ORIGINAL RESEARCH article
Sec. Social Physics
Model of Continuous Random Cascade Processes in Financial Markets
- 1Department of Economics, Seijo University, Tokyo, Japan
- 2Graduate School of Integrated Basic Sciences, Nihon University, Tokyo, Japan
This article presents a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade: one multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, the model parameters are estimated by the application to an actual stock price time series. Numerical calculation of the Fokker–Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters. The results reproduce the probability density function of the empirical volatility, the multifractality of the time series, and other empirical facts.
In financial time series, past coarse-grained measures of volatility correlate better to future fine-scale volatility than the reverse process. Such a causal structure of financial time series was first reported by Müller et al. . Since then, the causal structure between time scales, the flow of information from a long-term to a short-term scale, was investigated empirically in financial markets; it has been supported by multiple studies [2, 3] as a stylized fact of financial time series . The asymmetric flow of information resembles an energy cascade found in conditions of turbulence. In a developed turbulent flow, the energy injected from the outside at macroscopic spatial scales is transferred to smaller scales and finally dissipated as heat at microscopic spatial scales [5–9]. Gashghaie et al. investigated details of the self-similar transformation rule of the probability density function of price fluctuations and the nonlinear scaling law of the structure function (nth moment of fluctuations), signifying the multifractality of the time series, in their study of the time series of foreign exchange. They pointed out the similarity of price changes in the financial time series to the velocity difference between two spatial points in turbulence [10, 11]. The intermittency in turbulence is a phenomenon characterized by the sudden temporal change of the statistical feature of fluctuations and the spatial coexistence of large and small fluctuations. Such intermittency, which is frequently encountered in heterogeneous complex systems, is well known in financial markets as volatility clustering [4, 12]. Intermittency at each time scale produces a characteristic hierarchical structure designated as multifractality [8, 9].
In the developed turbulence, the process by which mechanically generated vortices on a macroscale deform and destabilize according to the Navier–Stokes equation and then split into smaller vortices is regarded as an energy cascade. A similar idea of modeling multifractal time series by a recursive random multiplication process from a coarse-grained scale to a microscopic scale has offered an attractive means of describing financial time series [13, 14]. Chen et al. verified the statistics of multiplier factors in the random multiplication process of turbulent flow by empirical studies using measured data and numerical experiments of Navier–Stokes equations . Results show that the multiplier factors connecting two adjacent layers follow a Cauchy distribution in which all moments diverge and show that they are not independent. They show strongly negative correlation between the multiplier factors of adjacent layers. The authors verified the statistics of multipliers calculated backward from actual stock price fluctuations, finding a Cauchy distribution of multiplier factors and also the strongly negative correlation between the multiplier factors in financial markets. Results show that the discrete cascade model using the random multiplication process did not reproduce the statistical property of the multiplier factors. Therefore, as an alternative model, a discrete random multiplicative cascade process with additional additive stochastic processes [16–18], or a model formulated as the Fokker–Planck equation considering the cascade process as a continuous Markov process [19–23] was proposed. Those models have been applied to stock market or foreign exchange market data, yielding empirical results including the statistics of multipliers.
This study examines a continuous cascade model of volatility formulated as a stochastic differential equation including two independent modes of Brownian motion: one has multiplicative coupling with volatility; the other has additive coupling as in the discrete random multiplicative cascade process with additional additive stochastic processes described above. The model parameters are estimated by its application to the stock price time series. Numerical calculation of the Fokker–Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters resulting in successful reproduction of the pdf of the empirical volatility and the multifractality of the time series.
Materials and Methods
Continuous Random Cascade Model
Stochastic Differential Equation
These analyses examine the following wavelet transform of the variation of the logarithmic stock price denoted by
where the function ψ is designated as the analyzing wavelet. When using the delta function
In general, by using the nth derivative of the function having asymptotic fast decay as the analyzing wavelet, one can remove the local trend of mth order
In actual financial market, the price fluctuation is nonstationary and the volatility is not observable. The quantity used herein is the absolute value of the wavelet transform
The following stochastic equation is used to start.
In that equation,
The solution is obtained easily using Ito’s formula as .
The power law behavior of the qth moment
The multifractality of signal
We introduce an additional additive stochastic process as we have done in the discrete cascade model. We first consider the following stochastic differential equation.
The equation is produced on the assumption that Brownian motions
To solve the stochastic differential Eq. 7, we consider the following stochastic differential equation:
the solution of Eq. 7 is expressed as shown below:
Statistics of Multipliers
We have mentioned the statistics of multipliers in Introduction:
(1) The stochastic multiplier
(2) When considering the three scales
Here, we show property (1) and infer the existence of correlation between adjacent multipliers under some reasonable approximations. The parameter
Therefore, the difference
By defining the differences
Relation to Discrete Random Cascade Model
we obtain the discrete random cascade equation as
shows that deviation of the quadratic curve from the origin results from the parameter
Constraint Condition From the pdf of
A remarkable feature of the probability density function (pdf) of the quantity
FIGURE 1. Scaling properties of
FIGURE 2. Results of multifractal analysis. (A)
The additional additive stochastic process in model Eq. 7 is expected to be a small perturbation to basic model Eq. 4 to avoid violating multifractality. We also impose the following condition for all scales s:
The power law scaling shown in Figure 1,
and condition Eq. 16 show the following constraint condition:
We can derive the Fokker–Planck equation for the stochastic process
Therein, the functions
The kth moment of the change
Therein, we used the identity
We analyze the normalized average of the logarithmic stock prices of the constituent issues of the FTSE 100 Index listed on the London Stock Exchange for November 2007 through January 2009, which includes the Lehman shock of September 15, 2008 and the market crash of October 8, 2008.
First, we calculate the average deseasonalized return of each issue
The data size L is
FIGURE 3. Regression of
First, we analyze the multifractal properties of the path
where x in a neighborhood of
For multifractal paths, the Hölder exponent α is distributed in a range; for paths of the Brownian motion, which are fractal,
Muzy, Bacry, and Arneodo proposed the wavelet transform modulus maxima (WTMM) method based on continuous wavelet transform of function to calculate the singular spectrum
aA and γM
where the standard errors are in parentheses. The estimated exponent
where the standard error is the value in the parenthesis.
FIGURE 4. Estimation of the parameter
FIGURE 5. Estimation of the parameter
FIGURE 6. Regression of
bA and σM
Similarly, we estimate parameters
As shown in Figure 6A, the second moment is well fitted by a quadratic function. Fitting of this kind is applied to various
where the standard errors are in parentheses. The estimated exponent
Therein, the standard error is in the parenthesis.
FIGURE 7. Estimation of the parameter
FIGURE 8. Estimation of the parameter
FIGURE 9. Fitting of
Similarly, it is possible to show the kth (
FIGURE 10. Pdf of measured
Numerical Calculation of the Fokker–Planck Equation
We confirmed that estimation of the parameter
where ϵ is a small parameter. The consistent range of ϵ found by estimation of Eq 29 and Eq 32 is
Using results of the numerical calculation of the pdf obtained at each scale, we calculate the scaling exponent
The result is presented in Figure 11. No difference exists between the two numerical calculation results. Both curves are convex upward, indicating multifractal properties. Comparison with measured values is also good. These results, when combined with consideration of the statistics of multipliers given in 2.1.2, underscore the effectiveness of the continuous cascade model Eq. 7 with additive stochastic processes proposed.
The random cascade model has evolved as a model of developed turbulence. The original model, in which the stochastic process that connects each layer of the spatial scale is an independent random multiplication process, contradicts results obtained through empirical research. Therefore, an improved discrete random multiplicative cascade model with additional additive stochastic processes was proposed along with a model formulated as a Fokker–Planck equation by considering cascade processes as a continuous Markov process. Moreover, those models have been applied to data analysis of the stock market and the foreign exchange market, where they have been successful. Herein, we propose a continuous cascade model formulated as a stochastic differential equation of volatility including two independent modes of Brownian motion: one has multiplicative coupling with volatility; the other has additive coupling, as in an improved discrete cascade model for the stock market, with effectiveness clarified by results of earlier research . The model parameters were estimated by application to a stock price time series. The Fokker–Planck equation was derived from the stochastic differential equation as a master equation with the transition probability density function of volatility. Furthermore, the model parameters were estimated by its application to the average stock price time series made from FTSE 100 constituents listed on the London Stock Exchange. At that time, as an alternative variable of volatility, the wavelet transform coefficient with the second derivative of the Gaussian function as an analyzing wavelet was used. Numerical calculation of the Fokker–Planck equation was conducted using the estimated parameter values. The results reported herein faithfully reproduce the results of an earlier empirical study. This model includes information about neither the time axis nor the sign of the price fluctuation, which is necessary for a model of price fluctuations. The actual stock market exhibits well-known properties that break symmetry with respect to the time axis, such as the causal structure from long-term to short-term scale volatility described first in Introduction and price–volatility correlation (leverage effect) . Therefore, the extension of the random cascade model to encompass these phenomena remains as a subject for future work.
Data Availability Statement
Publicly available datasets were analyzed in this study. This data can be found here: Historic Price Service (HPS) provided by the London Stock Exchange (https://www.londonstockexchange.com/products-and-services/reference-data/hps/hps.htm).
JM conducted the empirical and numerical study of the model based on the dataset. KK theoretically analyzed the model. All authors agree to be accountable for the contents of the work.
This research was partially supported by a Grant-in-Aid for Scientific Research (C) No. 16K01259.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relation that could be construed as a potential conflict of interest.
One author, JM, expresses special appreciation for support by a Seijo University Special Research Grant.
Derivation of Eq 15
We introduce some notation for simplification of the description:
From Eq. 13, we have
We also have
Because of the coincidence of the expected value and the standard deviation, we have
The WTMM method builds a partition function from the modulus maxima of the wavelet transform defined at each scale s as the local maxima of
The partition function is defined by the maxima lines as
Assuming power law behavior of the partition function
one can define the exponents
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Keywords: Fokker–Planck equation, intermittency, Langevin equation, Markovian process, multifractal
Citation: Maskawa J-i and Kuroda K (2020) Model of Continuous Random Cascade Processes in Financial Markets. Front. Phys. 8:565372. doi: 10.3389/fphy.2020.565372
Received: 25 May 2020; Accepted: 29 October 2020;
Published: 27 November 2020.
Edited by:Wei-Xing Zhou, East China University of Science and Technology, China
Reviewed by:Haroldo V. Ribeiro, State University of Maringá, Brazil
Joachim Peinke, University of Oldenburg, Germany
Copyright © 2020 Maskawa and Kuroda. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Jun-ichi Maskawa, email@example.com