This is a short review in honor of B. Mandelbrot's 80st birthday, to appear in Wilmott magaz... more This is a short review in honor of B. Mandelbrot's 80st birthday, to appear in Wilmott magazine. We discuss how multiplicative cascades and related multifractal ideas might be relevant to model the main statistical features of financial time series, in particular the intermit-tent, ...
Recently, Ghashghaie et al. have shown that some statistical aspects of fully developed turbulenc... more Recently, Ghashghaie et al. have shown that some statistical aspects of fully developed turbulence and exchange rate fluctuations exhibit striking similarities [1]. The authors then suggested that the two problems might be deeply connected, and speculated on the existence of an 'information cascade' which would play the role in finance of the well known Kolmogorov energy cascade in turbulence [2].
We introduce a class of multifractal processes, referred to as Multifractal Random Walks (MRWs). ... more We introduce a class of multifractal processes, referred to as Multifractal Random Walks (MRWs). To our knowledge, it is the first multifractal processes with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascade-like multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by few parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.
We use the wavelet transform microscope to explore the intricate fractal geometry of Witten and S... more We use the wavelet transform microscope to explore the intricate fractal geometry of Witten and Sander's diff'usion-limited aggregates. We report on the discovery of Fibonacci sequences in the internal hierarchical structure of these clusters. We discuss the relevance of the golden mean arithmetic to the numerically well established statistical self-similarity of these aggregates.
... (a) “One-scale” correlation functions C Dx,a, a for a 4 (), 16 (), and 64 (). (b) “Two-scale”... more ... (a) “One-scale” correlation functions C Dx,a, a for a 4 (), 16 (), and 64 (). (b) “Two-scale” correlation functions C Dx,a,a0 for a a0 16 (), a 4, a0 16 (), and a 16, a0 64 (). The solid lines represent fits of the data with the log-normal prediction [Eq. (9)] using the parameters l2 0.03 ...
... In the framework of this wavelet-based multifractal formalism, r(q) is the Legendre transform... more ... In the framework of this wavelet-based multifractal formalism, r(q) is the Legendre transform of the singularity spectrum D(h) de-fined as the Hausdorff dimension of the set of points x where the Holder exponent is h. Homogeneous fractal functions (ie, functions with a unique ...
In this paper we discuss the problem of the estimation of extreme event occurrence probability fo... more In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framemork that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed "inverse cubic law" of the return pdf tails. PACS numbers: 0.250.-r, 05.40.-a, 05.45.-a, 89.65.Gh
Modelling accurately financial price variations is an essential step underlying portfolio allocat... more Modelling accurately financial price variations is an essential step underlying portfolio allocation optimization, derivative pricing and hedging, fund management and trading. The observed complex price fluctuations guide and constraint our theoretical understanding of agent interactions and of the organization of the market. The gaussian paradigm of independent normally distributed price increments has long been known to be incorrect with many
International Journal of Bifurcation and Chaos, 1994
The multifractal formalism originally introduced for singular measures is revisited using the wav... more The multifractal formalism originally introduced for singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that the f(α) singularity spectrum can be ...
Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are... more Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the Wavelet Transform Modulus Maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of Fractional Auto-Regressive Integrated Moving Average (FARIMA) processes, we advocate the use of approximately adapted wavelet estimators. For these "ideal" processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series, if an adapted decomposition is used. But in practical situations, there generally exists a cut-off scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet-based estimators with respect to that cut-off scale as well as to the specific density of the underlying law of the process. In all situations the WTMM estimator is shown to be the best or among the best estimators in terms of the mean square error. We also compare the wavelet estimators with the Detrended Fluctuation Analysis (DFA) estimator which was recently proved to be among the best estimators which are not wavelet-based estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior.
This is a short review in honor of B. Mandelbrot's 80st birthday, to appear in Wilmott magaz... more This is a short review in honor of B. Mandelbrot's 80st birthday, to appear in Wilmott magazine. We discuss how multiplicative cascades and related multifractal ideas might be relevant to model the main statistical features of financial time series, in particular the intermit-tent, ...
Recently, Ghashghaie et al. have shown that some statistical aspects of fully developed turbulenc... more Recently, Ghashghaie et al. have shown that some statistical aspects of fully developed turbulence and exchange rate fluctuations exhibit striking similarities [1]. The authors then suggested that the two problems might be deeply connected, and speculated on the existence of an 'information cascade' which would play the role in finance of the well known Kolmogorov energy cascade in turbulence [2].
We introduce a class of multifractal processes, referred to as Multifractal Random Walks (MRWs). ... more We introduce a class of multifractal processes, referred to as Multifractal Random Walks (MRWs). To our knowledge, it is the first multifractal processes with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascade-like multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by few parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.
We use the wavelet transform microscope to explore the intricate fractal geometry of Witten and S... more We use the wavelet transform microscope to explore the intricate fractal geometry of Witten and Sander's diff'usion-limited aggregates. We report on the discovery of Fibonacci sequences in the internal hierarchical structure of these clusters. We discuss the relevance of the golden mean arithmetic to the numerically well established statistical self-similarity of these aggregates.
... (a) “One-scale” correlation functions C Dx,a, a for a 4 (), 16 (), and 64 (). (b) “Two-scale”... more ... (a) “One-scale” correlation functions C Dx,a, a for a 4 (), 16 (), and 64 (). (b) “Two-scale” correlation functions C Dx,a,a0 for a a0 16 (), a 4, a0 16 (), and a 16, a0 64 (). The solid lines represent fits of the data with the log-normal prediction [Eq. (9)] using the parameters l2 0.03 ...
... In the framework of this wavelet-based multifractal formalism, r(q) is the Legendre transform... more ... In the framework of this wavelet-based multifractal formalism, r(q) is the Legendre transform of the singularity spectrum D(h) de-fined as the Hausdorff dimension of the set of points x where the Holder exponent is h. Homogeneous fractal functions (ie, functions with a unique ...
In this paper we discuss the problem of the estimation of extreme event occurrence probability fo... more In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framemork that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed "inverse cubic law" of the return pdf tails. PACS numbers: 0.250.-r, 05.40.-a, 05.45.-a, 89.65.Gh
Modelling accurately financial price variations is an essential step underlying portfolio allocat... more Modelling accurately financial price variations is an essential step underlying portfolio allocation optimization, derivative pricing and hedging, fund management and trading. The observed complex price fluctuations guide and constraint our theoretical understanding of agent interactions and of the organization of the market. The gaussian paradigm of independent normally distributed price increments has long been known to be incorrect with many
International Journal of Bifurcation and Chaos, 1994
The multifractal formalism originally introduced for singular measures is revisited using the wav... more The multifractal formalism originally introduced for singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that the f(α) singularity spectrum can be ...
Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are... more Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the Wavelet Transform Modulus Maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of Fractional Auto-Regressive Integrated Moving Average (FARIMA) processes, we advocate the use of approximately adapted wavelet estimators. For these "ideal" processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series, if an adapted decomposition is used. But in practical situations, there generally exists a cut-off scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet-based estimators with respect to that cut-off scale as well as to the specific density of the underlying law of the process. In all situations the WTMM estimator is shown to be the best or among the best estimators in terms of the mean square error. We also compare the wavelet estimators with the Detrended Fluctuation Analysis (DFA) estimator which was recently proved to be among the best estimators which are not wavelet-based estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior.
Uploads
Papers by J.-f. Muzy