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PurposeThis paper applies a volume-price probability wave differential equation to propose a conceptual theory and has innovative behavioral interpretations of intraday dynamic market equilibrium price, in which traders' momentum, reversal and interactive behaviors play roles.Design/methodology/approachThe authors select intraday cumulative trading volume distribution over price as revealed preferences. An equilibrium price is a price at which the corresponding cumulative trading volume achieves the maximum value. Based on the existence of the equilibrium in social finance, the authors propose a testable interacting traders' preference hypothesis without imposing the invariance criterion of rational choices. Interactively coherent preferences signify the choices subject to interactive invariance over price.FindingsThe authors find that interactive trading choices generate a constant frequency over price and intraday dynamic market equilibrium in a tug-of-war between momentum and reversal traders. The authors explain the market equilibrium through interactive, momentum and reversal traders. The intelligent interactive trading preferences are coherent and account for local dynamic market equilibrium, holistic dynamic market disequilibrium and the nonlinear and non-monotone V-shaped probability of selling over profit (BH curves).Research limitations/implicationsThe authors will understand investors' behaviors and dynamic markets through more empirical execution in the future, suggesting a unified theory available in social finance.Practical implicationsThe authors can apply the subjects' intelligent behaviors to artificial intelligence (AI), deep learning and financial technology.Social implicationsUnderstanding the behavior of interacting individuals or units will help social risk management beyond the frontiers of the financial market, such as governance in an organization, social violence in a country and COVID-19 pandemics worldwide.Originality/valueIt uncovers subjects' intelligent interactively trading behaviors.

In this paper, we study individual trading behaviors by cumulative trading volume distribution over a price range. We select intraday volume distribution as individual revealed preferences over a price range and determine beliefs by the maximum volume price in stock market. We propose a coherent preferences hypothesis in stock market where momentum, reversal, and interactive trading behaviors coexist in the framework of a price-volume probability wave differential equation. Then, we examine it by a set of explicit models of coherent preferences in interaction using tick by tick high frequency trading data of an ETF in Chinese stock market against a large number of the price-volume distributions. We evidence that subject individuals behave coherent preferences significantly. Finally, we find that individuals behave disposition effect or inverse disposition effect, and explain them by asymmetry and nonlinear coherent preferences. The bounded rational hypothesis combines individual preferences, beliefs, and cognitive limits, and has a set of unified and better models available in behavioral finance. It can help us to understand excessive price change or bubble and study nonlinear equilibrium and risk in financial market.

What is the nature of the price formation process? Is it purely random or not? If so, does that mean inefficiency? With this paper we attempt to answer those questions by providing evidence of a structural, non-random, predictable behavior in financial markets, for all asset classes, at any frequency, proposing a theoretical framework in which such evidence can coexist with a broader efficient market hypothesis. We treat the price formation process as a mixture of patterns, or two-dimensional objects, made by a non-divisible combination of price changes over time changes, where both quantities are random variables. In this context, the analysis of returns becomes an overly simplistic measure to understand market behavior, since it alters the object by imposing a deterministic structure to its time dimension, linked to the data generating process. By developing a methodology that is able to extract trend lines from prices and identify their breakout moments, we try to capture and analyze the full price-time dimensional structure of the most elementary price pattern, i.e. market trends. We show that strong memory in trends can coexist with absence of memory in returns over the same time series of prices, separating de facto the concept of efficiency from the concept of randomness. Consequently, the autocorrelation of returns may be considered a proof of market efficiency only and strictly under the agents’ rationality assumption, but it cannot be considered a proof of market randomness or unpredictability. Finally we show, through the implementation of a trading strategy, how the evidence found can be used to obtain systematic extra returns on any asset class, at any frequency, suggesting the presence of a universal, structural, intrinsic behavior in financial markets.

This paper examines the influence that trading activity has over volatility at intraday level. The correlation between the waiting times between consecutive trades and the amplitude of price movements has been analyzed, confirming the results found by Engle (2000), and Doufur and Engle (2000), according to which volatility is positively correlated with the frequency of the trades, and so, negatively correlated with the waiting times. However, this study demonstrates that these findings are strongly influenced by the presence of deterministic patterns in both waiting times and volatility. In fact, both time series are highly non-stationary and autocorrelated. In order to analyze an unbiased correlation between the two quantities, a detrending procedure with different orders of the polynomial has been used. The results show that, with an appropriate detrending order, the correlation between trading activity and volatility is reduced during the central part of the day, while completely disappearing at the beginning and at the end of the trading session. The explanation may lie in the activity of informed vs uninformed traders. As Admati and Pfleiderer (1988) pointed out, in the central part of the day the liquidity traders are staying away leaving a high proportion of informed traders, which may translate in a direct information impact from the trades to the price formation process. On the other side, the higher activity of noise traders in the first and in the last 20 minutes of the trading session may destroy the relationship between transactions’ frequency and price movements.

The properties of q-dependent cross-correlation matrices of stock market have been analyzed by using the random matrix theory and complex network. The correlation structures of the fluctuations at different magnitudes have unique properties. The cross-correlations among small fluctuations are much stronger than those among large fluctuations. The large and small fluctuations are dominated by different groups of stocks. We use complex network representation to study these q-dependent matrices and discover some new identities. By utilizing those q-dependent correlation-based networks, we are able to construct some portfolio by those most independent stocks which consistently perform the best. The optimal multifractal order for portfolio optimization is approximately $q=2$. These results have deepened our understanding about the collective behaviors of the complex financial system., 25 pages, 16 figures

In this paper, we study individual trading behaviors by cumulative trading volume distribution over a price range. We select intraday volume distribution as individual revealed preferences over a price range and determine beliefs by the maximum volume price in stock market. We propose a coherent preferences hypothesis in stock market where momentum, reversal, and interactive trading behaviors coexist in the framework of a price-volume probability wave differential equation. Then, we examine it by a set of explicit models of coherent preferences in interaction using tick by tick high frequency trading data of an ETF in Chinese stock market against a large number of the price-volume distributions. We evidence that subject individuals behave coherent preferences significantly. Finally, we find that individuals behave disposition effect or inverse disposition effect, and explain them by asymmetry and nonlinear coherent preferences. The bounded rational hypothesis combines individual preferences, beliefs, and cognitive limits, and has a set of unified and better models available in behavioral finance. It can help us to understand excessive price change or bubble and study nonlinear equilibrium and risk in financial market.