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The time function of stock price

This paper tends to define the quantitative relationship between the stock price and time as a time function. Based on the empirical evidence that the log-return of a stock is the series of white noise, a mathematical model of the integral white noise is established to describe the phenomenon of stock price movement. A deductive approach is used to derive the auto-correlation function, displacement formula and power spectral density of the stock price movement, which reveals not only the characteristics and rules of the movement but also the predictability of the stock price. The deductive fundamental is provided for the price analysis, prediction and risk management of portfolio investment.

As early as 1900, a French mathematician, who is the founder of the quantitative finance, Louis Bachelier's PhD thesis "The Theory of Speculation" derives a probabilistic method to study the time-varying rule of the stock price (Courtault et al., 2000;Jovanovic, 2012;Weatherall, 2017). Bachelier discovers that the change of the stock price is completely stochastic and defines the price at each time point as a random variable. Then, he establishes a Brownian model to describe the stock price movement.
Later in 1958, a high energy physicist from the Naval Research Laboratory (NRL) finds that the stock price Brownian model may have negative values and develops it into the geometric Brownian model (Osborne, 1959). Owing to the assumption of the random variable, the geometric Brownian model also shares a zero mean and fails to explain the long-run linear trend existing in the stock price movement. To cover this, Samuelson (1965a) embeds a linear drift in the model. A geometric Brownian model with the drift part is founded (Szpiro, 2014).
With the development of the mathematical model of stock price (Bachelier, 1964;Osborne, 1959;Samuelson, 1965a), a paradigm of quantitative finance is produced. The paradigm states that the quantitative relationship between stock prices and time can be abstracted as a random variable. Meanwhile, it is believed to use the probabilistic approach to study how the stock price will change as time changes. The definition of a random variable states that it is a real function defined in a sample space rather than a function with respect to time variables, while the relationship between the stock price and time is commonly considered as the random variable. As a result, it is mistakenly concluded that the variance of stock price is proportional to time, breaching the real life.
Besides, the random variable captures all the sample functions in all states rather than a single sample function in only one state. The danger of misusing random variables will be discussed later.
This paper redefines the quantitative relationship between stock prices and time as a certain time function, based on the observation that stock prices and time has one-toone correspondence. It uses an analytical approach to investigate the time-varying process and rules of stock price and obtains the time-and frequency-domain characteristics of stock price movement.

The integral white noise model of stock price
Let ( ) be the price of the stock at time . For each determined time , there is a certain ( ). Therefore, ( ) is a deterministic function of time .

(I) Log-return of stock
Assume that ( ) = ln ( ( )) is the logarithmic stock price (or, stock price).
Then, the log-return of stock is Empirical analyses from Working (1934), Kendall (1953), Osborne (1959), Samuelson (1965b) and Fama (1965) demonstrate that the short-term log-return of stock prices is random, and the stock price is subject to a random walk with incremental white noise.

(II) Integral white noise model
According to "the short-term log-return of stock prices is white noise", we make the following basic hypothesis (law): In the time interval (0, +∞) , the first order condition ∆ ( ) of the stock price ( ) at the minute time change ∆ is where ( ) is a white noise sample function with mean zero.
The above equation can be regarded as a discretized differential equation. Let (0) = 0, and the stock price can be calculated as Obviously, the stock price ( ) is the variable-limit integral of the white noise sample function ( ) and, therefore, the integral white noise model is non-linear and time-varying.
The equation (3), the white noise integral model, has the following characteristics: Firstly, it can accurately calculate historical data of ( ) based on historical data of ( ). Besides, the time domain and frequency domain characteristics of ( ) in the past, present and future can be analyzed according to the time domain and frequency domain characteristics of ( ) . Therefore, the mathematical model can be used to describe and interpret the phenomenon, characteristics and laws of the stock price ( ) fluctuation.

(III) Time-and frequency-domain characteristics of white noise
In the basic assumption of equation (2), the white noise sample function ( ) is defined as follows: denotes the auto-correlation function of ( ) , 0 a positive constant and ( ) a unit impulse function such that mean of the sample function is zero, and the PSD is uniformly distributed in the entire frequency axis (−∞, +∞). Note that there is no probability distribution involving the white noise sample function. The distribution of ( ) can have different forms, for example, it can have a Gaussian form and then the equation (3) is the Wiener process (Brownian motion).
Equation (4) and (5) are idealized mathematical models because its PSD is "constant" and the autocorrelation function is an "impact function". Therefore, it has the advantages of simple processing and convenient calculating. It is an essential part of mathematical phenomenon study in the theoretical research.

The system model of stock price
Equation (3)  Meanwhile, the function of integrator is to perform an integral of ( ) and yield the output. Besides, the integrator is the transfer function model of the system.
The mathematical model of the switch can be expressed as linear trend term will be formed in ( ). The existence of the switch is based on a hypothesis that a listed company may run for infinitely long and its stock may also exist forever, though, the time data of the stock price is finite, which means that the available data begin at the time when the shares issued, i.e. at time = 0, and end today, i.e. at time = . The finite data also convey that the stock price has a discrete time interval.
If we arbitrarily define that the time interval of a stock is infinite, we allow the sustainability of the company and its stock. The equity of a sustainable company is certainly risk-free in the long run, while no financial asset is completely risk-free in reality, even the government bonds. In fact, the financial market mainly consists of various risky assets.
The integrator in the system has low-pass filtering characteristics. It will amplify the low frequency components in ( ) and reduce the high frequency components. Thus, the system output signal ( ) is mainly composed of slowly varying lowfrequency components, superimposed thereon. In addition, the integrator is memory, so that the output of the current moment of the system is not only related to the input at present, but also related to the input at all times before. Therefore, stock prices have "memory" or "relevance". (7) (8)

The Characteristic of Time Domain (I) Time autocorrelation function
The autocorrelation function of stock price ( ) is given by where is the lag of time . As ( ) has a domain of [0, ], | | ≤ . Since the autocorrelation function ( ) is relevant with time , the stock price ( ) is a nonstationary stochastic process.
The graph of ( ) is demonstrated as follows.
As can be seen, the graph has a very wide pattern, which means that ( ) changes The autocorrelation function essentially describes a certain dependence between (9) Figure 2 The autocorrelation function of stock price the historical data of the stock price ( ) and the future data ( + ) , which is illustrated by equation (9). In other words, the historical data can be used to predict the future data. However, the correlation between ( + ) and ( ) decreases linearly to zero with increasing horizons .

(II) Stock Price Displacement Formula
Suppose that in physics, stock price ( ) is regarded as the displacement of the particle in the time interval [0, ], then the average speed of ( ) in the interval [0, ] is given by Thus, we can rewrite the equation (3) as The displacement of stock price ( ) is equivalent to the product of average speed ( ) ������ and the time , that is, stock price has a positive relationship with time.
( ) ������ is the arithmetic mean of the white noise sampled signal ( ) , and in physics, it represents the DC component in ( ), which reflects the deterministic part of ( ) . As time increases, the range of fluctuation of ( ) ������ will gradually decrease and ( ) ������ will stabilize. Meanwhile, ( ) increases linearly with time.

The characteristics of frequency domain
is able to provide a certain formula, for example, the equation (5)  The frequency-domain characteristics of stock price are as follows: (1) ( ) is continuous with respect to and ( ) is a nonperiodic signal in time domain.
(2) The harmonic amplitude of ( ) is inversely proportional to the frequency , indicating that the stock price is 1/f distributed and structurally invariant (selfsimilar) under the scale conversion.
( Andreadis (2000) calculates the PSD of the S&P500 index from 1988.12.1 to1998.4.1 and obtains the empirical result that the log-return of S&P500 is inversely proportional to the squared frequency, corresponding with the implication in this paper.

Conclusion
This paper defines the quantitative relationship between stock prices and time as a certain time function. Based on the basic law that "The first order condition of the log-return of stock price is equal to white noise", an integral white noise model is established to describe the stock price movement, revealing the characteristics and rules of stock price by deriving its autocorrelation function, displacement function and power spectral density (PSD). The predictability and long-run linear trend of stock price have been proved in a theoretical framework, reaching a conclusion that the PSD is inversely proportional to the squared frequency. This paper can correctly explain the past and present stock price movement and experiential facts, meanwhile, it can describe and forecast the phenomenon, characteristics and rules of the future movement.