NEW PERSPECTIVES IN THE COMPLEXITIES AND CHAOS OF STOCK PRICE FLUCTUATIONS IN THE CAPITAL MARKET

The complexities and chaotic nature of fluctuations of stock price of Financial Institutions in the stock market is a known phenomenon. In this thesis we investigate some new perspectives in the fluctuations, chaos and complexities of the stock market using a combination of the Picard’s Iterative Method, the Melnikov’s Method and the Lyapunov’s Methods to create a novel way of tracking down the fluctuations in the market, so as to restore the investors’ confidence in the market. The results show that the approximate solution and chaos exist. Arising from this work also is the stability analysis of the equilibrium points of the market fluctuations in prices. The relevance of this thesis can be found in its application to policy formulation and in high risk businesses to minimize risk.

 

The problem of instability in the capital market due to the stock price fluctuations cannot be over emphasised. This problem needs solution since it is a set-back for the financial institutions. This work attempts to investigate some new perspectives in the fluctuations, chaos and complexities in share prices by using a combination of Picard’s Iterative method, the Melnikov’s method, and the Lyapunov’s method.

The mathematical modelling of many real life phenomena by reason of random perturbation are not usually done by ordinary differential equations. This is because such differential equations cannot be solved analytically hence the study of Duffings Oscillator Model which has cubic nonlinearity. Most problems that occur in real life are nonlinear in nature. The study of such are designed to perform certain order of accuracy in many fields of endeavour like science, engineering and finance.

Researches have shown the importance of stock market price fluctuations in the finance literature especially after the 1987 stock market crash (Kambouroudis, 2011). Although many researchers in the economic sector (like Fama and French (1989)) have developed models and key parameters on improving stock price fluctuations, finance research has still to reach a consensus on this matter. Thus this Dissertation joins the on-going debate and conduct research by investigating some new perspectives in the fluctuations, chaos and complexities of the stock price, using a combination of Picard’s method, Melnikov’s method and Lyapunov’s methods to create a novel way of tracking down the fluctuations in the market so as to restore the share-holder’s hope in the market. This Dissertation also proposed further key parameters which are used in improving the accuracy of the Duffing oscillator model.

Accurate modelling of the stock price fluctuations is of significant importance to investors in the stock market. Stock price fluctuations are associated with risk. The chaos of the stock price fluctuation is because the securities are not fairly priced.

Financial markets are important parts of the business sector since most big time companies invest in them and spend huge amount of money with the expectation of huge profits. Sometimes, because of the huge risk involved, huge losses are incurred instead.

Stock price fluctuations are naturally occurring phenomena. Existing models to study these fluctu- ations and the associated complexities and chaos are scanty and not robust. It therefore does not encourage investors to rely on them to make proper decisions and financial planning. In order to boost the knowledge in literature and also address these pertinent challenges, there is need for a more reliable model for this Dissertation.

The Model or the Governing Equation of this study is:

with the initial conditions as:

 

u (0) = u0,

 

This study aims at investigating some new perspectives in the fluctuations, chaos, and complexities of the stock price fluctuations using a combination of the Picard’s method, the Melnikov’s method and the Lyapunov’s method to study the Duffing Oscillator Model for Dissertation study.

The specific objectives are to:

 

The purpose of this study is to examine the new perspectives in the complexities and chaos of stock price fluctuations in the capital market. Greenwald et al., (2015) listed some reasons for the fluctua- tions. Studies in previous researches show that several methods have been used to track down these fluctuations but up till now there are still evidences of investors losing their investments due to these fluctuations. The problem associated with these fluctuations still persists. As the investors face the increased risk of loss in business caused by the stock price fluctuations, there arises the need for this study which is a way forward in the solution of the persisting problem. Many of the resent studies show evidence of excessive fluctuations (Scott, 1991). The results of this project may provide insight into the solution of the chaos in the stock market.

Investors have wondered how to better speculate the stock price fluctuation in the stock market. Even though Muller and Eisler (2007), Kambouroudis (2011) and Khositkulporn (2013) in their Ph.D thesis studied stock price volatility. They used different methods such as the GARCH type models, data collection etc., but the problem still lingers on. This calls for more research in this area. In this Dissertation we examined the stock price fluctuation by reducing the problem to a simple system (by formulating a market model), that is, the Duffing Oscillator Model and using different methods to check the existence, the solution itself, the chaotic behaviour and its stability analysis. The results bring us close to solution of the problem of stock price fluctuations and how they can better be used to quickly track down these fluctuations. This work is inspired by problems of financial institutions as regards the fluctuations in prices of commodities in the capital market. These fluctuations in stock prices make the risk encountered in the business high thus causing investors’ huge loss. The works of Von Bre Men et al. (1997) were studied and it was found that the Lyapunov exponent can be used to investigate the complexities and fluctuations in the prices of stock.

 

The findings of this study will be of benefit to the society considering the fact that stock price fluctuations play a vital function in the stock market today. The great demand for the formulation of strategies to minimize the risk taken by domestic investors justifies the need for an efficient market model that captures the different parts of the market. Thus the investors that apply the results of this work will be able to minimize the risk in the market better. The policy administrators will be guided by the outcome of this study on their policy formulation. This research will help to uncover critical areas associated with the business process that many researchers were not able to explore. Thus a new way of tracking down the stock price fluctuations is arrived at.

The study covers the investigation of the existence and approximate solution of the cubic nonlinear oscillators (market model) by Picard’s method, the determination of chaos in the oscillators by the Melnikov method and the study of its behaviour by the use of Lyapunov methods.

A period with falling stock prices. It is a condition in which securities prices fall 20% or more from recent heights and widespread pessimism and negative investor sentiment. A market that is bullish depicts a financial market condition in which securities price rise.

are natural or standard transformation of the trajectories in a state space of a system. It is the transactions taking place in the market.

 

is a fixed income investment in which an investor loans money to an entity. It is an investment of indebtedness.

is a market where buyers and sellers engage in trade of financial securities like bonds, stocks etc.

in the stock price fluctuations is the behaviour of the stock market as the stock price fluctuates. Chaos theory is a branch of Mathematics focusing on the dynamical systems that are highly sensitive to initial conditions.

occur when the motion between attractors cannot be easily described as simple combinators. Chaotic attractors occur when the movement (up and down) of the stock prices cannot be easily described.

is a condition where components put together are different to find answer.

is the action that reduces the oscillation degree in electrical or mechanical device signal intensity or prevents it from increasing. Example sound proof gadgets reduce the vibration of sounds.

describes a geometrical space where all the points in a function are time dependent. It describes a market where the stock price fluctuation is a function of time.

 

in differential equations describes solutions of the differential equation that are constant. In business, equilibrium point is the maximum or lowest point of commodities in which the amount generated produces equal demand and supply.

in finance, equity or owner’s equity is someone’s total of assets minus the value of his liability.

is a strategy aimed at stabilizing the stock activity and is typically applied at an uncertain dominant frequency of time signal. However, a change in the strength of this external signal may result in a period of changing from chaotic behaviour to regular behaviours or vice-versa.

describes a place where financial securities and derivatives are traded at a low transaction cost. Secu- rities include stocks and bonds, and derivatives include precious metal.

are continuous irregular variation or change.

is a function that shows its total energy i.e. its kinetic energy (energy of motion) plus its potential energy (energy of position).

are trajectories that show the direction of the dynamical system that adds a saddle equilibrium point to itself.

 

Inflation is a quantitative measure of the rate at which the average price level of a basket of selected goods and services in an economy increases over a period of time.

tells us the rate of convergence or divergence of nearby trajectories- a key component in chaos dynam- ics. In one-dimensional map, the exponent is the average Dlog[ df ]E over the dynamics.

is used to determine a measure of distance between stable and unstable manifolds in the Poincare map. When this measure is equal to zero, by the method, these manifolds cross each other transversely and from that crossing, the system becomes chaotic.

is a tool to identify chaos that occurs in dynamical systems under periodic perturbation.

is a new way of expressing something with different viewpoints.

is a conservative force or a function of position. Examples are elastic potential energy, springs, rubber, slingshots, etc.

In pull marketing traders are drawn to the product to buy and sell or consume.

 

is any asset that carries a degree of risk. It has a high degree of price volatility. Examples are equities, commodities, and currencies.

is a local bifurcation in which two fixed points (or equilibra) collide and annihilate each other. Example of a differential equation with saddle node bifurcation is dx r+x2, where x is a state variable and r is the bifurcation parameter, r < 0 represents two equilibrium points r = 0 represents the bifurcation point, the only one equilibrium point called saddle-node fixed point. r > 0 represents no equilibrium point.

is a tradable finance asset. Example is equity security, i.e. common stocks.

are trajectories in systems with multiplecritical points that serve to separate regions with different qualitative behaviours.

is a collection of all possible states of the system. Each coordinate is a state variable and the state variables completely describe the state of the system.