THE STATISTICAL MECHANICS OF FINANCIAL MARKETS

Abstract

Chapter 1 contains a brief introduction to the subject to put the problems and investigations in proper perspective. It also provides a brief introduction to the study, motivation for the research, objectives of the research and an outline of organization of this research work with a Chapter wise summary. Chapter 2 reviews the literature relevant to this research. Literature review is focused on the contemporary work being done towards convergence of the disciplines of physics and finance. Relevant literature on quantum mechanics has also been reviewed, in particular, those areas that are relevant to the research being envisaged. Literature on the financial applications of various types of stochastic processes including Gaussian processes and levy processes has also be studied to identify gaps in existing knowledge in the field. The Black Scholes model of option pricing constitutes the cornerstone of contemporary valuation theory. However, the model presupposes the existence of several unrealistic assumptions including the lognormal distribution of stock market price processes. There, now, subsists abundant empirical evidence that this is not the case. Consequently, several generalizations of the basic model have been attempted with relaxation of some of the underlying assumptions. In Chapter 3 we postulate a generalization that contemplates a statistical feedback process for the stochastic term in the Black Scholes partial differential equation. Several interesting implications of this modification emanate from the analysis and are explored. The Black Scholes model also assumes constancy of the return on the "hedge portfolio". In Chapter 4, we attempt one such generalisation based on the assumption that the return process on the "hedge portfolio" follows a stochastic process similar to the Vasicek model of short-term interest rates. The return process of stock markets has also been modeled as a Levy process in several studies relating to valuation of contingent claims. In Chapter 5, we attempt a generalization of such results through a deformation of the underlying Levy process. The cardinal contribution of physicists to the world of finance came from Fischer Black & Myron Scholes through the option pricing formula which bears their epitaph and which won them the Nobel Prize for economics in 1997 together with Robert Merton.- They obtained closed form expressions for the pricing of financial derivatives by converting the problem firstly, to a partial differential equation and then to a heat equation and solving it for specific boundary conditions. In Chapter 6, we apply the well-entrenched group theoretic methods to obtain various solutions of the Black Scholes equation for the pricing of contingent claims. We also examine the infinitesimal symmetries of the said equation and explore group transformation properties. The structure of the Lie algebra of the Black Scholes equation is also studied. In Chapter 7, we apply the well entrenched methods of quantum mechanics and quantum field theory to the modeling of the financial markets and the behaviour of stock prices. After defining the various constituents of the model including creation & annihilation operators and buying & selling operators for securities, we examine the time evolution of the financial markets and obtain the Hamiltonian for the trading activities of the market. We finally obtain the probability distribution of stock prices in terms of the propagators of the evolution equations. Chapter 8 is devoted- to an empirical study of the Indian capital markets with data over the last ten years and it is shown that stock return processes deviate significant from normality. Performance of R/S analysis on the data also showed that memory effects are prevalent in the price time series with a possibility of nonlinearities and chaos. Chapter 9 contains major findings and significant contributions of the research duly summarized followed by the set of recommendations. The thesis finally ends with the limitations of the study and suggestions for further