Alexandre Pantanella

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.

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.

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.