Abstracts

Power law appears in finance in distributions of ranked data such as stock market capitalizations, volatility clusters, market returns, volumes, etc. In particular, ranked market shares, known as capital distribution curves are known to possess certain shape, which is stable over periods of time. Despite different natures of these phenomena it is possible to look at them as random division of a constrained resource under some symmetric measure.
In the first part of this paper we propose to model distribution of ranked market capitalizations by means of the two-parameter Poisson-Dirichlet measure, since it provides greater flexibility over Dirichlet and stable distributions. Definition of the Poisson-Dirichlet process by subordination according to propositions 14 and 21 in Pitman, Yor [1997] generalizes Kingman's ranked jumps analysis of random measures.
Second part of this paper is devoted to application of fractional operator calculus for analytic representation of subordinators. Fractional derivative operator is related to generalized Laplace transform and allows simplified treatment of special functions expressed in power series, such as Bessel and Mittag-Leffler functions, as well as divergent series. Particularly, we show that cumulative density functions of stable and tempered stable subordinators admit simple closed form expressions in terms of fractional derivative operator applied to unity. Consequently, Levy measure appears naturally as probability distribution of jumps over infinitesimal time increments and also has fractional calculus representation. This also allows to obtain Levy measure of composition of two subordinators.
Stochastic volatility can be also considered as random division of some finite activity in limited period of time, where non-normalized two-parameter Poisson-Dirichlet process provides subordinator, which sits between gamma subordinator (VG model) and tempered stable one (CGMY model). The model can be generalized by consideration of alternative subordinators and by application to other types of financial data exhibiting power laws.