The time function of stock price
An integral white noise model and its time- and frequency- domain characteristics
Shengfeng Mei
University of Glasgow
Hong Gao
Tsinghua University bstract
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 (PSD) 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.
Key words:
Stock price; frequency-domain characteristics; predictability s 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-to-one 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 𝑠𝑠 ( 𝑡𝑡 ) be the price of the stock at time 𝑡𝑡 . For each determined time 𝑡𝑡 , there is a certain 𝑠𝑠 ( 𝑡𝑡 ) . Therefore, 𝑠𝑠 ( 𝑡𝑡 ) is a deterministic function of time 𝑡𝑡 . (I) Log-return of stock
Assume that 𝑦𝑦 ( 𝑡𝑡 ) = ln ( 𝑠𝑠 ( 𝑡𝑡 )) is the logarithmic stock price (or, stock price). Then, the log-return of stock is 𝑟𝑟 ( 𝑡𝑡 ) = 𝑦𝑦 ( 𝑡𝑡 + ∆𝑡𝑡 ) − 𝑦𝑦 ( 𝑡𝑡 ). 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 ∆𝑦𝑦 ( 𝑡𝑡 ) of the stock price 𝑦𝑦 ( 𝑡𝑡 ) at the minute time change ∆𝑡𝑡 is ∆𝑦𝑦 ( 𝑡𝑡 ) = 𝑦𝑦 ( 𝑡𝑡 + ∆𝑡𝑡 ) − 𝑦𝑦 ( 𝑡𝑡 ) = 𝑥𝑥 ( 𝑡𝑡 ) where 𝑥𝑥 ( 𝑡𝑡 ) is a white noise sample function with mean zero. The above equation can be regarded as a discretized differential equation. Let 𝑦𝑦 (0) = 0 , and the stock price can be calculated as 𝑦𝑦 ( 𝑡𝑡 ) = � 𝑥𝑥 ( 𝑡𝑡 ) 𝑑𝑑𝑡𝑡 𝑡𝑡0 . Obviously, the stock price 𝑦𝑦 ( 𝑡𝑡 ) is the variable-limit integral of the white noise sample function 𝑥𝑥 ( 𝑡𝑡 ) 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 𝑦𝑦 ( 𝑡𝑡 ) based on historical data of 𝑥𝑥 ( 𝑡𝑡 ) . Besides, the time domain and frequency domain characteristics of 𝑦𝑦 ( 𝑡𝑡 ) in the past, present and future can be analyzed according to the time domain and frequency domain characteristics of 𝑥𝑥 ( 𝑡𝑡 ) . Therefore, the mathematical model can be used to (1) (2) (3) escribe and interpret the phenomenon, characteristics and laws of the stock price 𝑦𝑦 ( 𝑡𝑡 ) fluctuation. (III) Time- and frequency-domain characteristics of white noise
In the basic assumption of equation (2), the white noise sample function 𝑥𝑥 ( 𝑡𝑡 ) is defined as follows: lim 𝑇𝑇→∞ 𝑇𝑇 � 𝑥𝑥 ( 𝑡𝑡 ) 𝑑𝑑𝑡𝑡 = 0 𝑇𝑇−𝑇𝑇 𝑅𝑅 𝑥𝑥 ( 𝜏𝜏 ) = 𝑁𝑁 𝛿𝛿 ( 𝑡𝑡 ) where 𝑅𝑅 𝑥𝑥 ( ∙ ) denotes the auto-correlation function of 𝑥𝑥 ( 𝑡𝑡 ) , 𝑁𝑁 a positive constant and 𝛿𝛿 ( 𝑡𝑡 ) a unit impulse function such that 𝛿𝛿 ( 𝑡𝑡 ) = � + ∞ , 𝑡𝑡 = 00, 𝑡𝑡 ≠ � 𝛿𝛿 ( 𝑡𝑡 ) 𝑑𝑑𝑡𝑡 𝑡𝑡0 = 1 . It is clear that 𝑥𝑥 ( 𝑡𝑡 ) has autocorrelation if and only if the lag 𝜏𝜏 = 0 . In other words, 𝑥𝑥 ( 𝑡𝑡 ) has no auto-correlation as the lag 𝜏𝜏 ≠ . Thus, the signal waveform of the white noise 𝑥𝑥 ( 𝑡𝑡 ) in the time domain is a series of random pulses with infinitely narrow width and extremely fast fluctuations. The white noise 𝑥𝑥 ( 𝑡𝑡 ) is a wide-sense stationary (WSS) process. According to the Wiener-Khinchin theorem, the auto-correlation function 𝑅𝑅 𝑥𝑥 ( 𝜏𝜏 ) and the power spectral density (PSD) of 𝑥𝑥 ( 𝑡𝑡 ) form a Fourier transform pair, giving the expression of PSD as 𝑃𝑃 𝑥𝑥 ( 𝜔𝜔 ) = 𝑁𝑁 where 𝜔𝜔 = 2 𝜋𝜋𝜋𝜋 with 𝜋𝜋 representing frequency (e.g. if time is measured in seconds, then frequency is in hertz) and, thus, 𝜔𝜔 the angular frequency. 𝑁𝑁 is a positive real constant, indicating that the PSD of white noise 𝑥𝑥 ( 𝑡𝑡 ) has a uniform distribution throughout the frequency axis ( −∞ , + ∞ ) . The physical meaning of 𝑁𝑁 represents the average power produced by the white noise signal on the unit resistance. The definition of white noise above is merely defined in the time domain. The (4) (5) (6) ean 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 𝑥𝑥 ( 𝑡𝑡 ) 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) shows that the stock price 𝑦𝑦 ( 𝑡𝑡 ) is the integral of white noise sample 𝑥𝑥 ( 𝑡𝑡 ) within the interval [0, 𝑡𝑡 ] . From the perspective of signal analysis and processing, stock price 𝑦𝑦 ( 𝑡𝑡 ) is the output produced when the white noise signal 𝑥𝑥 ( 𝑡𝑡 ) excites the nonlinear time-varying system shown in the following figure, Since the PSD of white noise 𝑥𝑥 ( 𝑡𝑡 ) is constant, the PSD of output 𝑦𝑦 ( 𝑡𝑡 ) completely depends on the transfer function of the system. So far, the study of the random walk of stock price can be equally transferred to the study of the characteristics of a certain system. The system shown contains two components: a switch and an integrator. The function of switch is to cut off the white noise input signal 𝑥𝑥 ( 𝑡𝑡 ) defined in the interval ( −∞ , + ∞ ) in order to obtain the sampling signal 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) defined in the interval [0, 𝑡𝑡 ] . Meanwhile, the function of integrator is to perform an integral of 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) 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 Figure 1 The system model of stock price ( 𝑡𝑡 ) = �
1, 0 ≤ 𝑡𝑡 ≤ 𝑇𝑇 𝑜𝑜𝑡𝑡ℎ𝑒𝑒𝑟𝑟𝑒𝑒𝑖𝑖𝑠𝑠𝑒𝑒 Apparently, 𝐾𝐾 ( 𝑡𝑡 ) is a non-linear function and the closing process of the switch changes dynamically with time. Consequently, the sampling signal can be given by 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) = 𝐾𝐾 ( 𝑡𝑡 ) ∙ 𝑥𝑥 ( 𝑡𝑡 ). As can be seen above, the sampled signal 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) is the product of the white noise signal, 𝑥𝑥 ( 𝑡𝑡 ) , and the switch, 𝐾𝐾 ( 𝑡𝑡 ) . The process of truncating the 𝑥𝑥 ( 𝑡𝑡 ) into 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) is equivalent to adding a rectangular window function to 𝑥𝑥 ( 𝑡𝑡 ) . Owing that 𝐾𝐾 ( 𝑡𝑡 ) is not possible to perform a full-cycle truncation of the harmonic components of all frequencies in the white noise signal 𝑥𝑥 ( 𝑡𝑡 ) , it will generate rate leakage effect in the frequency domain and cause DC component in 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) . As the integrator operates, a linear trend term will be formed in 𝑦𝑦 ( 𝑡𝑡 ) . 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 𝑡𝑡 = 0 , and end today, i.e. at time 𝑡𝑡 = 𝑇𝑇 . 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 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) and reduce the high frequency components. Thus, the system output signal 𝑦𝑦 ( 𝑡𝑡 ) is mainly composed of slowly varying low-frequency 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 𝑦𝑦 ( 𝑡𝑡 ) is given by 𝑅𝑅 𝑦𝑦 ( 𝜏𝜏 ) = 𝑦𝑦 ( 𝑡𝑡 − 𝜏𝜏 ) 𝑦𝑦 ( 𝑡𝑡 ) ���������������� = � � 𝑥𝑥 ( 𝑢𝑢 ) 𝑥𝑥 ( 𝑣𝑣 ) 𝑑𝑑𝑢𝑢𝑑𝑑𝑣𝑣 𝑡𝑡0𝑡𝑡−𝜏𝜏0 = � � 𝑁𝑁 𝛿𝛿 ( 𝑢𝑢 − 𝑣𝑣 ) 𝑑𝑑𝑢𝑢𝑑𝑑𝑣𝑣 𝑡𝑡0𝑡𝑡−𝜏𝜏0 = � 𝑁𝑁 𝑑𝑑𝑢𝑢 = 𝑁𝑁 ( 𝑡𝑡 − 𝜏𝜏 ) where 𝜏𝜏 is the lag of time 𝑡𝑡 . As 𝑦𝑦 ( 𝑡𝑡 ) has a domain of [0, 𝑡𝑡 ] , | 𝜏𝜏 | ≤ 𝑡𝑡 . Since the autocorrelation function 𝑅𝑅 𝑦𝑦 ( 𝜏𝜏 ) is relevant with time 𝑡𝑡 , the stock price 𝑦𝑦 ( 𝑡𝑡 ) is a non-stationary stochastic process. The graph of 𝑅𝑅 𝑦𝑦 ( 𝜏𝜏 ) is demonstrated as follows. As can be seen, the graph has a very wide pattern, which means that 𝑦𝑦 ( 𝑡𝑡 ) changes slowly over time and exists to have a large inertia or correlation. It indicates that there are laws that can be identified and utilized in stock price fluctuations, which are predictable. Zhuang et al. (2001) empirically analyze the autocorrelation function based on the Shanghai and Shenzhen CSI index from 1990.12.19 to 2000.6.1 and find a similar result as Figure 2. The autocorrelation function essentially describes a certain dependence between (9) Figure 2 The autocorrelation function of stock price he historical data of the stock price 𝑦𝑦 ( 𝑡𝑡 ) and the future data 𝑦𝑦 ( 𝑡𝑡 + 𝜏𝜏 ) , which is illustrated by equation (9). In other words, the historical data can be used to predict the future data. However, the correlation between 𝑦𝑦 ( 𝑡𝑡 + 𝜏𝜏 ) and 𝑦𝑦 ( 𝑡𝑡 ) decreases linearly to zero with increasing horizons 𝜏𝜏 . (II) Stock Price Displacement Formula
Suppose that in physics, stock price 𝑦𝑦 ( 𝑡𝑡 ) is regarded as the displacement of the particle in the time interval [0, 𝑡𝑡 ] , then the average speed of 𝑦𝑦 ( 𝑡𝑡 ) in the interval [0, 𝑡𝑡 ] is given by 𝑣𝑣 ( 𝑡𝑡 ) ������ = 1 𝑡𝑡 � 𝑥𝑥 ( 𝑡𝑡 ) 𝑑𝑑𝑡𝑡 𝑡𝑡0 Thus, we can rewrite the equation (3) as 𝑦𝑦 ( 𝑡𝑡 ) = � 𝑡𝑡 � 𝑥𝑥 ( 𝑡𝑡 ) 𝑑𝑑𝑡𝑡 𝑡𝑡0 � 𝑡𝑡 = 𝑣𝑣 ( 𝑡𝑡 ) ������ ∙ 𝑡𝑡 The displacement of stock price 𝑦𝑦 ( 𝑡𝑡 ) is equivalent to the product of average speed 𝑣𝑣 ( 𝑡𝑡 ) ������ and the time 𝑡𝑡 , that is, stock price has a positive relationship with time. 𝑣𝑣 ( 𝑡𝑡 ) ������ is the arithmetic mean of the white noise sampled signal 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) , and in physics, it represents the DC component in 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) , which reflects the deterministic part of 𝑥𝑥 𝑘𝑘 ( 𝑡𝑡 ) . As time 𝑡𝑡 increases, the range of fluctuation of 𝑣𝑣 ( 𝑡𝑡 ) ������ will gradually decrease and 𝑣𝑣 ( 𝑡𝑡 ) ������ will stabilize. Meanwhile, 𝑦𝑦 ( 𝑡𝑡 ) increases linearly with time. The characteristics of frequency domain
Stock price volatility is a time-domain signal that changes over time, so it is intuitive, simple, and easy to understand and analyze the structural characteristics of stock prices in the time domain. However, time-domain analysis studies the volatility as a whole. It is unable to reflect the intensity distribution of harmonic components of different frequencies (or cycles). Besides, it cannot distinguish the effect of harmonic component of different frequencies (or cycles) on the overall volatility. Therefore, it is impossible to effectively reveal the characteristics and laws of stock price volatility. Frequency-domain analysis can fill the gap of time-domain analysis. This analysis (10) (11) s able to provide a certain formula, for example, the equation (5) express the PSD of white noise in frequency domain. Some certain rules and characteristics hid in stochastic events are relatively easy to reveal as the frequency-domain analysis decomposes the stock price volatility into harmonic components of different frequencies and studies the intensity distribution in the frequency domain to find out the main frequency components that generate stock price volatility. The analysis provides a strong evidence for clarifying the internal mechanism of stock price fluctuations, trend forecasting and risk management. Within the interval [0, 𝑡𝑡 ] , the average power of 𝑦𝑦 ( 𝑡𝑡 ) is finite and the autocorrelation function 𝑅𝑅 𝑦𝑦 ( 𝜏𝜏 ) is absolutely integrable. According to the Wiener-Khinchin theorem, the PSD of 𝑦𝑦 ( 𝑡𝑡 ) , 𝑆𝑆 𝑦𝑦 ( 𝜔𝜔 ) , is the Fourier transformation of its autocorrelation function, given the expression as 𝑆𝑆 𝑦𝑦 ( 𝜔𝜔 ) = � 𝑅𝑅 𝑦𝑦 ( 𝜏𝜏 ) 𝑒𝑒 −𝑗𝑗𝑗𝑗𝜏𝜏 𝑑𝑑𝜏𝜏 +∞−∞ = 𝑁𝑁 sin ( 𝜔𝜔𝑇𝑇 ) 𝜔𝜔 = 𝑁𝑁 𝑇𝑇 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠 ( 𝜔𝜔𝑇𝑇 ) in which 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠 ( ∙ ) is a Sinc function with formula 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠 ( 𝑥𝑥 ) = sin ( 𝑥𝑥 ) 𝑥𝑥 , 𝑥𝑥 ≠ Graphically, 𝑆𝑆 𝑦𝑦 ( 𝜔𝜔 ) 𝑁𝑁 𝑇𝑇 (12) Figure 3 The PSD of stock price he frequency-domain characteristics of stock price are as follows: (1) 𝑆𝑆 𝑦𝑦 ( 𝜔𝜔 ) is continuous with respect to 𝜔𝜔 and 𝑦𝑦 ( 𝑡𝑡 ) is a nonperiodic signal in time domain. (2) The harmonic amplitude of 𝑦𝑦 ( 𝑡𝑡 ) is inversely proportional to the frequency 𝜔𝜔 , indicating that the stock price is 1/f distributed and structurally invariant (self-similar) under the scale conversion. (3) 𝑆𝑆 𝑦𝑦 (0) = 𝑁𝑁 𝑇𝑇 means that in the time domain, 𝑦𝑦 ( 𝑡𝑡 ) has a linear trend line proportional with time 𝑡𝑡 and 𝑦𝑦 ( 𝑡𝑡 ) fluctuates around the line. (4) The main lobe of Sinc function ( −𝜋𝜋 / 𝑇𝑇 ≤ 𝜔𝜔 ≤ 𝜋𝜋 / 𝑇𝑇 ) concentrates more than 90% of the fluctuation energy, so 𝑦𝑦 ( 𝑡𝑡 ) is, in principle, pink noise, revealing the fact that the volatility of stock price is mainly low-frequency, i.e. the movement has a large inertance (or correlation), ensuring the original trend and state under certain time and conditions. 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. eferences
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