DEPARTMENT OF MATHEMATICS

This study explores the methodology underlying the pricing of call options using continuous-time models. Beginning with the Wiener process as a foundational model, the research extends into stochastic calculus and the derivation of the Black-Scholes equation. The analysis delves into the fundamental properties of Brownian motion, the normal distribution, and stochastic differential equations to establish a rigorous mathematical framework for asset price movements.

A key focus is placed on the arbitrage argument, which ensures that financial markets remain free from riskless profit opportunities. The no-arbitrage condition is then used to derive the Black-Scholes partial differential equation, which governs the pricing of options. Through a series of transformations, the equation is reduced to the heat equation, allowing for an analytical solution to be obtained. Finally, the study applies this methodology to derive explicit pricing formulas for European call and put options, highlighting the impact of volatility, risk-free interest rates, and time to maturity on option values. By formalizing the theoretical framework with stochastic calculus and arbitrage pricing theory, this research provides a robust foundation for the application of call option models in financial engineering.

Enterohepatic Circulation (EHC) is the process by which bile acid are secreted from the liver into the bile, excreted into the small intestine and then reabsorbed back into the liver. This efflux process is spurred by drug saturation, which is a condition in which the rate of absorption of the drug is limited by the rate of transport to the liver or the rate of secretion into the bile. EHC plays a crucial role for several liver and gastrointestinal functions such as bile flow, solubilization and excretion of cholesterol, clearance of toxic molecules, intestinal absorption of lipophilic nutrients, as well as metabolic and antimicrobial effects. Despite its positive impact in human homeostasis,it is known that EHC can increase toxicity of drugs(due to incomplete elimination during recycling), increased risk of gallstones which result to systemic diseases such as cholelithiasis, bile duct cancer, pancreatic cancer and hepatotoxicity(drug liver injury). In the formulation of a Physiologically Based Pharmacokinetic Model of EHC Drugs with Saturation Kinetics is formulated. The model is affected by secreted drug in the hepatocyte and gastrointestinal compartment with delay effect on metabolites. The drug toxicity threshold parameter and delay effect accounting for gallbladder and intestine disorder(alter the rate of bile circulation) will be discussed. The model is rigorously analyzed on Drug Free Equilibria, Drug Saturation Equilibria, Toxicity Equilibria and Drug Reabsorption Equilibria. Threshold value for Pathological parameter for which there exist a trans from Hoph bifurcation to periodic system was established.
The direction of Stability (super critical and subcritical) was also established. Global and Local stabilities were also investigated. The results from the analysis showed that drug saturation induces toxicity in the
absence of pathological defect parameters when Drug Toxicity Number (DTN) is xi greater than one .Whereas in the presence of pathological parameters (Mild Case), Drug Toxicity does not annul the physiological state of the compartments hence cannot effect drug reabsorption. There exist a threshold for pathological parameters for which drug reabsorption occurs, and defect in physiological compartment progresses from mild to acute case when pathological parameters exceed this threshold i.e τ1 + τ2 > v2+2m2 η2 Hoph bifurcation analysis on the Drug Free and Drug Saturation Equilibria showed that there exist an upper bound for which the system remains asymptotically stable. Numerical results obtained from this work will provide a framework for Pharmaceutical Policies and decisions on EHC.Enterohepatic Circulation (EHC) is the process by which bile acid are secreted from the liver into the bile, excreted into the small intestine and then reabsorbed back into the liver. This efflux process is spurred by drug saturation, which is a condition in
which the rate of absorption of the drug is limited by the rate of transport to the liver or the rate of secretion into the bile. EHC plays a crucial role for several liver and gastrointestinal functions such as bile flow, solubilization and excretion of cholesterol, clearance of toxic molecules, intestinal absorption of lipophilic nutrients, as well as metabolic and antimicrobial effects. Despite its positive impact in human homeostasis,
it is known that EHC can increase toxicity of drugs(due to incomplete elimination during recycling), increased risk of gallstones which result to systemic diseases such as cholelithiasis, bile duct cancer, pancreatic cancer and hepatotoxicity(drug liver injury). In the formulation of a Physiologically Based Pharmacokinetic Model of EHC Drugs with Saturation Kinetics is formulated. The model is affected by secreted drug in the hepatocyte and gastrointestinal compartment with delay effect on metabolites. The drug toxicity threshold parameter and delay effect accounting for gallbladder and intestine disorder(alter the rate of bile circulation) will be discussed. The model is rigorously analyzed on Drug Free Equilibria, Drug Saturation Equilibria, Toxicity Equilibria and Drug Reabsorption Equilibria. Threshold value for Pathological parameter for which there exist a trans from Hoph bifurcation to periodic system was established. The direction of Stability (super critical and subcritical) was also established. Global and Local stabilities were also investigated.
The results from the analysis showed that drug saturation induces toxicity in the absence of pathological defect parameters when Drug Toxicity Number (DTN) is xi greater than one .Whereas in the presence of pathological parameters (Mild Case), Drug Toxicity does not annul the physiological state of the compartments hence cannot effect drug reabsorption. There exist a threshold for pathological parameters
for which drug reabsorption occurs, and defect in physiological compartment progresses from mild to acute case when pathological parameters exceed this threshold i.e τ1 + τ2 > v2+2m2 η2. Hoph bifurcation analysis on the Drug Free and Drug Saturation Equilibria showed that there exist an upper bound for which the system remains asymptotically stable. Numerical results obtained from this work will provide a framework for Pharmaceutical Policies and decisions on EHC.

Enterohepatic Circulation (EHC) is the process by which bile acid are secreted from the liver into the bile, excreted into the small intestine and then reabsorbed back into the liver. This efflux process is spurred by drug saturation, which is a condition in
which the rate of absorption of the drug is limited by the rate of transport to the liver or the rate of secretion into the bile. EHC plays a crucial role for several liver and gastrointestinal functions such as bile flow, solubilization and excretion of cholesterol,
clearance of toxic molecules, intestinal absorption of lipophilic nutrients, as well as metabolic and antimicrobial effects. Despite its positive impact in human homeostasis, it is known that EHC can increase toxicity of drugs(due to incomplete elimination
during recycling), increased risk of gallstones which result to systemic diseases such as cholelithiasis, bile duct cancer, pancreatic cancer and hepatotoxicity(drug liver injury). In the formulation of a Physiologically Based Pharmacokinetic Model of EHC Drugs
with Saturation Kinetics is formulated. The model is affected by secreted drug in the hepatocyte and gastrointestinal compartment with delay effect on metabolites. The drug toxicity threshold parameter and delay effect accounting for gallbladder and intestine disorder(alter the rate of bile circulation) will be discussed. The model is rigorously analyzed on Drug Free Equilibria, Drug Saturation Equilibria, Toxicity Equilibria and Drug Reabsorption Equilibria. Threshold value for Pathological parameter for
which there exist a trans from Hoph bifurcation to periodic system was established. The direction of Stability (super critical and subcritical) was also established. Global and Local stabilities were also investigated. The results from the analysis showed that drug saturation induces toxicity in theabsence of pathological defect parameters when Drug Toxicity Number (DTN) is
greater than one .Whereas in the presence of pathological parameters (Mild Case), Drug Toxicity does not annul the physiological state of the compartments hence cannot effect drug reabsorption. There exist a threshold for pathological parameters
for which drug reabsorption occurs, and defect in physiological compartment progresses from mild to acute case when pathological parameters exceed this threshold τ_1+τ_2>(v_2+2m_2)/η_2 . Hoph bifurcation analysis on the Drug Free and Drug Saturation Equilibria showed that there exist an upper bound for which the system remains asymptotically stable. Numerical results obtained from this work will provide a framework for Pharmaceutical Policies and decisions on EHC.

The derivation and solution of the celebrated Black-Scholes OptionPricing Formula is set out in rather more detail than has appeared in the literature so far. One problem with the Black-Scholes analysis is that the mathematical skills required in the derivation and particularly in the solution of the model are fairly advanced and probably unfamiliar to most economists. In this project, we will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that the stock price is log-normally distributed and that the universe is risk neutral. Then, using Ito’s Lemma, we will justif5’ the use of the risk-neutral rate in these initial calculations. Finally, we will prove put-call parity in order to price European put options, and apply the concepts of the Black-Scholes formula to value an option with pricing equity

Laplace transforms involves the mathematical study of Laplaces, or using it to obtain solution of differential equations. The formation of differential equations is, of course, a common phenomenon that occurs whenever the demand to solve systems of differential equations. Principles regarding the various approaches to it, which may be made frequent in technology like the transfer system used. In this project work, we look at some different system of different equations and consider such procedure as how long it takes on contribution to technology to pass through the system of introduction to application. The kinds of method with which we are all familiar in mathematical field, mechanics. However, the result of Laplace theory have equal if not more important applications in computer such as Data transfer system, Data and signal impulse response and needing to join a Laplace system to be considered. Hence the need in technology is of another usefulness

The primary objective of Random Walk Theory is that it suggests the changes in a
particular element e.g. stock price, having the same distribution and are independent of each other. Therefore, it assumes that the past movement or trend of maybe a stock price cannot be used to predict its future movement. Random Walk Theory have crucial point of conversation in ballot theorem, Markov process and gambler’s ruin. Though, various kind of Random Walks are of interest, which can vary in more than one way. The term itself most often refer to an extraordinary class of markov chain. Random Walk theory can actually take place in variety of spaces: usually concentrated ones include graphs, others on the integers or the real line in the plane or higher dimensional vector spaces on curved surfaces. Next we discussed the application of Random Walk Theory in relation to gambler`s ruin, we furthermore look into this and then concluded.

Numerical methods for solving initial and boundary value problems play a crucial role in various fields of science and engineering.The objective of this project is to present a numerical iterative method for solving initial and boundary value problems to ordinary differential equations . This iterative method is based on the use of the Euler's method and the finite difference method (FDM) in solving initial and boundary value problems respectively. The project begins with a comprehensive literature review on numerical methods for solving IVPs and BVPs, emphasizing the theoretical foundations and practical applications of Euler's and
Finite difference methods. The mathematical formulations and algorithmic procedures of both methods are discussed in detail, highlighting their similarities, and differences. Furthermore, the Euler's method and the finite difference method enables us to approximate the solutions of an ordinary differential equation at a given initial value problem and boundary value problem respectively.
Indeed, two numerical examples are provided to illustrate the effectiv
ness of the Euler's and Finite difference methods. Results obtained show that the numerical method is very effective and convenient for solving ordinary differential equations with initial and boundary value
problems.

Brief history, origin and relevant roles of some numerical methods in the solution of the Hamiltonian Differential Equation with the help of some definitions and theorem. Some important results, are incorporated in this work, extending, specifically the Runge-Kutta Methods. We study the application of Runge-Kutta schemes to Hamiltonian systems. Basic principles are illustrated by means of examples. This work has been selected carefully so that the work is useful for study in this area of research. Particularly, a survey of the effectiveness of the Runge-kutta Methods. The numerical methods developed are primarily intended for use with Hamiltonian systems, but many find uses in solving other forms of ordinary differential equations. Almost all the real conservative physical processes can be cast in suitable Hamiltonian formulation in phase spaces with symplectic structure, which has the advantages to make the intrinsic properties and symmetries of the underlying processes more explicit than in other mathematically equivalent formulations, so I choose the Hamiltonian formalism as the basis, together with the mathematical and physical motivations of our symplectic approach for the purpose of numerical simulation of dynamical evolutions.

The population of man is a germane thing to know and analyze, from the north of the world to the South, the world is compose of huge numbers of human. From the East of the world to west of it, it is comprises of enormous numbers of man. knowing the world population is not enough but also being able to project its future reality. In this project we consider using Malthusian model in estimating relative population reality by the 2050. It’s very interesting because we really present the estimation in two categories; first, by continent, and then globally. This project serve its aim as it give a relative result that the present population can work with to expect and probably tackle the challenges that may arises.