Brief Review of Financial Time-Series Models
As discussed above, the naive random walk model is too simple to describe the complexity of price dynamics. The fractional Brownian motion (Mandelbrot and Van Ness 1968), which is the natural extension of the random walk that accounts for long memory, can not be directly applied for modeling due to the presence of memory both in volatility and signed returns (violation of the “no arbitrage hypothesis”). The most obvious way to account for the absence of linear autocorrelation, but preserve structure in volatility is to separate noise term from volatility term in the equation for returns in the following multiplicative manner:
where ijt structure-less represents iid innovations (often considered to be Gaussian) and a, represents volatility of the process. For a, = a0 = const one recovers the simple random walk model. Depending on the structure of a, models are typically classified into two groups: stochastic volatility models, where a, is modeled as an independent from r, and stochastic process, and conditional volatility models where a, is defined as a functional form of the past values of rx and £x (for x < г).
One of the most well-known models of conditional volatility family are Autoregressive Conditional Heteroscedasticity model (ARCH) (Engle 1982) and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) (Bollerslev 1986) models. They are successfully used for reproducing volatility clustering and non-trivial (but though sufficiently short) memory in volatility. The ARCH/GARCH models gave birth to the whole family that accounts for more than 50 different models (see review in Bollerslev (2010)), the most popular of which are: t-GARCH with innovations having t-Student distribution that reproduces heavy tails of returns distribution; EGARCH (Exponential GARCH) and T-GARCH (Threshold GARCH) that model leverage effect; FIGARCH (Fractional Integrated GARCH), MS-GARCH (Markov Switching GARCH) and LM-GARCH (Long Memory GARCH) that account for long memory in volatility and some others. Without spending time on discussion of all of them we suggest a number of reviews and handbooks with details, such as (Engle 2001; Ai’t-Sahalia and Hansen 2009; Zakoian and Francq 2010). Being very flexible with respect to modifications, ARCH/GARCH family is constrained with its autoregressive form and does not allow to easily and parsimoniously combine different stylized facts within one model. However due to its simplicity and sufficient robustness the whole family is being very widely used nowadays.
In the present paper we focus on another class of models—stochastic volatility models, and in particular at its subclass of so-called multifractal models. The theory of multifractal random processes started with rethinking and generalization of cascade models that was introduced by Richardson (1961) and Kolmogorov (1941, 1962). Being proposed to model velocity in turbulence, they reflect the fact, that in turbulent gas or fluid flow energy is transferred from large-scale vortices to small-scale vortices by cascades where structures at different scales are similar to each other (resulting in self-similarity of the whole system). Similar cascade structures for returns at different time scales were observed at financial markets as well (Ghashghaie et al. 1996), and the idea of self-similar cascades were used in several models, most successful of which are Multiplicative Cascades Model (MCM) (Breymann et al. 2000) and Markov Switching Multifractal (MSM) model (Calvet and Fisher 2008, 2004). When MCM model successfully reproduced heavy tails of returns distribution, long memory in absolute returns and volatility clustering, practical application of this model is limited due to nontransparent parametrization and absence of robust method of parameters estimation. In contrast, when calibration of the MSM model is relatively simple and sufficiently robust, the model describes heavy tails and long memory only in the limit of infinite number of components. However despite these drawbacks and unclear economic underpinning and a rather artificial discrete hierarchical structure, MSM model was shown to be much better in terms of volatility forecasting than GARCH and some of its siblings (Calvet and Fisher 2004).
The multifractal random walk (MRW) (Bacry et al. 2000, 2001) is the only continuous stochastic stationary causal process with exact multifractal properties and Gaussian infinitesimal increments. Being first introduced within stochastic volatility framework (2), later it was shown to have also exact cascade representation (Bacry et al. 2008; Bacry and Muzy 2003). The exact multifractality comes with a cost of a delicate tuning to a critical point associated with logarithmic decay of the correlation function of the log-increment up to an integral scale. As a consequence, the moments of the increments of the MRW process become infinite above some finite order, which depend on the intermittency parameter of the model. The extension of MRW—Quasi-Multifractal (QMF) model (Saichev and Filimonov 2008,2007; Saichev and Sornette 2006)—was free of these drawbacks. Rather than insisting on the exact multifractal properties, QMF model described process that was approximately multifractal within the finite range of scales. This approximation makes the model more flexible and removes above contradictions. Being very successful in reproducing many stylized facts, most of multifractal models failed to describe leverage and gain-loss asymmetry effects. To account for asymmetry effects, the Skewed MRW (Pochart and Bouchaud 2002) explicitly introduce the negative correlations between returns and volatility. More parsimonious way was implemented in the so-called Self-Excited Multifractal (SEMF) model (Filimonov and Sornette 2011), which describes the self-reinforcing feedback mechanism (explicit dependence of the future returns on the dynamics of past returns) in a manner similar to autoregressive models.