Scaling symmetry, renormalization, and time series modeling: The case of financial assets dynamics

Abstract

We present and discuss a stochastic model of financial assets dynamics based on the idea of an inverse renormalization group strategy. With this strategy we construct the multivariate distributions of elementary returns based on the scaling with time of the probability density of their aggregates. In its simplest version the model is the product of an endogenous autoregressive component and a random rescaling factor designed to embody also exogenous influences. Mathematical properties like increments' stationarity and ergodicity can be proven. Thanks to the relatively low number of parameters, model calibration can be conveniently based on a method of moments, as exemplified in the case of historical data of the S&P500 index. The calibrated model accounts very well for many stylized facts, like volatility clustering, power-law decay of the volatility autocorrelation function, and multiscaling with time of the aggregated return distribution. In agreement with empirical evidence in finance, the dynamics is not invariant under time reversal, and, with suitable generalizations, skewness of the return distribution and leverage effects can be included. The analytical tractability of the model opens interesting perspectives for applications, for instance, in terms of obtaining closed formulas for derivative pricing. Further important features are the possibility of making contact, in certain limits, with autoregressive models widely used in finance and the possibility of partially resolving the long- and short-memory components of the volatility, with consistent results when applied to historical series.

Cocaine-Induced Changes in Low-Dimensional Attractors of Local Field Potentials in Optogenetic Mice

“…For example, a simple Gaussian random walk is time-reversal invariant (Weiss, 1975 ). Practical implementations of temporal asymmetry measures use, for example, the difference between the probability density functions of the original and time-reversed series, or of their corresponding variances (Zumbach, 2009 , 2012 ), or temporal-based correlation measures over different temporal windows, or between the past and future data blocks of the same temporal length (Zamparo et al, 2013 ), or by using Granger causality (Winkler et al, 2016 ). Time irreversibility is a strong signature of nonlinearity (Schreiber and Schmitz, 2000 ) and is commonly related to entropy production by the underlying (often unknown) mechanism that generated the time series (see Vladimirov and Petersen, 2010 ; Roldán, 2014 and references therein).…”

Option pricing with non-Gaussian scaling and infinite-state switching volatility

“…The main contribution of the present paper is the derivation of closedform formulae for option pricing and associated hedging strategy, based on an equivalent martingale measure. This is obtained by means of a model recently introduced in statistical mechanics by Zamparo et al (2013) whose advantage is to be consistent with the observed features of financial data, thus including non-Gaussian scaling behaviour. The general results provided here include as a special case the BS pricing formula.…”

“…Upon explicit integration over σ, the endogenous component becomes in this case a genuine ARCH process (see Zamparo et al, 2013) described by…”

Self-Organized Criticality in Economic Fluctuations: The Age of Maturity

“…[26] and systematically exposed in Ref. [27]. It is grounded on a "fine-graining" procedure obtained "inverting" the real space renormalization group (RG hereafter) flow in the space of return probability distributions.…”

“…The merits and challenges of this approach are extensively discussed in Ref. [27]. This approach seems to offer the best setup to frame the probabilistic analysis of (possibly selforganized) critical behavior in time series.…”