Augusto Pianese

Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity

This paper deals with the problem of estimating the pointwise regularity of multifractional Brownian motion, assumed as a model of stock price dynamics. We (a) correct the shifting bias affecting a class of absolute moment-based estimators and (b) build a data-driven algorithm in order to dynamically check the local Gaussianity of the process. The estimation is therefore performed for three stock indices: the Dow Jones Industrial Average, the FTSE 100 and the Nikkei 225. Our findings show that, after the correction, the pointwise regularity fluctuates around 1/2 (the sole value consistent with the absence of arbitrage), but significant deviations are also observed.

MULTIFRACTIONAL PROPERTIES OF STOCK INDICES DECOMPOSED BY FILTERING THEIR POINTWISE HÖLDER REGULARITY

We propose a decomposition of financial time series into Gaussian subsequences characterized by a constant Holder exponent. In (multi)fractal models this condition is equivalent to the subsequences themselves being stationarity. For the different subsequences, we study the scaling of the variance and the bias that is generated when the Holder exponent is re-estimated using traditional estimators. The results achieved by both analyses are shown to be strongly consistent with the assumption that the price process can be modeled by the multifractional Brownian motion, a nonstationary process whose Holder regularity changes from point to point.

Modelling stock price movements: multifractality or multifractionality?

The scaling properties of two alternative fractal models recently proposed to characterize the dynamics of stock market prices are compared. The former is the Multifractal Model of Asset Return (MMAR) introduced in 1997 by Mandelbrot, Calvet and Fisher in three companion papers. The latter is the multifractional Brownian motion (mBm), defined in 1995 by Péltier and Lévy Véhel as an extension of the very well-known fractional Brownian motion (fBm).  We argue that, when fitted on financial time series, the partition function as well as the scaling function of the mBm, i.e. of a generally non-multifractal process, behave as those of a genuine multifractal process. The analysis, which concerns the daily closing prices of eight major stock indexes, suggests to evaluate prudently the recent findings about the multifractal behaviour in finance and economics.

Scaling Laws in Stock Markets: An Analysis of Prices and Volumes

The scaling behaviour of both log-price and volume is analyzed for three stock indexes. The traditional approach, mainly consisting of the evaluation of particular moments such as variance or higher absolute moments, is replaced by a new technique which allows the estimation of the self-similarity parameter on the whole empirical distribution designed by any time horizon. In this way, the method we propose attaches its own scaling parameter to any two given time lags, so defining a scaling surface whose properties give information about the nature of the analyzed process. We conclude that, for the log-price process, self-similarity is rejected with a frequency much larger than that assumed by the confidence interval and, when not rejected, the scaling parameter heavily changes with the considered pair of time horizons. Opposite evidence is provided for the volumes, characterized by (generally low) self-similarity parameters which are somewhat uniform with respect to the pairs of time horizons.

Fractal Properties of Some European Electricity Markets

This paper investigates the fractal behaviour of the electric spot prices traded in some European markets. Whereas the analysis leads to exclude the presence of multifractality, we provide evidence supporting the conclusion that the multifractional Brownian motion can represent a good candidate to model the dynamics of electricity prices.

Fast and unbiased estimator of the time-dependent Hurst exponent

We combine two existing estimators of the local Hurst exponent to improve both the goodness of fit and the computational speed of the algorithm. An application with simulated time series is implemented, and a Monte Carlo simulation is performed to provide evidence of the improvement.

Asset Price Modeling: From Fractional to Multifractional Processes

This contribution surveys the main characteristics of two stochastic processes that generalize the fractional Brownian motion: the multifractional Brownian motion and the multifractional processes with random exponent. A special emphasis will be devoted to the meaning and to the applications that they can have in finance. If fractional Brownian motion is by now very well-known and studied as a model of the price dynamics, multifractional processes are yet widely unknown in the field of quantitative finance, mainly because of their nonstationarity. Nonetheless, in spite of their complex structure, such processes deserve consideration for their capability to seize the stylized facts that most of the current models cannot account for. In addition, their functional parameter provides an insightful and parsimonious interpretation of the market mechanism, and is able to unify in a single model two opposite approaches such as the theory of efficient markets and the behavioral finance.

Self-Similarity Parameter Estimation for K-Dimensional Processes

An algorithm is proposed that allows to estimate the self-similarity parameter of a fractal k-dimensional stochastic process. Our technique greatly improves the processing times of a distribution-based estimator, that–introduced years ago–efficiently worked only in the one-dimensional distribution case.