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.     

Long term memory in extreme returns of financial time series

It is well known that while daily price returns of financial markets are uncorrelated, their absolute values (‘volatility’) are long-term correlated. Here we provide evidence that certain subsequences of the returns themselves also exhibit long-term memory. These subsequences consist of maxima (or minima) of returns in consecutive time windows of R days. Our analysis shows that for both stocks and currency exchange rates, long-term correlations are significant for R≥4. We argue that this long-term memory which is similar to that observed in volatility clustering sheds further insight on price dynamics that might be used for risk estimation.     

Coupled continuous-time random walk approach to the Rachev-Rüschendorf model for financial data

In this paper we expand the Rachev-Rüschendorf asset-pricing model introducing a coupled continuous-time-random-walk-(CTRW)-like form of the random number of price changes. Such a form results from the concept of the random clustering procedure (that resembles the coarse-graining methods of statistical physics) and, on the other hand, indicates applicability of the CTRW idea, widely used in physics to model anomalous diffusion, for describing financial markets. In the framework of the proposed model we derive the limiting distributions of log-returns and the corresponding pricing formulas for European call option. In order to illustrate the obtained theoretical results we present their fitting with several sets of financial data.     

Analyzing the prices of the most expensive sheet iron all over the world: Modeling, prediction and regime change

The private car license plates issued in Shanghai are bestowed the title of “the most expensive sheet iron all over the world”, more expensive than gold. A citizen has to bid in a monthly auction to obtain a license plate for his new private car. We perform statistical analysis to investigate the influence of the minimal price P_min of the bidding winners, the quota of private car license plates, the number of bidders, as well as two external shocks including the legality debate of the auction in 2004 and the auction regime reform in January 2008 on the average price of all bidding winners. It is found that the legality debate of the auction had marginal transient impact on the average price in a short time period. In contrast, the change of the auction rules has significant permanent influence on the average price, which reduces the price by about 3020 yuan Renminbi. It means that the average price exhibits nonlinear behaviors with a regime change. The evolution of the average price is independent of the number of bidders in both regimes. In the early regime before January 2008, the average price was influenced only by the minimal price P_min in the preceding month with a positive correlation. In the current regime since January 2008, the average price is positively correlated with the minimal price and the quota in the preceding month and negatively correlated with the quota in the same month. We test the predictive power of the two models using 2-year and 3-year moving windows and find that the latter outperforms the former. It seems that the auction market becomes more efficient after the auction reform since the prediction error increases.     

Tick size and stock returns

Tick size is an important aspect of the micro-structural level organization of financial markets. It is the smallest institutionally allowed price increment, has a direct bearing on the bid-ask spread, influences the strategy of trading order placement in electronic markets, affects the price formation mechanism, and appears to be related to the long-term memory of volatility clustering. In this paper we investigate the impact of tick size on stock returns. We start with a simple simulation to demonstrate how continuous returns become distorted after confining the price to a discrete grid governed by the tick size. We then move on to a novel experimental set-up that combines decimalization pilot programs and cross-listed stocks in New York and Toronto. This allows us to observe a set of stocks traded simultaneously under two different ticks while holding all security-specific characteristics fixed. We then study the normality of the return distributions and carry out fits to the chosen distribution models. Our empirical findings are somewhat mixed and in some cases appear to challenge the simulation results.     

A unified econophysics explanation for the power-law exponents of stock market activity

We survey a theory (first sketched in Nature in 2003, then fleshed out in the Quarterly Journal of Economics in 2006) of the economic underpinnings of the fat-tailed distributions of a number of financial variables, such as returns and trading volume. Our theory posits that they have a common origin in the strategic trading behavior of very large financial institutions in a relatively illiquid market. We show how the fat-tailed distribution of fund sizes can indeed generate extreme returns and volumes, even in the absence of fundamental news. Moreover, we are able to replicate the individually different empirical values of the power-law exponents for each distribution: 3 for returns, 3/2 for volumes, 1 for the assets under management of large investors. Large investors moderate their trades to reduce their price impact; coupled with a concave price impact function, this leads to volumes being more fat-tailed than returns but less fat-tailed than fund sizes. The trades of large institutions also offer a unified explanation for apparently disconnected empirical regularities that are otherwise a challenge for economic theory.     

On the probability distribution of stock returns in the Mike-Farmer model

Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not only the well-known power-law tails, together with several other important stylized facts. There are three key ingredients in the Mike-Farmer (MF) model: the long memory of order signs characterized by the Hurst index H_s, the distribution of relative order prices x in reference to the same best price described by a Student distribution (or Tsallis’ q-Gaussian), and the dynamics of order cancelation. They showed that different values of the Hurst index H_s and the freedom degree α_x of the Student distribution can always produce power-law tails in the return distribution f_r(r) with different tail exponent α_r. In this paper, we study the origin of the power-law tails of the return distribution f_r(r) in the MF model, based on extensive simulations with different combinations of the left part L(x) for x < 0 and the right part R(x) for x > 0 of f_x(x). We find that power-law tails appear only when L(x) has a power-law tail, no matter R(x) has a power-law tail or not. In addition, we find that the distributions of returns in the MF model at different timescales can be well modeled by the Student distributions, whose tail exponents are close to the well-known cubic law and increase with the timescale.     

Effect of impurity clustering on ferromagnetism in dilute magnetic semiconductors

We establish a model to investigate the effect of clustering of impurities on the ferromagnetism in dilute magnetic semiconductors (DMS). The Curie temperature Tc is calculated by the mean-field theory on a lattice with randomly distributed clusters of magnetic impurities which are interacting with each other by carrier mediated RKKY exchange coupling together with the nearest-neighbor (NN) direct exchange interaction. We consider different types and sizes of the clusters and find that the clustering of impurities can either enhance or reduce Tc, depending on the type and strength of the NN exchange interaction. If the NN interaction is antiferromagnetic and strong compared with the RKKY interaction, the clustering will reduce Tc. On the other hand, if it is ferromagnetic interaction or weak antiferromagnetic one, the clustering can enhance Tc. The trend of enhancing Tc is magnified if the average size of clusters increases. The clustering also changes the distribution of polarizations of impurities. The obtained results provide natural explanations on the fact that the ferromagnetism of DMS samples depends on the preparing and annealing processes even though the density of the magnetic impurities is kept the same.     

Continuous Time Random Walk with Correlated Waiting Times. The Crucial Role of Inter-Trade Times in Volatility Clustering

In many physical, social, and economic phenomena, we observe changes in a studied quantity only in discrete, irregularly distributed points in time. The stochastic process usually applied to describe this kind of variable is the continuous-time random walk (CTRW). Despite the popularity of these types of stochastic processes and strong empirical motivation, models with a long-term memory within the sequence of time intervals between observations are rare in the physics literature. Here, we fill this gap by introducing a new family of CTRWs. The memory is introduced to the model by assuming that many consecutive time intervals can be the same. Surprisingly, in this process we can observe a slowly decaying nonlinear autocorrelation function without a fat-tailed distribution of time intervals. Our model, applied to high-frequency stock market data, can successfully describe the slope of decay of the nonlinear autocorrelation function of stock market returns. We achieve this result without imposing any dependence between consecutive price changes. This proves the crucial role of inter-event times in the volatility clustering phenomenon observed in all stock markets.     

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.     

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.     

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.     

Long memory properties in return and volatility: Evidence from the Korean stock market

In this paper, we study the dual long memory property of the Korean stock market. For this purpose, the ARFIMA-FIGARCH model is applied to two daily Korean stock price indices (KOSPI and KOSDAQ). Our empirical results indicate that long memory dynamics in the returns and volatility can be adequately estimated by the joint ARFIMA-FIGARCH model. We also found that the assumption of a skewed Student-t distribution is better for incorporating the tendency of asymmetric leptokurtosis in a return distribution.     

Resonances in a circular dielectric cavity

We study resonance distributions in a circular dielectric cavity. It is shown that the decay-rate distribution has a peak structure and the details of the peak are consistent with the classical survival probability time distribution. We also investigate the behavior of the complex resonance positions at the small opening limit (n→∞, n is the refractive index of the cavity). At the large n limit, the real part of complex resonance positions approaches the solutions with different m of Dirichlet problem with a scale n⁻² and the imaginary part goes zero as n−2m for TM and n−2(m+1) for TE polarization, where m is the order of the resonance.     

Frequency clustering of coupled phase oscillators on small-world networks

We analyze the phenomenon of frequency clustering in a system of coupled phase oscillators. The oscillators, which in the absence of coupling have uniformly distributed natural frequencies, are coupled through a small-world network, built according to the Watts-Strogatz model. We study the time evolution and determine variations in the transient times depending on the disorder of the network and on the coupling strength. We investigate the effects of fluctuations in the average frequencies, and discuss the definition of the threshold for synchronization. We characterize the structure of clusters and the distribution of cluster sizes in the synchronization transition, and define suitable order parameters to describe the aggregation of the oscillators as the network disorder and the coupling strength change. The non-monotonic behavior observed in some order parameters is related to fluctuations in the mean frequencies.     

Anderson transition in 1D systems with spatial disorder

A simple Kronig-Penney model for 1D mesoscopic systems with δ peak potentials is used to study numerically the influence of spatial disorder on conductance fluctuations and distribution at different regimes. The Lévy laws are used to investigate the statistical properties of the eigenstates. It is found that an Anderson transition occurs even in 1D meaning that the disorder can also provide constructive quantum interferences. The critical disorder W_c for this transition is estimated. In these 1D systems, the metallic phase is well characterized by a Gaussian conductance distribution. Indeed, the results relative to conductance distribution are in good agreement with the previous works in 2D and 3D systems for other models. At this transition, the conductance probability distribution has a system size independent shape with large fluctuations in good agreement with previous works.     

Volatility return intervals analysis of the Japanese market

We investigate scaling and memory effects in return intervals between price volatilities above a certain threshold q for the Japanese stock market using daily and intraday data sets. We find that the distribution of return intervals can be approximated by a scaling function that depends only on the ratio between the return interval τ and its mean 〈τ〉. We also find memory effects such that a large (or small) return interval follows a large (or small) interval by investigating the conditional distribution and mean return interval. The results are similar to previous studies of other markets and indicate that similar statistical features appear in different financial markets. We also compare our results between the period before and after the big crash at the end of 1989. We find that scaling and memory effects of the return intervals show similar features although the statistical properties of the returns are different.     

The space-time pattern of price waves

We investigate the way price fluctuations are transmitted between spatially separated markets. More specifically we show that the correlation patterns of wheat prices exhibit definite regularities some of which appear to be at variance with intuitive reasoning. Such patterns can be explained in the framework of a wave propagation model based on the so-called spatial arbitrage assumption. In 19th century France the velocity of price waves was of the order of 100 km/month. The economic implications of such an order of magnitude are discussed. In the concluding section we emphasize that what gives this problem its importance is its relative “simplicity”, a word for which we propose an operational definition. Received 1 September 1998 and Received in final form 3 October 1998     

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.     

Market memory and fat tail consequences in option pricing on the expOU stochastic volatility model

The expOU stochastic volatility model is capable of reproducing fairly well most important statistical properties of financial markets daily data. Among them, the presence of multiple time scales in the volatility autocorrelation is perhaps the most relevant which makes appear fat tails in the return distributions. This paper wants to go further on with the expOU model we have studied in Ref. [J. Masoliver, J. Perelló, Quant. Finance 6 (2006) 423] by exploring an aspect of practical interest. Having as a benchmark the parameters estimated from the Dow Jones daily data, we want to compute the price for the European option. This is actually done by Monte Carlo, running a large number of simulations. Our main interest is to “see” the effects of a long-range market memory from our expOU model in its subsequent European call option. We pay attention to the effects of the existence of a broad range of time scales in the volatility. We find that a richer set of time scales brings the price of the option higher. This appears in clear contrast to the presence of memory in the price itself which makes the price of the option cheaper.