Additive-multiplicative stochastic models of financial mean-reverting processes

We investigate a generalized stochastic model with the property known as mean reversion, that is, the tendency to relax towards a historical reference level. Besides this property, the dynamics is driven by multiplicative and additive Wiener processes. While the former is modulated by the internal behavior of the system, the latter is purely exogenous. We focus on the stochastic dynamics of volatilities, but our model may also be suitable for other financial random variables exhibiting the mean reversion property. The generalized model contains, as particular cases, many early approaches in the literature of volatilities or, more generally, of mean-reverting financial processes. We analyze the long-time probability density function associated to the model defined through an Itô-Langevin equation. We obtain a rich spectrum of shapes for the probability function according to the model parameters. We show that additive-multiplicative processes provide realistic models to describe empirical distributions, for the whole range of data.

Figure 1

Diagram of the asymptotic behavior of the PDF given by Eq. (3), in the $(s,r)$ plane. Unshadowed regions and dotted borders identify regions excluded by the normalization condition. At the positive $r$ axis, the PDF is finite at the origin. Tilted lines denote the marginal cases $r=2s(p=0)$, with pure exponential tail and power-law growth at the origin, and $r=2s-1(p=-1)$, with power-law tail and exponential of $1∕x$ growth at the origin (the threshold points of these lines have coordinates $s=[1+{\gamma }_{\mu }\theta ]∕2$ and $[1-{\gamma }_{\mu }]∕2$, respectively). Parameter $a>0$, in the exponential formulas, as well as the power-law exponents, depend on model parameters. Symbols correspond to the special processes: Hull-White (HW), Heston (H), Ornstein-Ulhenbeck (OU), and geometric OU (GOU).

Figure 2

Diagram of the asymptotic behavior of the PDF defined by Eq. (12), in the $(s,r)$ plane. Unshadowed regions and dotted borders are regions excluded by the normalization requirement. At both positive semiaxes, the growth at the origin is power law. On dark gray lines, tails are power law, with the tilted line corresponding to $r=2s-1(p=-1)$. Dashed lines correspond to pure exponential tails, with the tilted line corresponding to $r=2s(p=0)$. In the formulas, $a>0$, as well as the power-law exponents, generically depend on model parameters, moreover $p\equiv r-2s$. Symbols correspond to the special processes (A) [Eq. (16)] and (B) [Eq. (18)].

Figure 3

PDF of normalized volatility of stocks traded in US equity market (data from Ref. 6). The full line corresponds to a fitting by the theoretical PDF given by expression (20). Fitting parameters are $(q,\nu ,v_o^2)\simeq (1.178,2.20,0.097)$. Insets: linear-linear and log-log representation of the same data.