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Motivated by how transaction amount constrain trading volume and price volatility in stock market, we, in this paper, study the relation between volume and price if amount of transaction is given. We find that accumulative trading volume gradually emerges a kurtosis near the price mean value over a trading price range when it takes a longer trading time, regardless of actual price fluctuation path, time series, or total transaction volume in the time interval. To explain the volume–price behavior, we, in terms of physics, propose a transaction energy hypothesis, derive a time-independent transaction volume–price probability wave equation, and get two sets of analytical volume distribution eigenfunctions over a trading price range. By empiric test, we show the existence of coherence in stock market and demonstrate the model validation at this early stage. The volume–price behaves like a probability wave.

Although there are many trading price models in financial market, none of them has the explicit price formation mechanism that is expressed by an analytical expression. Fama [1] and Ross [2] launched efficient market hypothesis and arbitrage pricing theory, respectively, based on rational trading assumption. Black and Scholes [3], together with Merton [4], derived a Black–Scholes–Merton model in terms of Samuelson's log-normal process or economic Brownian motion [5] that could be traced to Bachelier's dissertation regarding an option pricing problem [6]. In addition, Engle [7] formulated ARCH model, which was later developed to GARCH model by Bollerslev [8], to estimate price volatility error or non-linear item. In recent years, some econophysicists begin using formulation in physics to develop asset pricing models in financial market. For example, McCauley and Gunaratne [9] showed how the Fokker-Plank formulation of fluctuations can be used with a local volatility to generate an exponential distribution for asset return. In the past 20 years, there was a growing body of research studying price and volume. Gallant et al. [10] undertook a comprehensive investigation of price and volume co-movement using daily New York Stock Exchange data from 1928 to 1987. Gervais et al. [11] claimed the existence of high-volume return premium in stock market. Moreover, Zhang [12], an econophysicist, presented an argument for a square-root relationship between price changes and demand. Over the last few years, spin models are used in studying price and volume as the most popular models in econophysics [13]. Plerou et al. [14] applied a spin model and empirically addressed how stock prices respond to changes in demand. They found that large price fluctuations occur when demand is very small. Ausloos and Ivanova [15] studied price and volume by introducing the notion of a generalized kinetic energy. A generalized momentum is also borrowed from classical mechanics. It is defined as the product of normalized volume transaction and the average rate of price change during a price moving average period. They emphasized at the close that these concepts might also serve in a dynamic equation framework. Wang and Pandey [16] followed the same terminology with somewhat different definitions. They defined trading momentum as the product of relative price velocity and a time-dependent “mass”, a normalized trading volume in a time interval, i.e., the volume liquidity. Up to date, the literature on price and volume mainly focuses on the correlation between return and total volume (over a trading price range) in a given time interval. Some scholars attempted to explain the behavior of price and volume. Admati and Pfleiderer [17] developed a theory in which concentrated-trading patterns arise endogenously as a result of the strategic behavior of liquidity traders and informed traders. Wang [18] used ICAPM to establish a theoretical links between prices and volume. Econophysicists, for example, Gabaix et al. [19] proposed a theory to provide a unified way to understand the power-law tailed distributions of return and volume, the non-normal distributions that have been caught much attention by econophysicists since Mandelbrot's finding [20]. Current theories credited the correlation between price and volume to a variety of factors, for examples (optimal) trading motive and information quality etc. “What is surprising is how little we really know about trading volume” [21]. Soros [22] guessed: In natural sciences, the phenomenon most similar to that in financial economics probably exists in quantum physics, in which scientific observation generates Heisenberg's uncertainty principle… Unfortunately, he added, it is impossible for economics to become science… Among econophysicists, however, Schaden [23] prudently discussed the possible generic aspects of quantum finance to model secondary financial markets and the challenges we probably had to face in this potential interdisciplinary field. Piotrowski and Sladkowski [24] published a series of papers on quantum finance and currently proposed the price model that uses complex amplitudes whose squared modules describe price movement probabilities, inspired by quantum mechanical evolution of physical particles. Kleinert [25] applied path integrals to the price fluctuation of assets, considering the prices as a function of time. Baaquie et al. [26] developed a derivative pricing model based on a Hamiltonian formulation, and wrote that it was too difficult to solve Merton–Garman Hamiltonian analytically in most pricing problems [27]. Jimenez and Moya [28], on the other hand, showed that it is possible to obtain quantum mechanics principles using information and game theory, etc. These researches are based on existed formulation and principle in quantum mechanics. There is a celebrated dictum in Walls Street: Cash is king. Inspired by Soros's guess and motivated by how transaction amount constrain trading volume and price volatility in stock market, we study the relation between volume and price through amount of transaction in terms of physics. Stock market appears a complex system because of a variety of interacted and coupled trading agents in this open and fully competitive market. Thus, it is probably a key for us to model it successfully how we observe the system and find a simplified methodology. Price is volatile upward and downward to its mean value in intraday transactions on individual stock. We observe that accumulative trading volume gradually emerges a kurtosis near the price mean value over a trading price range when it takes a longer trading time, regardless of actual price fluctuation path, time series, or total transaction volume in the time interval [29] and [30]. Moreover, the volumes are not distributed normally. Whereas some of the distributions appear to be normal, others appear to be wave, and the others exhibit to be exponent. These phenomena cannot be explained by a current economic and finance mainstream theory—both a rational trading assumption and a price volatility random walk hypothesis. Is the volume–price behavior driven by a kind of restoring force? Unlike previous literature, this paper investigates how total trading volume distributes over a trading price range in a given time interval and why it does in market dynamic perspective. Solving the volume–price probability wave equation that is derived from a transaction energy hypothesis rather than an existed formulation in finance or physics, we find two sets of analytical volume distribution eigenfunctions over a trading price range in a given time interval (volume liquidity distribution eigenfunctions). By empiric test, we demonstrate our model validation at this early stage. The rest of the paper is organized as follows. In Section 2, we use the absolute of zero-order Bessel eigenfunction model to fit and test the volume distributions over a trading price range and draw a major conclusion that our observation holds true. In order to explain our observation and volume–price behavior, we propose a transaction energy hypothesis and use a potential function to describe price volatility in Section 3; We, then, assume that there is the restoring force that drives trading towards an equilibrium price, and identify the linear potential (energy) that can represent for effective supply–demand quantity restoring force in stock market in Section 4; Section 5 constructs the Hamiltonian that is equal to the sum of transaction dynamic energy and potential energy, derives a time-independent security transaction volume–price probability wave equation, and gets two sets of analytical solutions; in Section 6 are analyses, discussions, and possible applications (practical uses); and finally are summaries and conclusions.

In this subsection, we mainly discuss transaction momentum, two sets of analytical solutions, and transaction energy hypothesis (together with methodology). 6.2.1. Transaction momentum (the volume liquidity) and transaction force According to Eq. (19), transaction momentum does be transaction volume liquidity at a price. In stock market, there is no price change at all if there is no actual trading (volume). The volume plays an exclusive role in determining price change and its liquidity controls the rise or drop of an equilibrium price by its weight. This can explain why large price fluctuations occur when demand is very small [14]. When demand is very small, a small trading volume can produce big impact on its equilibrium price, and even cause its step or jump change. The market is much more unstable. In addition, transaction momentum rate or transaction volume acceleration is transaction force. The force is defined by Eq. (20). It describes how we make trading volume, namely, trading behavior in stock market. A skeptical reader may argue why transaction volume acceleration is not acceleration. Of course, one may define the volume acceleration to be acceleration rather than force. Let us see what happens if we do so. (1) The transaction volume acceleration is acceleration if and only if we choose the volume as an independent generalized coordinate. In this way, Eq. (8) is not a holonomic constraint in the study of volume and price because it includes generalized velocity. Not all generalized force can be derivable by differentiating (to the volume) function(s) in this non-holonomic constraint system [32]. (2) If the volume is chosen as an independent generalized coordinate, we would study price as a function of volume. It would be another research topic. Osborne [36], a pioneer in econophysics, observed that price as a function of volume does not exist empirically, and then explained why volume is the function of price, and this is not invertible. McCauley [37] showed that price as a function of volume does not exist mathematically. (3) If the volume is chosen as an independent generalized coordinate, we probably have to study the correlation between the volume and price. In this approach, no analytical solution can be obtained. (4) If the volume is chosen as an independent generalized coordinate, it would be inconvenient for us to observe the volume–price behavior because we choose price as an independent coordinate in financial market. Thus, it is reasonable for us to choose price rather than volume as an independent variable. The transaction volume acceleration is transaction force rather than acceleration in Eq. (4)