Abstract

The developing of the financial market contributes to paying attention to analyses its dynamics and quality of properties of the variables. The results of the analysis showed that among the complex technical applications the geometry approach in this area is an advanced method to explain the financial market behaviour. This paper expands the idea that the portfolio is balanced for quotation in marketing is equivalent to the number of points that lie on the quotation hyperplane in projective geometry has been extended in geometric concepts. In this paper, the new geometric approach was proposed to estimate the number of times the portfolio is balanced for quotation. In fact, the calculations were made in terms of the volume of the Halmos-Thomson (as the bridge between Feinsler geometry, integral geometry, and symplectic geometry). The highest number of risks in market equilibrium was mentioned in the geometric concepts mentioned. In this way, the proposed geometric approach allows to analyse situations with a portfolio under different conditions: financial market equilibrium, reduce the risk of investment risk, and others developments in the stock market. The author calculates the parameters through the surface area of a convex body of the corresponding projective spaces and Holmes-Thompson volume in notions of integral geometry (Radon and Fourier transform), Finsler Geometry (and Minkowski spaces) and symplectic geometry. Contrary to existing numerical methods, this approach allows one to reach the analytic solution and also, concludes that the highest number of risks in market equilibrium can be obtained by minimality of the introduced volume.

Keywords: financial market, risk, Minkowski space, Finsler metric, Fourier transform.

JEL Classification: B16, D81, G32.

Cite as: Hasan-Zaden, A. (2018). Geometric modelling of portfolio and risk in market equilibrium. Marketing and Management of Innovations, 2, 210-217. https://doi.org/10.21272/mmi.2018.2-17

This work is licensed under a Creative Commons Attribution 4.0 International License

References

1. Henry-Labordère, P., Analysis, Geometry, and Modeling in Finance Advanced Methods in Option Pricing, CRC Press, Taylor & Francis Group, 2009.
2. Swishchuk, A. and Islam, S., Random Dynamical Systems in Finance, Taylor and Francis Group, 2013.
3. Farinelli, S., Geometric arbitrage theory and market dynamics, American Institute of Mathematical Sciences, 7(4), 2015, 431-471.
4. Zabreiko, P.P. and Lebedev, A.V., Banach geometry of financial market models. Doklady Mathematics, 95(2), 2017, 164-167.
5. Chile, S., Random Geometric Analysis in the Stochastic Volatility: Financial Markets States Degeneracy, Analysis and Computations Journal, Forthcoming, 2017, SSRN: https://ssrn.com/abstract=2968295.
6. Piotrowski, E.W. and Stankowski, J., Geometry of Financial Markets – Towards Information Theory Model of Markets, Physica A: Statistical Mechanics and its Applications, 382(1), 2007, 228-234.
7. Piotrowski, E.W. and Stankowski, J., The merchandising mathematician model: profit intensities, Physica A: Statistical Mechanics and its Applications, 318(3-4), 2003, 496-504.
8. Holmes, R.D. and Thompson, A.C., N-dimensional area and content in Minkowski spaces, Pacific Journal of Mathematics, 85(1), 1979, 77-110.
9. Paiva, J.C.A. and Fernandes, E., Fourier transforms and Holmes-Thompson volume of Finsler manifolds,  International Mathematics Research Notices, 1999(19), 1999, 1031-1042.
10. Paiva, J.C.A. and Thompson, A.C., Volumes on normed and Finsler spaces, Chapter book of a Sampler of Riemann-Finsler Geometry, Cambridge University Press, 2004, 1-48
11. Paiva, J.C.A., Some problems on Finsler geometry, Handbook of Differential Geometry, vol. 2, 2006, 1-33
12. Koldobsky, A., Fourier Analysis in Convex Geometry, American Mathematical Society, vol. 116, 2006.
13. Benjani, A.A., Finsler geometry and applications, Ellis Horwood, 1996.
14. Besse, A., Manifolds of all whose geodesics are closed, Springer-Verlag, New York, 1978.
15. Gelfand, I.M and Smirnov, M., Lagrangians satisfying Crofton formulas, Radon transforms, and nonlocal differentials, Advances in Mathematics, 109(2), 1994, 188-227.
16. Gromov, M., Filling Riemannian manifolds, Journal of Differential Geometry, 18(1), 1998, 1-147.
17. Rudin, W., Functional Analysis, Mc Graw-Hil. New York, 1973.
18. McDuff, D. and Salamon, D., Introduction to symplectic topology, Oxford Mathematical Monographs, Clarendon Press, 1998.
19. Dym, H. and ‎McKean H.P., Fourier series and Integrals, Academic Press, New York, 1972.
20. Schneider, R., On integral geometry in projective Finsler spaces, Izvestiya Natsional’noĭ Akademii Nauk Armenii. Matematika, 37, 2002, 34-51.
21. Schneider, R. and Wieacker, J.A., Integral geometry in Minkowski spaces, Advances in Mathematics, 129(2), 1997, 222-260.
22. Paiva, J.C.A. and Fernandes, E., Crofton formulas in Projective Finsler spaces, Electronic Research Announcements of the American Mathematical Society, 4, 1998, 91-100.