Fractal Activity Time Models for Risky Asset with Dependence and Generalized Hyperbolic Distributions

Abstract

Risky asset models with the dependence through fractal activity time are described. The construction of the fractal activity time is implemented via superpositions of Ornstein-Uhlenbeck type processes driven by Lévy noise. The model features both tractable dependence structure and desired marginal distributions of the returns from the generalized hyperbolic class: the Variance Gamma and normal inverseGaussian. These distributions provide good fit to real financial data. Pricing formulae for the proposed models are derived.

Weak dependence and GMM estimation of supOU and mixed moving average processes

“…Assuming that f is not equal to zero π-almost everywhere and that f ≥ 0 or f ≤ 0, the asymptotic variance in Theorems 4.1 and 4.2 is not degenerate. This is the case for example when working with the supOU process (37). Moreover, it is worthy to observe that the assumptions in Theorem 4.2 clearly indicate that we obtain asymptotic normality of the sample mean for a causal MMA process just in the short memory case.…”

“…Let us consider a supOU process X defined as in (37) such that the underlying Lévy process L is a subordinator. It can be shown that the process is adapted and càdlàg under the assumptions (ii) and (iii) of [10,Theorem 3.12].…”

“…For example, its autocorrelations can show a polynomial decay depending on the probability distribution π. Moreover, when a discrete probability distribution π for the random parameter A is considered, we obtain a popular model used, for example, in stochastic volatility models [3], in modeling fractal activity times [36,37] and in astrophysics [35].MMA processes can also be used, under suitable conditions, as building blocks for more complex models. We study in this paper the class of MMA stochastic volatility models.…”

Fractional Skellam processes with applications to finance

“…Meerschaert et al [23] showed that the same fractional Poisson process can also be obtained via an inverse stable time change. Using a time change in models for financial data has been popularized in the last decade based on the idea that it is not the calendar time that drives the changes in price, but rather the information flow or activity time is what matters for modeling of the prices [15,14,18]. In this paper we define fractional Skellam processes via time changes in ordinary Skellam processes.…”

An Inverse Gamma Activity Time Process with Noninteger Parameters and a Self-Similar Limit

“…For example, Leonenko et al (2012) constructed gamma and inverse Gaussian processes, based on a superposition of Ornstein-Uhlenbecktype processes, which converge to Brownian motion (in the case of a finite superposition) or fractional Brownian motion (in the case of an infinite superposition). Finlay and Seneta (2007) also reviewed a number of different approaches to constructing activity time processes.…”