Fractional Brownian Motion Model Research Articles

Time scale and fractionality in financial time series

Purpose– Turvey (2007, Physica A) introduced a scaled variance ratio procedure for testing the random walk hypothesis (RWH) for financial time series by estimating Hurst coefficients for a fractional Brownian motion model of asset prices. The purpose of this paper is to extend his work by making the estimation procedure robust to heteroskedasticity and by addressing the multiple hypothesis testing problem.Design/methodology/approach– Unbiased, heteroskedasticity consistent, variance ratio estimates are calculated for end of day price data for eight time lags over 12 agricultural commodity futures (front month) and 40 US equities from 2000-2014. A bootstrapped stepdown procedure is used to obtain appropriate statistical confidence for the multiplicity of hypothesis tests. The variance ratio approach is compared against regression-based testing for fractionality.Findings– Failing to account for bias, heteroskedasticity, and multiplicity of testing can lead to large numbers of erroneous rejections of the null hypothesis of efficient markets following an independent random walk. Even with these adjustments, a few futures contracts significantly violate independence for short lags at the 99 percent level, and a number of equities/lags violate independence at the 95 percent level. When testing at the asset level, futures prices are found not to contain fractional properties, while some equities do.Research limitations/implications– Only a subsample of futures and equities, and only a limited number of lags, are evaluated. It is possible that multiplicity adjustments for larger numbers of tests would result in fewer rejections of independence.Originality/value– This paper provides empirical evidence that violations of the RWH for financial time series are likely to exist, but are perhaps less common than previously thought.

Statistics of Natural Stochastic Textures and Their Application in Image Denoising.

Natural stochastic textures (NSTs), characterized by their fine details, are prone to corruption by artifacts, introduced during the image acquisition process by the combined effect of blur and noise. While many successful algorithms exist for image restoration and enhancement, the restoration of natural textures and textured images based on suitable statistical models has yet to be further improved. We examine the statistical properties of NST using three image databases. We show that the Gaussian distribution is suitable for many NST, while other natural textures can be properly represented by a model that separates the image into two layers; one of these layers contains the structural elements of smooth areas and edges, while the other contains the statistically Gaussian textural details. Based on these statistical properties, an algorithm for the denoising of natural images containing NST is proposed, using patch-based fractional Brownian motion model and regularization by means of anisotropic diffusion. It is illustrated that this algorithm successfully recovers both missing textural details and structural attributes that characterize natural images. The algorithm is compared with classical as well as the state-of-the-art denoising algorithms.

Modelling and measuring the irrational behaviour of agents in financial markets: Discovering the psychological soliton

Following a Geometrical Brownian Motion extension into an Irrational fractional Brownian Motion model, we re-examine agent behaviour reacting to time dependent news on the log-returns thereby modifying a financial market evolution. We specifically discuss the role of financial news or economic information positive or negative feedback of such irrational (or contrarian) agents upon the price evolution. We observe a kink-like effect reminiscent of soliton behaviour, suggesting how analysts' forecasts errors induce stock prices to adjust accordingly, thereby proposing a measure of the irrational force in a market.

Numerically pricing American options under the generalized mixed fractional Brownian motion model

In this paper, we introduce a robust numerical method, based on the upwind scheme, for the pricing of American puts under the generalized mixed fractional Brownian motion (GMFBM) model. By using portfolio analysis and applying the Wick–Itô formula, a partial differential equation (PDE) governing the prices of vanilla options under the GMFBM is successfully derived for the first time. Based on this, we formulate the pricing of American puts under the current model as a linear complementarity problem (LCP). Unlike the classical Black–Scholes (B–S) model or the generalized B–S model discussed in Cen and Le (2011), the newly obtained LCP under the GMFBM model is difficult to be solved accurately because of the numerical instability which results from the degeneration of the governing PDE as time approaches zero. To overcome this difficulty, a numerical approach based on the upwind scheme is adopted. It is shown that the coefficient matrix of the current method is an M-matrix, which ensures its stability in the maximum-norm sense. Remarkably, we have managed to provide a sharp theoretic error estimate for the current method, which is further verified numerically. The results of various numerical experiments also suggest that this new approach is quite accurate, and can be easily extended to price other types of financial derivatives with an American-style exercise feature under the GMFBM model.

Geometric Fractional Brownian Motion Perturbed by Fractional Ornstein-Uhlenbeck Process and Application on KLCI Option Pricing

This paper presents an enhanced model of geometric fractional Brownian motion where its volatility is assumed to be stochastic volatility model that obeys fractional Ornstein-Uhlenbeck process. The method of estimation for all parameters (α, β, m, μ, H1, and H2) in this model is derived. We calculated the value of European call option using the estimates based on the methods of Masnita [1] [2] and Kukush [3], traditional Black-Scholes European option price, in addition to proposed model in order to make comparison study.

Efficient Rotation-Scaling-Translation Parameter Estimation Based on the Fractal Image Model

This paper deals with area-based subpixel image registration under the rotation-isometric scaling-translation transformation hypothesis. Our approach is based on parametrical modeling of geometrically transformed textural image fragments and maximum-likelihood estimation of the transformation vector between them. Due to the parametrical approach based on the fractional Brownian motion modeling of the local fragments' texture, the proposed estimator ML <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fBm</sub> (ML stands for “maximum likelihood” and fBm stands for “fractal Brownian motion”) has the ability to better adapt to real image texture content compared with other methods relying on universal similarity measures such as mutual information or normalized correlation. The main benefits are observed when assumptions underlying the fBm model are fully satisfied, e.g., for isotropic normally distributed textures with stationary increments. Experiments on both simulated and real images and for high and weak correlations between registered images show that the ML <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fBm</sub> estimator offers significant improvement compared with other state-of-the-art methods. It reduces translation vector, rotation angle, and scaling factor estimation errors by a factor of about 1.75-2, and it decreases the probability of false match by up to five times. In addition, an accurate confidence interval for ML <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fBm</sub> estimates can be obtained from the Cramér-Rao lower bound on rotation-scaling-translation parameter estimation error. This bound depends on texture roughness, noise level in reference and template images, correlation between these images, and geometrical transformation parameters.

Satellite modulation transfer function estimation from natural scenes

We propose an in-orbit modulation transfer function (MTF) statistical estimation algorithm based on natural scene, called SeMTF. The algorithm can estimate the in-orbit MTF of a sensor from an image without specialized targets. First, the power spectrum of a satellite image is analyzed, then a two-dimensional (2-D) fractal Brownian motion model is adopted to represent the natural scene. The in-orbit MTF is modeled by a parametric exponential function. Subsequently, the statistical model of satellite imaging is established. Second, the model is solved by the improved profile-likelihood function method. In order to handle the nuisance parameter in the profile-likelihood function, we divided the estimation problem into two minimization problems for the parameters of the MTF model and nuisance parameters, respectively. By alternating the two iterative minimizations, the result will converge to the optimal MTF parameters. Then the SeMTF algorithm is proposed. Finally, the algorithm is tested using real satellite images. Experimental results indicate that the estimation of MTF is highly accurate.

Pricing Model for Convertible Bonds: A Mixed Fractional Brownian Motion with Jumps

AbstractA mathematical model to price convertible bonds involving mixed fractional Brownian motion with jumps is presented. We obtain a general pricing formula using the risk neutral pricing principle and quasi-conditional expectation. The sensitivity of the price to changing various parameters is discussed. Theoretical prices from our jump mixed fractional Brownian motion model are compared with the prices predicted by traditional models. An empirical study shows that our new model is more acceptable.

Construction of maximum likelihood estimator in the mixed fractional–fractional Brownian motion model with double long-range dependence

We construct an estimator of the unknown drift parameter $\theta\in {\mathbb{R}}$ in the linear model \[X_t=\theta t+\sigma_1B^{H_1}(t)+\sigma_2B^{H_2}(t),\;t\in[0,T],\] where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $H_1$ and $H_2$ satisfying the condition $\frac{1}{2}\leq H_1<H_2<1.$ Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular one, and this representation allows us to prove the compactness of the corresponding integral operator. This, in turn, allows us to establish an existence--uniqueness result for the sequence of the equations on the increasing intervals, to construct accordingly a sequence of statistical estimators, and to establish asymptotic consistency.

OPTION PRICING FOR PROCESSES DRIVEN BY MIXED FRACTIONAL BROWNIAN MOTION WITH SUPERIMPOSED JUMPS

We propose a geometric mixed fractional Brownian motion model for the stock price process with possible jumps superimposed by an independent Poisson process. Option price of the European call option is computed for such a model. Some special cases are studied in detail.

Statistic estimation and validation of in-orbit modulation transfer function based on fractal characteristics of remote sensing images

This paper deals with the estimation of an in-orbit modulation transfer function (MTF) by a remote sensing image sequence, which is often difficult to measure because of a lack of suitable target images. A model is constructed which combines a fractal Brownian motion model that describes natural images stochastic fractal characteristics, with an inverse Fourier transform of an ideal remote sensing image amplitude spectrum. The model is used to decouple the blurring effect and an ideal natural image. Then, a model of MTF statistical estimation is built by standard deviation of the image sequence amplitude spectrum. Furthermore, model parameters are estimated by the ergodicity assumption of a remote sensing image sequence. Finally, the results of the statistical MTF estimation method are given and verified. The experimental results demonstrate that the method is practical and effective, and the relative deviation at the Nyquist frequency between the edge method and the method in this paper is less than 5.74%. The MTF estimation method is applicable for remote sensing image sequences and is not restricted by the characteristic target of images.

Alveolar bone-loss area localization in periodontitis radiographs based on threshold segmentation with a hybrid feature fused of intensity and the H-value of fractional Brownian motion model

Background and objectivePeriodontitis involves progressive loss of alveolar bone around the teeth. Hence, automatic alveolar bone-loss (ABL) measurement in periapical radiographs can assist dentists in diagnosing such disease. In this paper, we propose an effective method for ABL area localization and denote it as ABLIfBm. MethodABLIfBm is a threshold segmentation method that uses a hybrid feature fused of both intensity and texture measured by the H-value of fractional Brownian motion (fBm) model, where the H-value is the Hurst coefficient in the expectation function of a fBm curve (intensity change) and is directly related to the value of fractal dimension. Adopting leave-one-out cross validation training and testing mechanism, ABLIfBm trains weights for both features using Bayesian classifier and transforms the radiograph image into a feature image obtained from a weighted average of both features. Finally, by Otsu's thresholding, it segments the feature image into normal and bone-loss regions. ResultsExperimental results on 31 periodontitis radiograph images in terms of mean true positive fraction and false positive fraction are about 92.5% and 14.0%, respectively, where the ground truth is provided by a dentist. The results also demonstrate that ABLIfBm outperforms (a) the threshold segmentation method using either feature alone or a weighted average of the same two features but with weights trained differently; (b) a level set segmentation method presented earlier in literature; and (c) segmentation methods based on Bayesian, K-NN, or SVM classifier using the same two features. ConclusionOur results suggest that the proposed method can effectively localize alveolar bone-loss areas in periodontitis radiograph images and hence would be useful for dentists in evaluating degree of bone-loss for periodontitis patients.

Fractal Analysis Of Colors And Shapes For Natural And Urbanscapes URBANSCAPES

Abstract. Fractal analysis has been applied in many fields since it was proposed by Mandelbrot in 1967. Fractal dimension is a basic parameter of fractal analysis. According to the difference of fractal dimensions for images, natural landscapes and urbanscapes could be differentiated, which is of great significance. In this paper, two methods were used for two types of landscape images to discuss the difference between natural landscapes and urbanscapes. Traditionally, a box-counting method was adopted to evaluate the shape of grayscale images. On the other way, for the spatial distributions of RGB values in images, the fractal Brownian motion (fBm) model was employed to calculate the fractal dimensions of colour images for two types of landscape images. From the results, the fractal dimensions of natural landscape images were lower than that of urbanscapes for both grayscale images and colour images with two types of methods. Moreover, the spatial distributions of RGB values in images were clearly related with the fractal dimensions. The results indicated that there was obvious difference (about 0.09) between the fractal dimensions for two kinds of landscapes. It was worthy to mention that when the correlation coefficient is 0 in the semivariogram, the fractal dimension is 2, which means that when the RGB values are completely random for their locations in the colour image, the fractal dimension becomes 3. Two kinds of fractal dimensions could evaluate the shape and the color distributions of landscapes and discriminate the natural landscapes from urbanscapes clearly.

Research on Cutting Tool Wear Based on Fractional Brownian Motion

Cutting tool wear is a very complex process. Various factors have a direct or indirect effect on cutting tool wear, resulting in uncertainty, so it is difficult for experimental data and result to have good stability. However, Vibration analysis is a very important means for condition monitoring and fault diagnosis. This paper aims to study the methods of tool vibration signal processing, pattern recognition and trend prediction. Collected on tool vibration signal at different times, wavelet noise reduction is used to pretreat the vibration signals. Then, for the self-similar vibration signals, we propose the fractional Brownian motion (FBM) theory with long-range dependence (LRD). Combined with Wigner-Ville spectrum, characteristic parameter can be extracted, so the cutting tool wear state can be determined according to fractal dimension and average slope of the fitting curve of the logarithm power spectrum. Finally, we use FBM model to predict the trend of tool vibration signals. Experiments show that the methods have a good effect on tool wear state recognition and trend prediction.

Numerical study on gas diffusion in isotropic and anisotropic fractal porous media (gas diffusion in fractal porous media)

Abstract A binary mixture lattice Boltzmann model (LBM) is used to simulate the diffusion process in stochastic fractal porous media generated by the fractional Brownian motion model. The isotropic porous media is reconstructed from fractional Brownian rough surface using the random midpoint displacement algorithm and the binary processing. In addition, the 2D and 3D anisotropic porous media is generated by transforming the isotropic fractional Brownian rough surface with a structure filter. LBM is applied to simulate the diffusion processes of reconstructed isotropic and anisotropic porous media for various fractal dimensions. The results indicate that the increasing fractal dimension can lead to the reduction of effective diffusion coefficients at the same porosity. The degrees of anisotropy for diffusion of anisotropic porous media are presented with the different anisotropic strength.

Fluctuation Modeling of Macroscopic Cross Section with Spatial Correlation

Fluctuation modeling of the macroscopic cross section is studied in the framework of a continuously distributed stochastic medium. In particular, spatial correlation is approached by fractional Brownian motion (FBM) and randomized Weierstrass function (RWF). Here, FBM is capable of modeling correlation due to coordinate increments while RWF has the same property as FBM on a small scale, is able to confine the influence of correlation within a certain range of increments, and is globally under a fixed variance. In numerical experiments, first flights of neutral particles are examined using Woodcock tracking. Results obtained indicate that the attenuation of an uncollided beam becomes slower than the exponential law of the corresponding nonstochastic homogeneous medium as the spatial correlation changes from negative to positive; this departure to the slower side is very small or negligible in the full antipersistency limit of negative correlation. It is also shown that the departure from the exponential law of attenuation is nearly negligible if the influence of correlation is confined within the mean free path (mfp) determined by the macroscopic cross section of the corresponding nonstochastic homogeneous medium. However, the mfp’s for individual realizations of the medium distribute widely. FBM turns out not to be feasible for modeling positive correlation. Overall, RWF virtually eliminates the risk of negative values of the macroscopic cross section inherent in the FBM modeling.

Risky Asset Models with Tempered Stable Fractal Activity Time

Two constructions of fractal activity time in the fractal activity time geometric Brownian motion (FATGBM) model for a risky asset are discussed. Both constructions produce tractable dependence structure that includes long-range dependence. One construction uses Ornstein-Uhlenbeck type processes and leads to stationary log returns with exact normal tempered stable distribution and pricing formula that is based on the asymptotic self-similarity of the activity time. The second construction uses convoluted subordinator with Holmgren-Liouville kernel and leads to exact pricing formula that is based on exact tempered stable distribution of the activity time. Option prices computed using our model and pricing formulae are remarkably close to the real market values.

A Precise Lower Bound on Image Subpixel Registration Accuracy

A new performance bound is proposed for analyzing parametric image registration methods objectively. This original bound is derived from the Cramer-Rao lower bound on the estimation error of parameters involved in a geometric transformation assumed between reference and template images (pure translation in this work) and parameters describing the texture of these images. For describing local fragments of both the reference and the template images, the parametric fractional Brownian motion (fBm) model has been chosen. Experimental results, obtained first on pure fBm data with full matching of the data to the texture model assumption, give evidence that the proposed bound describes more adequately the performance of conventional estimators than two other bounds previously proposed in the literature. This holds with respect to the signal-to-noise ratio value of both images, the roughness of their texture, their correlation, and the actual value of translation parameters between their grids. Then, one real Hyperion hyperspectral data set is considered to test the proposed bound behavior on real data. The proposed bound is demonstrated to describe more adequately the estimation accuracy of the translation parameters between different bands of this data set.

Mathematical Model of Stock Prices via a Fractional Brownian Motion Model with Adaptive Parameters

The paper presents a mathematical model of stock prices using a fractional Brownian motion model with adaptive parameters (FBMAP). The accuracy index of the proposed model is compared with the Brownian motion model with adaptive parameters (BMAP). The parameters in both models are adapted at any time. The ADVANC Info Service Public Company Limited (ADVANC) and Land and Houses Public Company Limited (LH) closed prices are concerned in the paper. The Brownian motion model with adaptive parameters (BMAP) and fractional Brownian motion model with adaptive parameters (FBMAP) are applied to identify ADVANC and LH closed prices. The simulation results show that the FBMAP is more suitable for forecasting the ADVANC and LH closed price than the BMAP.

Variance Estimation for Fractional Brownian Motions with Fixed Hurst Parameters

Some real-world phenomena in geo-science, micro-economy, and turbulence, to name a few, can be effectively modeled by a fractional Brownian motion indexed by a Hurst parameter, a regularity level, and a scaling parameter σ2, an energy level. This article discusses estimation of a scaling parameter σ2 when a Hurst parameter is known. To estimate σ2, we propose three approaches based on maximum likelihood estimation, moment-matching, and concentration inequalities, respectively, and discuss the theoretical characteristics of the estimators and optimal-filtering guidelines. We also justify the improvement of the estimation of σ2 when a Hurst parameter is known. Using the three approaches and a parametric bootstrap methodology in a simulation study, we compare the confidence intervals of σ2 in terms of their lengths, coverage rates, and computational complexity and discuss empirical attributes of the tested approaches. We found that the approach based on maximum likelihood estimation was optimal in terms of efficiency and accuracy, but computationally expensive. The moment-matching approach was found to be not only comparably efficient and accurate but also computationally fast and robust to deviations from the fractional Brownian motion model.