell orders. We show that (i) layers are not independent (in the sense that the mutual information is statistically larger than zero), (ii) the mutual information between layers is small (compared to the joint entropy), and (iii) the mutual information between layers increases when comparing the uppermost layers to the deepest layers analyzed (i.e., further away from the market price). Our findings, and our method for estimating mutual information, are relevant to developing trading strategies that attempt to utilize the information content of the limit order book.     

Studies of the limit order book around large price changes

We study the dynamics of the limit order book of liquid stocks after experiencing large intra-day price changes. In the data we find large variations in several microscopical measures, e.g., the volatility the bid-ask spread, the bid-ask imbalance, the number of queuing limit orders, the activity (number and volume) of limit orders placed and canceled, etc. The relaxation of the quantities is generally very slow that can be described by a power law of exponent ≈ 0.4. We introduce a numerical model in order to understand the empirical results better. We find that with a zero intelligence deposition model of the order flow the empirical results can be reproduced qualitatively. This suggests that the slow relaxations might not be results of agents’ strategic behaviour. Studying the difference between the exponents found empirically and numerically helps us to better identify the role of strategic behaviour in the phenomena.     

Fat tails and multi-scaling in a simple model of limit order markets

We use a simple model where traders submit limit orders which are cleared in a double auction market. The limit prices are set by traders randomly, for buyers around a long-term trend and for sellers in a narrow band around their purchase price. Orders which are not filled within a specific time frame are randomly assigned a new limit price. In this framework we find evidence for the endogenous emergence of fat tails in the distribution of returns and multi-scaling whose origin is attributed to the market structure.