Fibrations of financial events

Carfì, David (2009): Fibrations of financial events. Published in: Proceedings of the International Geometry Center - Proceedings of the International Conference “Geometry in Odessa 2009” - ISSN 2072-9812 , Vol. vol. 2, No. 3 (2009): pp. 7-18.

Abstract

In this paper we shall prove that the plane of financial events, introduced and applied to financial problems by the author himself (see [2], [3] and [4]) can be considered as a fiber bundle in two different ways. The first one, the natural one, reveals itself to be isomorphic to the tangent bundle of the real line, when the last one is considered as a differentiable manifold in the natural way; the second one is a fibration induced by the status of compound interest capitalization at a given rate i in the interval ] − 1, → [. Moreover, in the paper we define on the first vector bundle an affine connection, also in this case induced by the compound interest capitalization at a given rate i. The final goal of this paper is the awareness that all the effects determined by compound interest capitalization are nothing but the consequences of the fact that the space of financial events is a fibration endowed with a particular affine connection, so they are consequences of purely geometric properties, at last, depending upon the curvature determined by the connection upon the fibration. A natural preorder upon the set of fibers of the second fibration is considered. Some remarks about the applicability to economics and finance of the theories presented in the paper and about the possible developments are given in the directions followed in papers [1], [5], [6], [7], [8] of the author himself.

• Item Type: MPRA Paper
• Language: English
• Keywords: Financial event; compound interest; capital evolution; accumulation function; force of interest; fiber bundle; vector bundle; affine connection; Christoffel bilinear form
• Subjects: E - Macroeconomics and Monetary Economics > E6 - Macroeconomic Policy, Macroeconomic Aspects of Public Finance, and General Outlook; D - Microeconomics > D1 - Household Behavior and Family Economics; D - Microeconomics > D1 - Household Behavior and Family Economics > D14 - Household Saving; Personal Finance; G - Financial Economics > G2 - Financial Institutions and Services > G21 - Banks ; Depository Institutions ; Micro Finance Institutions ; Mortgages

References

[1] David Carf`i, Optimal boundaries for decisions, Atti dell’Accademia Peloritana dei Pericolanti - Classe di Scienze Fisiche, Matematiche e Naturali Vol. LXXXVI, (2008)

[2] David Carf`i, Structures on the space of financial events, Atti dell’Accademia Peloritana dei Pericolanti - Classe di Scienze Fisiche, Matematiche e Naturali, Vol. LXXXVI, (2008)

[3] David Carf`i, Giovanni Caristi, Financial dynamical systems, Differential Geometry - Dynamical Systems, Vol.10, (2008)

[4] David Carf`i, The family of transformations associated with a financial law, Atti dell’Accademia Peloritana dei Pericolanti - Classe di Scienze Fisiche, Matematiche e Naturali, Vol. LXXXI- LXXXII, (2004)

[5] David Carf`i, S-Linear Algebra in Economics and Physics, APPLIED SCIENCES Vol. 9, 2007

[6] David Carf`i, Prigogine approach to irreversibility for Financial and Physical applications, Sup- plemento Atti dell’Accademia Peloritana dei Pericolanti di Messina, Proceedings Thermocon’05. (2008)

[7] David Carf`i,Dyson formulas for Financial and Physical evolutions in S′, Communications to SIMAI congress, vol. 2 (2007)

[8] D. Carf`i, Feynmann’s transition amplitudes in the space S′, Atti della Accademia Peloritana dei Pericolanti, classe di scienze Fisiche Matematiche e Naturali. Volume on line LXXXIII (2005)

[9] J. A. Dieudonn`e, Element d’Analyse, vol. 3, Edition J. Gabay.