Littérature scientifique sur le sujet « Multiscaling of Moments »

We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate, and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation, and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with remarkable accuracy.

We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate, and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation, and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with remarkable accuracy.

Abstract. Simulations based on random multiplicative cascade models are used to investigate the uncertainty in estimates of parameters characterizing the multiscaling nature of rainfall time series. The principal parameters used and discussed are the spectral exponent, β, and the K(q) function which characterizes the scaling of the moments. By simulating a large number of series, the sampling variability of parameter estimates in relation to the length of the time series is assessed and found to be in excess of 10%-20% for fields less than ~104 points in length. The issue of long time series which may consist of physically distinct processes with different statistics is addressed and it is shown that highly variable data mixed with an equal amount of less variable data of similar strength is dominated entirely by the statistics of the highly variable data. The effects on the estimates of β and K(q) with the addition of white noise or the tipping bucket effect (quantization) can also be significant, particularly following gradient transformations. Some high resolution rainfall data are also analyzed to illustrate how a single instrumental glitch can strongly bias results and how mixing physically different processes together can lead to incorrect conclusions.

Abstract. We use multifractal analysis to estimate the Rényi dimensions of river basins by two different partition methods. These methods differ in the way that the Euclidian plane support of the measure is covered, partitioning it by using mutually exclusive boxes or by gliding a box over the plane. Images of two different drainage basins, for the Ebro and Tajo rivers, located in Spain, were digitalized with a resolution of 0.5 km, giving image sizes of 617×1059 pixels and 515×1059, respectively. Box sizes were chosen as powers of 2, ranging from 2×4 pixels to 512×1024 pixels located within the image, with the purpose of covering the entire network. The resulting measures were plotted versus the logarithmic value of the box area instead of the box size length. Multifractal Analysis (MFA) using a box counting algorithm was carried out according to the method of moments ranging from −5<q<5, and the Rényi dimensions were calculated from the log/log slope of the probability distribution for the respective moments over the box area. An optimal interval of box sizes was determined by estimating the characteristic length of the river networks and by taking the next higher power of 2 as the smallest box size. The optimized box size for both river networks ranges from 64×128 to 512×1024 pixels and illustrates the multiscaling behaviour of the Ebro and Tajo. By restricting the multifractal analysis to the box size range, good generalized dimension (Dq) spectra were obtained but with very few points and with a low number of boxes for each size due to image size restrictions. The gliding box method was applied to the same box size range, providing more consistent and representative Dq values. The numerical differences between the results, as well as the standard error values, are discussed.

Abstract. This paper shows how modern ideas of scaling can be used to model topography with various morphologies and also to accurately characterize topography over wide ranges of scales. Our argument is divided in two parts. We first survey the main topographic models and show that they are based on convolutions of basic structures (singularities) with noises. Focusing on models with large numbers of degrees of freedom (fractional Brownian motion (fBm), fractional Levy motion (fLm), multifractal fractionally integrated flux (FIF) model), we show that they are distinguished by the type of underlying noise. In addition, realistic models require anisotropic singularities; we show how to generalize the basic isotropic (self-similar) models to anisotropic ones. Using numerical simulations, we display the subtle interplay between statistics, singularity structure and resulting topographic morphology. We show how the existence of anisotropic singularities with highly variable statistics can lead to unwarranted conclusions about scale breaking. We then analyze topographic transects from four Digital Elevation Models (DEMs) which collectively span scales from planetary down to 50 cm (4 orders of magnitude larger than in previous studies) and contain more than 2×108 pixels (a hundred times more data than in previous studies). We use power spectra and multiscaling analysis tools to study the global properties of topography. We show that the isotropic scaling for moments of order ≤2 holds to within ±45% down to scales ≈40 m. We also show that the multifractal FIF is easily compatible with the data, while the monofractal fBm and fLm are not. We estimate the universal parameters (α, C1) characterizing the underlying FIF noise to be (1.79, 0.12), where α is the degree of multifractality (0≤α≤2, 0 means monofractal) and C1 is the degree of sparseness of the surface (0≤C1, 0 means space filling). In the same way, we investigate the variation of multifractal parameters between continents, oceans and continental margins. Our analyses show that no significant variation is found for (α, C1) and that the third parameter H, which is a degree of smoothing (higher H means smoother), is variable: our estimates are H=0.46, 0.66, 0.77 for bathymetry, continents and continental margins. An application we developped here is to use (α, C1) values to correct standard spectra of DEMs for multifractal resolution effects.

Abstract. The theory of scale similarity and breakdown coefficients is applied here to intermittent rainfall data consisting of time series and spatial rain fields. The probability distributions (pdf) of the logarithm of the breakdown coefficients are the principal descriptor used. Rain fields are distinguished as being either multiscaling or multiaffine depending on whether the pdfs of breakdown coefficients are scale similar or scale dependent, respectively. Parameter estimation techniques are developed which are applicable to both multiscaling and multiaffine fields. The scale parameter (width), σ, of the pdfs of the log-breakdown coefficients is a measure of the intermittency of a field. For multiaffine fields, this scale parameter is found to increase with scale in a power-law fashion consistent with a bounded-cascade picture of rainfall modelling. The resulting power-law exponent, H, is indicative of the smoothness of the field. Some details of breakdown coefficient analysis are addressed and a theoretical link between this analysis and moment scaling analysis is also presented. Breakdown coefficient properties of cascades are also investigated in the context of parameter estimation for modelling purposes.

Previous studies have examined the spatial, temporal or magnitude distributions of earthquakes. Moreover, others have shown that the spatial distribution of earthquakes is multiscaling. We extend these studies by incorporating the magnitude of the events when examining the scaling properties of the statistics of the earthquakes. We introduce seismic fields as deduced from the maximum ground motion of seismic events (i.e. earthquakes). We then show that these fields are multifractals. Moreover, using a technique called the double trace moment (DTM) analysis, we present here the estimates for the lower bound of the universal exponents of seismic fields: α = 1.1±0.1. We also estimate C1 = 1.35±0.05. It is suggested that while the value of C1 changes from year to year, the estimate of the Lévy index remains relatively constant.

Abstract. In this paper, we present evidence that intermittency of Eulerian and Lagrangian turbulence of ocean temperature and plankton fields is multifractal and furthermore can be analysed with the help of universal multifractals. We analyse time series of temperature and in vivo fluorescence taken from a drifter in the mixed coastal waters of the eastern English Channel. Two analysis techniques are used to compute the fundamental universal multifiractal parameters, which describe all the statistics of the turbulent fluctuations: the analysis of the scale invariant structure function exponent ζ(q) and the Double Trace Moment technique. At small scales, we do not detect any significant difference between the universal multifiractal behavior of temperature and fluorescence in an Eulerian framework. This supports the hypothesis that the latter is passively advected with the flow as the former. On the one hand, we show that large scale measurements are Lagrangian and indeed we obtain for temperature fluctuations a ω2 power spectrum corresponding to the theoretical scaling of a Lagrangian passive scalar. Furthermore, we show that Lagrangian temperature fluctuations are multiscaling and intermittent. On the other hand, the flatter slope at large scales of the fluorescence power spectrum points out that the plankton is at these scales a "biologically active" scalar.

The spatial description of flows in porous media is a main issue due to their influence on processes that take place inside. In addition to descriptive statistics, the multifractal analysis based on the Box-Counting fixed-size method has been used during last decade to study some porous media features. However, this method gives emphasis to domain regions containing few data points that spark the biased assessment of generalized fractal dimensions for negative moment orders. This circumstance is relevant when describing the flow velocity field in idealised three-dimensional porous media. The application of the Sandbox method is explored in this work as an alternative to the Box-Counting procedure for analyzing flow velocity magnitude simulated with the lattice model approach for six media with different porosities. According to the results, simulated flows have multiscaling behaviour. The multifractal spectra obtained with the Sandbox method reveal more heterogeneity as well as the presence of some extreme values in the distribution of high flow velocity magnitudes as porosity decreases. This situation is not so evident for the multifractal spectra estimated with the Box-Counting method. As a consequence, the description of the influence of porous media structure on flow velocity distribution provided by the Sandbox method improves the results obtained with the Box-Counting procedure.

In this thesis we discuss several aspects of the implied volatility surface. We first derive some model independent results, linking tail probabilities to option price and implied volatility. We then apply these results to a specific stochastic volatility model, obtaining a complete picture of the asymptotic volatility smile for bounded maturity. In Chapter 1 we present an extended summary of all the results obtained in this thesis. The details are contained in the following chapters, that are structured as follows. In Chapter 2 we show that, under general conditions satisfied by many models, the probability tails of the log-price under the risk-neutral measure determine the behavior of European option prices and of the implied volatility, in the regime of either extremes strike (with bounded maturity) or short maturity. Our results provide a powerful extension of previous work by Benaim and Friz (2009). We discuss the application to some popular models, including Carr-Wu finite moment logstable moment, Heston's model and Merton's jump diffusion model. In Chapter 3 we devote ourselves to the analysis of the implied volatility for a specific model, that has been recently proposed by Andreoli, Caravenna, Dai Pra and Posta (2012) to reproduce the multiscaling of moments and clustering of volatility observed in many financial indexes. Based on Chapter 2, this amounts to give sharp estimates on the tails of the log-price distribution. Although the moment generating function of the log-price is not known explicitly, we show that the tails can be well estimated via Large Deviation techniques, notably the Garter-Ellis theorem. In Chapter 4 we propose a possible enrichment of the model, adding jumps to the log-price in order to take account of the so called leverage effect. We prove some basic results and we describe a natural one-parameter family of martingale measures for this enriched model. We also show that the price of European options can be expressed through a generalization of the celebrated Hull & White formula, by averaging the usual Black & Scholes formula with respect to both a random volatility and a random spot price. Finally, in Chapter 5 we describe a numerical algorithm to price European option under the enriched model presented in Chapter 4, exploiting the generalized Hull & White formula. The algorithm uses a stratification method in order to improve the speed. Some preliminary results on the calibration of the model with real data, taken from the DAX index, are presented and discussed.