The origins of multifractality in financial time series and the effect of extreme events

Abstract

Abst ract . T his paper present s t he result s of mult ifract al t est ing of two set s of financial dat a: daily dat a of t he Dow J ones Indust rial Average (DJ IA) index and minut ely dat a of t he Euro St oxx 50 index. W here mult ifract al scaling is found, t he spect rum of scaling exponent s is calculat ed via Mult ifract al Det rended F luct uat ion Analysis. In bot h cases, furt her invest igat ions reveal t hat t he t emporal correlat ions in t he dat a are a more significant source of t he mult ifract al scaling t han are t he dist ribut ions of t he ret urns. It is also shown t hat t he ext reme event s which make up t he heavy t ails of t he dist ribut ion of t he Euro St oxx 50 log ret urns dist ort t he scaling in t he dat a set . T he most ext reme event s are inimical t o t he scaling regime. T his result is in cont rast t o previous findings t hat ext reme event s cont ribut e t o mult ifract ality. Int roduct ionMult ifract al analysis has proved t o be a valuable met hod of capt uring t he underlying scaling st ruct ure present in many types of syst ems via generalised dimensions [1] and f (α) spect ra [2]. T hese syst ems include diff usion limit ed aggregat ion [3][4][5], fluid flow t hrough random porous media [6], at omic spect ra of rare-eart h element s [7], clust erclust er aggregat ion [8] and t urbulent flow [9]. In physiology, mult ifract al st ruct ures have been found in heart rat e variability [10] and brain dynamics [11], and mult ifract al analysis has been helpful in dist inguishing between healthy and pathological patients [12]. Mult ifract al measures have also been found in man-made phenomena such as t he Int ernet [13], art [14] and the stock market [15][16][17].T he concept of mult ifract ality was first int roduced in t he cont ext of t urbulence. It was soon applied t o finance because of it s heavy t ails and long-t erm dependence. T hese two feat ures are also argued t o be present in financial dat a [18,19]. P erforming mult ifract al analysis helps t o increase our knowledge about t he financial syst em and furt her charact erise it . Many st udies have found mult ifract al scaling in financial dat a [20][21][22][23]. An underst anding of t his mult ifract al st ruct ure can enable deeper underst anding of t he dynamics of financial market s. If it is found t o be a universal feat ure of financial dat a, it provides an addit ional benchmark by which t o measure t he fitness of financial models. This in turn can help in the design of well performing port folios and in risk management [17]. a e-mail: elena.s.green@nuim.ieThe Multifractal Model of Asset Returns (MMAR) was int roduced by Mandelbrot et al. [24] as an explanat ion of t he volat ility clust ers in financial dat a and t o include "out liers", large deviat ions which make up t he fat t ails of t he ret urn dist ribut ion. T he MMAR was present ed as an alt ernat ive t o Aut oregressive Condit ional Het eroscedast icity (ARCH) models which were int roduced by Engle [25] t o account for volat ility clust er...

Measuring multiscaling in financial time-series

“…In [23] the authors studied again the Dow Jones Industrial Average taken on a daily basis plus the Dow Jones Euro Stoxx 50 sampled at one minute. In this case three analysis were performed:…”

“…The authors found that when shuffled, the dataset loses its multiscaling behaviour ( [23]). The shuffling of the intervals showed that the linearity of the scaling of the fluctuation functions worsen when the length of the interval is small and improves increasing it, thus according to the authors this should be regarded as a sign that the temporal correlations are the source of multiscaling.…”

Extreme value theory in the solar wind: the role of current sheets

“…Financial time series present some key characteristics found in solar wind time series, namely chaoticity (Hołyst,Żebrowska & Urbanowicz 2001), multifractality, extreme events (Green, Hanan & Heffernan 2014) and turbulence (Voit 2005), which encourages the use of the same tools in the analysis of solar wind data. Loosely, the effects of current sheets and magnetic reconnections in the solar wind are qualitatively analogous to the effects of economic crises in financial time series.…”

Asymptotic scaling properties and estimation of the generalized Hurst exponents in financial data

“…In a previous paper [42], solving an ongoing debate in the literature (see, for example, Refs. [43][44][45]), it has been clarified that the true source of the multifractal behavior found in empirical financial time series is their causal structure. However, it was also shown that the measure of multifractality performed via the scaling of the moments in log-log scale is aggregation horizon dependent and that the true multifractal scaling should be measured in the limit of an infinite aggregation horizon.…”