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This article was submitted to Social Physics, a section of the journal Frontiers in Physics

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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. [

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 [

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.

These analyses examine the following wavelet transform of the variation of the logarithmic stock price denoted by

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.

The solution is obtained easily using Ito’s formula as [

The power law behavior of the

We introduce an additional additive stochastic process as we have done in the discrete cascade model. We first consider the following stochastic differential equation.

To solve the stochastic differential

We have mentioned the statistics of multipliers in

The stochastic multiplier

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

By defining the differences

Assuming that

A remarkable feature of the probability density function (pdf) of the quantity

Scaling properties of

Results of multifractal analysis.

The additional additive stochastic process in model

The power law scaling shown in

We can derive the Fokker–Planck equation for the stochastic process

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

Regression of

First, we analyze the multifractal properties of the path

Muzy, Bacry, and Arneodo proposed the wavelet transform modulus maxima (WTMM) method based on continuous wavelet transform of function to calculate the singular spectrum

Parameters

Estimation of the parameter

Estimation of the parameter

Regression of

Similarly, we estimate parameters

Estimation of the parameter

Estimation of the parameter

Fitting of

Similarly, it is possible to show the

Pdf of measured

We confirmed that estimation of the parameter

Using results of the numerical calculation of the pdf obtained at each scale, we calculate the scaling exponent

Scaling exponent

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 [

Publicly available datasets were analyzed in this study. This data can be found here: Historic Price Service (HPS) provided by the London Stock Exchange (

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.

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.

We introduce some notation for simplification of the description:

The WTMM method builds a partition function from the modulus maxima of the wavelet transform defined at each scale