Residue Sum Formula for Pricing Options under the Variance Gamma Model

We present and prove a triple sum series formula for the European call option price in a market model where the underlying asset price is driven by a Variance Gamma process. In order to obtain this formula, we present some concepts and properties of multidimensional complex analysis, with particular emphasis on the multidimensional Jordan Lemma and the application of residue calculus to a Mellin–Barnes integral representation in C3, for the call option price. Moreover, we derive triple sum series formulas for some of the Greeks associated to the call option and we discuss the numerical accuracy and convergence of the main pricing formula.

On functions with the boundary Morera property in domains with piecewise-smooth boundary

The problem of holomorphic extension of functions defined on the boundary of a domain into this domain is actual in multidimensional complex analysis. It has a long history, starting with the proceedings of Poincaré and Hartogs. This paper considers continuous functions defined on the boundary of a bounded domain $ D $ in $ \mathbb C ^ n $, $ n> 1 $, with piecewise-smooth boundary, and having the generalized boundary Morera property along the family of complex lines that intersect the boundary of a domain. Morera property is that the integral of a given function is equal to zero over the intersection of the boundary of the domain with the complex line. It is shown that such functions extend holomorphically to the domain $ D $. For functions of one complex variable, the Morera property obviously does not imply a holomorphic extension. Therefore, this problem should be considered only in the multidimensional case $ (n> 1) $. The main method for studying such functions is the method of multidimensional integral representations, in particular, the Bochner-Martinelli integral representation.

SOME PRICING TOOLS FOR THE VARIANCE GAMMA MODEL

We establish several closed pricing formulas for various path-independent payoffs, under an exponential Lévy model driven by the Variance Gamma process. These formulas take the form of quickly convergent series and are obtained via tools from Mellin transform theory as well as from multidimensional complex analysis. Particular focus is made on the symmetric process, but extension to the asymmetric process is also provided. Speed of convergence and comparison with numerical methods (Fourier transform, quadrature approximations, Monte Carlo simulations) are also discussed; notable feature is the accelerated convergence of the series for short-term options, which constitutes an interesting improvement of numerical Fourier inversion techniques.

GIS-based risk monitoring of zoonotic cestodiasis in human

Objective of research: The target of the paper is to develop a model of GIS-based risk monitoring of zoonotic cestodiasis in human. Materials and methods: The use of geographic information systems (GIS) as an epizootiological and epidemiological method for the risk-based monitoring of human cestodiasis enables the development of a multi-level platform for solution of a wide range of tasks related to the control of this disease. The modern GIS tools use the methods of geoinformatics applying powerful software and hardware: open access geographic web servers, tools for multidimensional complex analysis, creating most accurate electronic and paper maps. Full-featured GIS contain a full set for processing geospatial data including acquisition of data, its integration and storage, automatic data processing, editing, creation and maintenance of topology, spatial analysis, access to the database management system (DBMS), visualization and creation of hard copies of any cartographic data. Results and discussion: The use of GIS enables to study more closely the regularities of epizootic process, geography of human cestodiasis and to improve the methodology both for short-term and long-term retrospective epizootiological analyses.