A general class of multifractional processes and stock price informativeness

Abstract

We introduce a general class of stochastic processes driven by a multifractional Brownian motion (mBm) and study the estimation problems of their pointwise Hölder exponents (PHE) based on a new localized generalized quadratic variation approach (LGQV). By comparing our suggested approach with the other two existing benchmark estimation approaches (classic GQV and oscillation approach) through a simulation study, we show that our estimator has better performance in the case where the observed process is some unknown bivariate function of time and mBm. Such multifractional processes, whose PHEs are time-varying, can be used to model stock prices under various market conditions, that are both timedependent and region-dependent. As an application to finance, an empirical study on modeling cross-listed stocks provides new evidence that the equity path's roughness varies via time and the stock price informativeness properties from global stock markets.Keywords: Multifractional process · pointwise Hölder exponent · LGQV estimation · stock price informativeness MSC (2010): 62F10 · 62F12 · 62M86 arXiv:1708.04217v3 [q-fin.MF]

Cluster Analysis on Locally Asymptotically Self-similar Processes with Known Number of Clusters

“…Through empirical study it is proved that these indexes returns exhibit the "long memory" path feature hence they can be modeled by self-similar processes or more generally by locally asymptotically self-similar processes such as fBms and mBms (see e.g. Bianchi and Pianese (2008) Bianchi et al (2013) and Peng and Zhao (2018)). Therefore similar to Section 5 we may cluster the increments of the indexes returns with the log * -transformed dissimilarity measure.…”

“…The idea of modeling stock returns by locally asymptotically self-similar processes (mBm) is pioneered by Bianchi and Pianese (2008). This time-varying self-similar feature of the financial markets stochastic processes is further convinced by Bianchi et al (2013) and Peng and Zhao (2018). We consider data from global financial markets to be perfect underlying stochastic processes of our proposed clustering algorithms.…”

Minimal model of diffusion with time changing Hurst exponent

“…Another very closely related class of processes are linear time series models with fractionally integrated noises (ARFIMA) and time dependent coefficients [67]. Inter alia MFBMs found applications in finance, where it is natural to expect a time-dependence of the market dynamics [68][69][70][71], and also network traffic [72], geometry of mountain ranges [73] or atmospheric turbulence [74], as well as heterogeneous diffusion [45]. Statistical methods for analyzing MFBM models include wavelet decomposition [75], covariance and MSD analysis and testing [76], or neural networks [77].…”

Fractional Brownian Motion: Local Modulus of Continuity with Refined Almost Sure Upper Bound and First Exit Time from One-sided Barrier

“…For examples, the upper bounds of E[C p ] for all p > 0 are used in [1,2] to obtain the asymptotic rates of the lower and upper bounds of the tail probabilities of the target processes' first exit time. They are also needed in [16,[27][28][29] to construct strongly consistent (a.s. convergent) estimators of the Hölder exponents of fractional and multifractional Gaussian processes. Motivated by this inconvenience, in this paper we show an enhanced almost sure upper bound C (t )|h| H log log |h| −1 of |B H (t +h)−B H (t )| as |h| is small, where for every t , the random multiplier C (t ) has finite moments of every order.…”