Multiplicative random cascades with additional stochastic process in financial markets

Abstract

Multiplicative random cascade model naturally reproduces the intermittency or multifractality, which is frequently shown among hierarchical complex systems such as turbulence and financial markets. As described herein, we investigate the validity of a multiplicative hierarchical random cascade model through an empirical study using financial data. Although the intermittency and multifractality of the time series are verified, random multiplicative factors linking successive hierarchical layers show a strongly negative correlation. We extend the multiplicative model to incorporate an additional stochastic term. Results show that the proposed model is consistent with all the empirical results presented here.

Appendix: WTMM method

This appendix briefly describes the WTMM method based on the continuous wavelet transform proposed earlier in the literature (Bacry et al. 1993; Muzy et al. 1993). The continuous wavelet transformation of the function f using the analyzing wavelet $\psi$ is defined as

$$\begin{aligned} W_{\psi }[f](x,s)=\frac{1}{s}\int _{-\infty }^{\infty }f(u)\psi \left(\frac{u-x}{s}\right)du, \end{aligned}$$

(16)

where parameters s and x respectively represent the dilation and the translation of the function $\psi$. The analyzing wavelet $\psi$ has been assumed to have $N_{\psi }>0$ vanishing moments. The successive derivative of the Gaussian function

$$\begin{aligned} \psi ^{(N_\psi )}(x)=\frac{d^{N_\psi } (e^{-x^2/2})}{d x^{N_\psi }} \end{aligned}$$

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has $N_{\psi }>0$ vanishing moments. Here we specify $N_{\psi }=2$ and use the second derivative of the Gaussian function as the analyzing wavelet. 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 $|W_{\psi }[f](x,s)|$ regarded as a function of x. Those maxima mutually connect across scales and form ridge lines designated as maxima lines. The set $\mathcal {L}(s_0)$ is the set of all the maxima lines l that satisfy

$$\begin{aligned} (x,s)\in l \Rightarrow s \le s_0,\,\forall s \le s_0 \Rightarrow \exists (x,s)\in l. \end{aligned}$$

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The partition function is defined by the maxima lined as

$$\begin{aligned} Z(q,s)=\sum _{l\in \mathcal {L}(s)}(\sup _{(x,s')\in l}|W_{\psi }[x,s']|)^q. \end{aligned}$$

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Assuming the power-law behavior of the partition function

$$\begin{aligned} Z(q,s)\sim s^{\tau (q)}, \end{aligned}$$

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one can define the exponents $\tau (q)$. The singular spectrum $D(\alpha )$ can be computed using the Legendre transform of $\tau (q)$:

$$\begin{aligned} D(\alpha )=\min _q (q\alpha -\tau (q)). \end{aligned}$$

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