Multiple time scales and the empirical models for stochastic volatility

The most common stochastic volatility models such as the Ornstein-Uhlenbeck (OU), the Heston, the exponential OU (ExpOU) and Hull-White models define volatility as a Markovian process. In this work we check of the applicability of the Markovian approximation at separate times scales and will try to answer the question which of the stochastic volatility models indicated above is the most realistic. To this end we consider the volatility at both short (a few days) and long (a few months)time scales as a Markovian process and estimate for it the coefficients of the Kramers-Moyal expansion using the data for Dow-Jones Index. It has been found that the empirical data allow to take only the first two coefficients of expansion to be non zero that define form of the volatility stochastic differential equation of Ito. It proved to be that for the long time scale the empirical data support the ExpOU model. At the short time scale the empirical model coincides with ExpOU model for the small volatility quantities only.Comment: 19 pages, 6 figure

Multiple time scales in volatility and leverage correlations: An stochastic volatility model

Financial time series exhibit two different type of non linear correlations: (i) volatility autocorrelations that have a very long range memory, on the order of years, and (ii) asymmetric return-volatility (or `leverage') correlations that are much shorter ranged. Different stochastic volatility models have been proposed in the past to account for both these correlations. However, in these models, the decay of the correlations is exponential, with a single time scale for both the volatility and the leverage correlations, at variance with observations. We extend the linear Ornstein-Uhlenbeck stochastic volatility model by assuming that the mean reverting level is itself random. We find that the resulting three-dimensional diffusion process can account for different correlation time scales. We show that the results are in good agreement with a century of the Dow Jones index daily returns (1900-2000), with the exception of crash days.Comment: 19 pages, 5 figure

The role and management of physical space in social innovation

The aim of this work is thus to investigate the role of physical space in social innovation activities and projects and its relevance in terms of costs - in the framework of business planning and modelling - and in terms of social impact generated. Despite of the increasing interesting in social innovation, previous studies missed to tackle space as a specific barrier and asset of social innovation activities. In this sense, this research aims at filling a gap in the conceptual framework of social innovation, analysing how physical space is managed in social innovation ventures and to which extent physical space represent not only a physical asset but might generate also intangible assets. The research performed consisted of literature review on social innovation definition, social impact measurement and practices and on the analysis of space as enabling factor of innovation. Through empirical observation of 52 social innovation projects and building of three case studies, the work provides insights on the business models for social innovation projects centered around the acquisition and maintenance of a physical space, the social impact generated by these projects and linked to the physical space and measurement efforts done to assess the social impact

Trading activity as driven Poisson process: comparison with empirical data

We propose the point process model as the Poissonian-like stochastic sequence with slowly diffusing mean rate and adjust the parameters of the model to the empirical data of trading activity for 26 stocks traded on NYSE. The proposed scaled stochastic differential equation provides the universal description of the trading activities with the same parameters applicable for all stocks.Comment: 9 pages, 5 figures, proceedings of APFA