By Valery A. Kholodnyi Beliefs-Preferences Gauge Symmetry Group and Replication of Contingent Claims in a General Market Environment studies the actual financial phenomena underlying the evaluation of financial derivatives, which is today virtually identified with and even replaced by the study of the mathematical aspects of stochastic calculus as a model for such phenomena. This book expresses the view that the study of financial phenomena is on the brink of a revolution similar to that of quantum physics in the 1920s. This volume introduces a fundamental symmetry, a gauge symmetry, between beliefs of market participants and their preferences in a general market environment for a market with exchange of an arbitrary number of arbitrary underlying securities. In particular, this beliefs-preferences gauge symmetry makes it possible to obtain an evolution equation that determines, in a general market environment, the values of European contingent claims independent of these beliefs and preferences. This evolution equation is obtained under the assumption that market participants have agreed upon the market values of a certain minimal set of European contingent claims. It is this set of European contingent claims that dynamically spans the market, and an equation is also obtained that determines the portfolios of these European contingent claims needed to dynamically replicate a given European contingent claim. Being able to dynamically replicate general European contingent claims in a general market environment makes it possible to dynamically replicate contingent claims of a general type, such as the universal contingent claims introduced by the author in 1995. In the particular case of the beliefs of market participants given by general multidimensional diffusion processes the evolution equation that determines, in a general market environment, the values of European contingent claims independent of the beliefs and preferences of market participants is nothing but the Black and Scholes equation. Moreover, the dynamically spanning set of European contingent claims can be chosen as the underlying securities themselves and a pure discount bond with a dynamically replicating portfolio chosen according to the standard delta hedging. The practical applications of the beliefs-preferences gauge symmetry are significant and far-reaching. They range from the detection of a new type of true arbitrage to the beliefs-preferences-independent valuation and dynamic replication of contingent claims in a general market environment. Readership: Finance practitioners, researchers, mathematicians, physicists and engineers. Contents: Introduction; Preliminaries; The Beliefs-Preferences Gauge Symmetry Group; Beliefs-Preferences Gauge Symmetry Group for a Markovian Generalized Market Populace with Generalized Beliefs Determined by Diffusion Processes; Beliefs-Preferences Gauge Symmetry Group for a Markovian Market Populace with Beliefs Determined by Diffusion Processes; Application of the Beliefs-Preferences Gauge Symmetry Group to the Dynamic Replication of European Contingent Claims; Application of the Beliefs-Preferences Gauge Symmetry Group to the Dynamic Replication of European Contingent Claims for a Markovian Market Populace with Beliefs Determined by Diffusion Processes; Method of Quasidifferential Operators for an Approximate Dynamic Replication of European Contingent Claims Based on the Beliefs Preferences Gauge Symmetry Group.
Valery A. Kholodnyi received his Ph.D. in Applied Mathematics from the Moscow Institute of Electronics and Mathematics in 1990. He has held university positions in various departments, such as the Department of Microwave and Quantum Electronics, the Department of Mathematical Modeling of Physical Processes, and the Department of Physics, in both Russia and the United States. He has authored or co-authored two other monographs and over 60 research papers in finance, mathematics, theoretical physics and engineering and has published in journals such as the Journal of Mathematical Physics, the Journal of Integral Equations and Applications, and Nonlinear Analysis, Theory, Methods and Applications. He was an Invited Speaker at the Second World Congress of Nonlinear Analysts and at numerous international and national conferences, as well as at research seminars in university departments and industry. Currently, he is the Vice-President of Research and Development for Integrated Energy Services, L.C., an independent research institute for financial capital markets.
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