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

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.

Figure 1

Model multiscaling behavior for $1\le t\le 31$ and $M\ge 30$. The couple $(D,\nu )$ determines the behavior.

Figure 2

Simple- ($q\lesssim 3$) and multi- ($q\gtrsim 3$) scaling behavior of S&P500 (log) returns PDF, analyzed for $t$ from 1 day to 2 months in 1950–2010. Notice the strong dependence of multiscaling features on the specific sample. In particular, data marked with an asterisk refer to the full interval 1950–2010 with an artificious alteration of a single datum: a reduction of $20\%$ of the strongest fluctuation.

Figure 3

Calibration outcome in terms of the scaling indicator $m_1^X$ for different values of $M$, also reported in the abscissa with different colors. (a) The model with $\rho$ given by Eq. (20); (b) the null model.

Figure 4

Calibration outcome in terms of the volatility autocorrelation $r_1^X$. (a) The model with $\rho$ given by Eq. (20); (b) the null model.

Figure 5

Single-variable PDF comparison between S&P500 and calibrated complete model (a) and between S&P500 and the null model (b) in linear scale. Symbols and lines color code as in the previous plots.

Figure 6

Single-variable PDF comparison between S&P500 and calibrated complete model (a) and between S&P500 and the null model (b) in log scale. Symbols and lines color code as in the previous plots.

Figure 7

Time series comparison between S&P500 (black) and complete model (a) and between S&P500 (black) and the null model (b). To facilitate the inspection, model time series are shifted by $-0.3$ ($M=63$), $-0.6$ ($M=42$), and $-0.9$ ($M=21$).

Figure 8

Multiscaling comparison between the S&P500 and the complete model (a) and between the S&P500 and the null model (b).

Figure 9

Level curves plot of $\overline{\chi }(t_h,t_r)$ or volatility mug shots. (a) Whole S&P500 time series ($T=15385$); (b) second half of S&P500 time series ($T=7694$); (c) model's prediction with the $M=42$ calibration.

Figure 10

A procedure to locate the restart. In black is depicted a $T=12000$-long time series generated by our model with the parameters $(\overline{D},\overline{\nu },\overline{\alpha },\overline{\beta })=(0.16,0.004,5.5,0.14)$, associated to the $M=63$ calibration. For convenience of inspection, the time series is vertically shifted by $-0.2$. In red $\mathbb{P}[I_t=1|X_n={\overline{x}}_n,|n-t|\le 2]$ is reported. The blue line corresponds to the threshold for which $\nu T$ restarts are detected; blue circles mark the true restarts generated by the dynamics.

Figure 11

Same as Fig. 10, but applied to the S&P500 historical time series. Here, of course, blue circles are absent.

Figure 12

Distribution of the long-memory component of the volatility: the continuous line is the theoretical prediction, Eq. (38); circles and squares refer to two time series (t.s.) of $T$ data generated by the model with the $M=63$ calibration parameter set; triangles are obtained from the time series with $T={10}^6$, with the time restarts chosen randomly.

Figure 13

Distribution of the long-memory component of the volatility for the S&P500 data set (circles), compared with the model-based prediction, Eq. (38) (full line).