Sergio Bianchi

PATHWISE IDENTIFICATION OF THE MEMORY FUNCTION OF MULTIFRACTIONAL BROWNIAN MOTION WITH APPLICATION TO FINANCE

We extend and adapt a class of estimators of the parameter H of the fractional Brownian motion in order to estimate the (time-dependent) memory function of a multifractional process. We provide: (a) the estimators distribution when H ∈ (0,3/4); (b) the confidence interval under the null hypothesis H = 1/2; (c) a scaling law, independent on the value of H, discriminating between fractional and multifractional processes. Furthermore, assuming as a model for the price process the multifractional Brownian motion, empirical evidence is offered which is able to conciliate the inconsistent results achieved in estimating the intensity of dependence in financial time series.

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.

A Cautionary Note on the Detection of Multifractal Scaling in Finance and Economics

The scaling properties of the multifractional Brownian motion (mBm), a generally not multifractal process is investigated, and it is argued that, when calibrated on actual financial time series, its partition function as well as its spectrum behave as those of genuine multifractal processes. The examples here provided, based on the analysis of two major stock indexes, are intended to solicit a prudent evaluation of the recent findings about the multifractal behaviour in finance and economics.

Sustainability of a pay-as-you-go pension system by dynamic immigration control

Abstract The sustainability of a pay-as-you-go pension system strongly depends on the underlying demographic process determining the inverse old-age dependency ratio (the proportion of the active subpopulation to pensioners), considered as a sustainability index. Based on the classical Leslie population model, a dynamic demographic model including a controlled immigration is set up. A convergent algorithm is given which steers the population towards a demographic equilibrium with a better sustainability index, and at the same time minimizes the yearly immigration. In addition, simulations are provided for Italian data, comparing the demographic dynamics corresponding to different decision scenarios.

Pointwise Regularity Exponents and Well-Behaved Residuals in Stock Markets

The article deals with a class of stochastic processes, the Multifractional Processes with Random Exponent (MPRE), recently introduced to gain flexibility in modeling many complex phenomena. We claim that MPRE can capture in a very parsimonious way most of the well known financial stylized facts. In particular, we prove that the process unconditional distributions are fat-tailed and high-peaked and show that, as it occurs for asset returns, the empirical autocorrelation functions of the process increments are close to zero whereas significant values are exhibited by squared (or absolute) increments. Furthermore, we provide evidence that the sole knowledge of functional parameter of the MPRE allows to calculate residuals that perform much better than those obtained by other discrete models such as the GARCH family.

A New Distribution-Based Test of Self-Similarity

In studying the scale invariance of an empirical time series a twofold problem arises: it is necessary to test the series for self-similarity and, once passed such a test, the goal becomes to estimate the parameter H0 of self-similarity. The estimation is therefore correct only if the sequence is truly self-similar but in general this is just assumed and not tested in advance. In this paper we suggest a solution for this problem. Given the process {X(t)}, we propose a new test based on the diameter d of the space of the rescaled probability distribution functions of X(t). Two necessary conditions are deduced which contribute to discriminate self-similar processes and a closed formula is provided for the diameter of the fractional Brownian motion (fBm). Furthermore, by properly chosing the distance function, we reduce the measure of self-similarity to the Smirnov statistics when the one-dimensional distributions of X(t) are considered. This permits the application of the well-known two-sided test due to Kolmogorov and Smirnov in order to evaluate the statistical significance of the diameter d, even in the case of strongly dependent sequences. As a consequence, our approach both tests the series for self-similarity and provides an estimate of the self-similarity parameter.

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 stock markets: International evidence of dynamical (in)efficiency

The last systemic financial crisis has reawakened the debate on the efficient nature of financial markets, traditionally described as semimartingales. The standard approaches to endow the general notion of efficiency of an empirical content turned out to be somewhat inconclusive and misleading. We propose a topological-based approach to quantify the informational efficiency of a financial time series. The idea is to measure the efficiency by means of the pointwise regularity of a (stochastic) function, given that the signature of a martingale is that its pointwise regularity equals 12. We provide estimates for real financial time series and investigate their (in)efficient behavior by comparing three main stock indexes.