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.     

Quantitative model of price diffusion and market friction based on trading as a mechanistic random process

We model trading and price formation in a market under the assumption that order arrival and cancellations are Poisson random processes. This model makes testable predictions for the most basic properties of markets, such as the diffusion rate of prices (which is the standard measure of financial risk) and the spread and price impact functions (which are the main determinants of transaction cost). Guided by dimensional analysis, simulation, and mean-field theory, we find scaling relations in terms of order flow rates. We show that even under completely random order flow the need to store supply and demand to facilitate trading induces anomalous diffusion and temporal structure in prices.     

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.     

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.     

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.     

Uncertainty relations in models of market microstructure

This paper presents a new interacting particle system and uses it as a spin model for financial market microstructure. The asymptotic analysis of this stochastic process exhibits a lower bound to the contemporaneous measurement of price and trading volume under the invariant measure in the frozen phase of the supercritical regime.     

Editorial - Constraints: From physical principles to molecular simulations and beyond

The two articles in this issue of the European Physical Journal Special Topics cover topics in Econophysics and GPU computing in the last years. In the first article [1], the formation of market prices for financial assets is described which can be understood as superposition of individual actions of market participants, in which they provide cumulative supply and demand. This concept of macroscopic properties emerging from microscopic interactions among the various subcomponents of the overall system is also well-known in statistical physics. The distribution of price changes in financial markets is clearly non-Gaussian leading to distinct features of the price process, such as scaling behavior, non-trivial correlation functions and clustered volatility. This article focuses on the analysis of financial time series and their correlations. A method is used for quantifying pattern based correlations of a time series. With this methodology, evidence is found that typical behavioral patterns of financial market participants manifest over short time scales, i.e., that reactions to given price patterns are not entirely random, but that similar price patterns also cause similar reactions. Based on the investigation of the complex correlations in financial time series, the question arises, which properties change when switching from a positive trend to a negative trend. An empirical quantification by rescaling provides the result that new price extrema coincide with a significant increase in transaction volume and a significant decrease in the length of corresponding time intervals between transactions. These findings are independent of the time scale over 9 orders of magnitude, and they exhibit characteristics which one can also find in other complex systems in nature (and in physical systems in particular). These properties are independent of the markets analyzed. Trends that exist only for a few seconds show the same characteristics as trends on time scales of several months. Thus, it is possible to study financial bubbles and their collapses in more detail, because trend switching processes occur with higher frequency on small time scales. In addition, a Monte Carlo based simulation of financial markets is analyzed and extended in order to reproduce empirical features and to gain insight into their causes. These causes include both financial market microstructure and the risk aversion of market participants.     

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.     

Crowded Trades,Market Clustering,and Price Instability

Crowded trades by similarly trading peers influence the dynamics of asset prices, possibly creating systemic risk. We propose a market clustering measure using granular trading data. For each stock, the clustering measure captures the degree of trading overlap among any two investors in that stock, based on a comparison with the expected crowding in a null model where trades are maximally random while still respecting the empirical heterogeneity of both stocks and investors. We investigate the effect of crowded trades on stock price stability and present evidence that market clustering has a causal effect on the properties of the tails of the stock return distribution, particularly the positive tail, even after controlling for commonly considered risk drivers. Reduced investor pool diversity could thus negatively affect stock price stability.     

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.     

Critical comparison of several order-book models for stock-market fluctuations

Far-from-equilibrium models of interacting particles in one dimension are used as a basis for modelling the stock-market fluctuations. Particle types and their positions are interpreted as buy and sel orders placed on a price axis in the order book. We revisit some modifications of well-known models, starting with the Bak-Paczuski-Shubik model. We look at the four decades old Stigler model and investigate its variants. One of them is the simplified version of the Genoa artificial market. The list of studied models is completed by the models of Maslov and Daniels et al. Generically, in all cases we compare the return distribution, absolute return autocorrelation and the value of the Hurst exponent. It turns out that none of the models reproduces satisfactorily all the empirical data, but the most promising candidates for further development are the Genoa artificial market and the Maslov model with moderate order evaporation.     

From market games to real-world markets

This paper uses the development of multi-agent market models to present a unified approach to the joint questions of how financial market movements may be simulated, predicted, and hedged against. We first present the results of agent-based market simulations in which traders equipped with simple buy/sell strategies and limited information compete in speculatory trading. We examine the effect of different market clearing mechanisms and show that implementation of a simple Walrasian auction leads to unstable market dynamics. We then show that a more realistic out-of-equilibrium clearing process leads to dynamics that closely resemble real financial movements, with fat-tailed price increments, clustered volatility and high volume autocorrelation. We then show that replacing the `synthetic' price history used by these simulations with data taken from real financial time-series leads to the remarkable result that the agents can collectively learn to identify moments in the market where profit is attainable. Hence on real financial data, the system as a whole can perform better than random. We then employ the formalism of Bouchaud in conjunction with agent based models to show that in general risk cannot be eliminated from trading with these models. We also show that, in the presence of transaction costs, the risk of option writing is greatly increased. This risk, and the costs, can however be reduced through the use of a delta-hedging strategy with modified, time-dependent volatility structure. Received 30 August 2000     

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.     

Self-organized model for information spread in financial markets

A self-organized model with social percolation process is proposed to describe the propagations of information for different trading ways across a social system and the automatic formation of various groups within market traders. Based on the market structure of this model, some stylized observations of real market can be reproduced, including the slow decay of volatility correlations, and the fat tail distribution of price returns which is found to cross over to an exponential-type asymptotic decay in different dimensional systems. Received 15 March 2000     

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.     

A statistical analysis of product prices in online markets

We empirically investigate fluctuations in product prices in online markets by using a tick-by-tick price data collected from a Japanese price comparison site, and find some similarities and differences between product and asset prices. The average price of a product across e-retailers behaves almost like a random walk, although the probability of price increase/decrease is higher conditional on the multiple events of price increase/decrease. This is quite similar to the property reported by previous studies about asset prices. However, we fail to find a long memory property in the volatility of product price changes. Also, we find that the price change distribution for product prices is close to an exponential distribution, rather than a power law distribution. These two findings are in a sharp contrast with the previous results regarding asset prices. We propose an interpretation that these differences may stem from the absence of speculative activities in product markets; namely, e-retailers seldom repeat buy and sell of a product, unlike traders in asset markets.     

Critical fluctuations of demand and supply

In order to describe price changes in open markets we introduce a virtual balanced price which is determined by the distribution of dealers’ expectation at a time. The dealers do not know directly the virtual balanced price but they can only guess it from the time series of market prices. By this assumption we derive a set of stochastic time evolution equations composed of the market price and the virtual balanced price as an extension of Langevin type equations.     

Effect of Trader Composition on Stock Market

In this study, we build a double auction market model, which containstwo types of agent traders, i.e., the noise traders and fundamentalists, to investigate the effect of the trader composition on the stock market. It is found that, the non-trivial Hurst exponent and the fat-tailed distribution of transaction prices can be observed at any ratio of the noise traders. Analyses on the price variation properties, including the Hurst exponent and the price variation region, show that these properties are stable when the ratio is moderate. However, the non-price variation properties, including the trading volume and the profitability of the two kinds of agents, do not keep stable untrivially in any interval of the ratio of noise traders.     

Trading model with pair pattern strategies

A simple trading model based on pair pattern strategy space with holding periods is proposed. Power-law behavior is observed for the return variance σ², the price impact H and the predictability K for both models, with linear and square root impact functions. The sum of the traders’ wealth displays a positive value for the model with a square root price impact function, and a qualitative explanation is given based on the observation of the conditional excess demand 〈A|u〉. The cumulative wealth distribution also obeys a power-law behavior with an exponent close to that of real markets. An evolutionary trading model is further proposed. The elimination mechanism effectively changes the behavior of traders, and a power-law behavior is observed in the measure of zero return distribution P(r=0). The trading model with other types of traders, e.g., traders with the MG’s strategies and producers, are also carefully studied.     

Intra-group and inter-group trading differentiation analysis: A new approach for detecting the effect of trading volume on stock market fluctuations

Analyzing statistical properties of stock market data using statistical physics has received much attention from physicists and economists in recent years. Although some statistical characteristics of stock market data such as power-low tails of stock returns have become established fact, behavior of other related variables such as trading volume are less studied. In this paper, in order to examine the impact of trading volume on statistical properties of stock market returns, different trading behavior of different traders in Tehran Stock Exchange is analyzed. We define a new coefficient which measures the equilibrium between these different forces affecting the market at any given trading day. By adjusting market returns by this coefficient, we also assessed the impact of these forces on the statistical properties of stock market returns.