Approximated maximum likelihood estimation in multifractal random walks

Abstract

We present an approximated maximum likelihood method for the multifractal random walk processes of [E. Bacry et al., Phys. Rev. E 64, 026103 (2001)]. The likelihood is computed using a Laplace approximation and a truncation in the dependency structure for the latent volatility. The procedure is implemented as a package in the r computer language. Its performance is tested on synthetic data and compared to an inference approach based on the generalized method of moments. The method is applied to estimate parameters for various financial stock indices.

Random cascade model in the limit of infinite integral scale as the exponential of a nonstationary1/fnoise: Application to volatility fluctuations in stock markets

“…(10) are far more reliable than those based on moment multiscaling (3) (see also [14,30] for additionnal results on the intermittency exponent estimation using GMM methods). Empirical evidence for the logarithmic nature of log-volatility correlations have been provided for different asset price time series over different markets [9,10,14,15,21]. The broad range of observed values of the integral scale in empirical studies leads us to ask the question of the interpretation of the integral scale value in financial markets.…”

“…9 below). Even if it is well admitted that a precise estimation of T can be hardly achieved [14,15], one can naturally wonder why one observes such a large range in the estimated integral scale values. Beyond the problem of the determination of T , a challenging question remains to understand the meaning of the integral scale in finance.…”

Consistency of detrended fluctuation analysis

“…where E denotes the expectation, i.e., the ensemble mean. Important examples of stochastic processes X(t) with scaling properties are self-similar and multifractal models, see e.g., [6]. For this large class of models, the existing q-moments satisfy E |X(t + t 0 ) − X(t 0 )| q ∝ t ζ(q) .…”

A Bayesian approach for the multifractal analysis of spatio-temporal data

“…The estimator (3) is widely used but is known to yield poor performance for small sample size [10], [11]. Alternative estimators have been described in, e.g., [12], [13], but they make assumptions, (e.g. fully parametric model, specific multifractal process), that are often too restrictive in real-world applications.…”