Pricing Exotic Options Using Some Lattice Procedures

In this work, we discuss some very simple and extremely efficient lattice models, namely, Binomial tree model (BTM) and Trinomial tree model (TTM) for valuing some types of exotic barrier options in details. For both these models, we consider the concept of random walks in the simulation of the path which is followed by the underlying stock price. Our main objective is to estimate the value of barrier options by using BTM and TTM for different time steps and compare these with the exact values obtained by the benchmark Black-Scholes model (BSM). Moreover, we analyze the convergence of these lattice models for these exotic options. All the results have been shown numerically as well as graphically. GANITJ. Bangladesh Math. Soc.41.1 (2021) 26-40

Factorizing a Quadratic Expression with Reasoning

It is not popularly realized that the factorization of a general quadratic expression basically requires solution to a hyperbola and a line. This fact is conspicuously pointed out and a few solutions to the problem are demonstrated that are scattered in the literature. In the high school level, the coefficients of a quadratic expression are mostly integers, and factorization is performed by the popular method of decomposition of the middle term. In this expository note, we have presented it in a simpler way that shows both insight and reasoning into the problem. Some other methods, namely, Linear Method, Average Method and Difference of two Squares Method are also discussed. Depending on the background, a reader may prefer a particular method to the others. The expository nature and artistry in presentation of the paper are expected to make the learning of the topic amusing and instructive. GANITJ. Bangladesh Math. Soc.41.1 (2021) 15-25

Galerkin Approximations for the Solution of Fredholm Volterra Integral Equation of Second Kind

In this research, we have introduced Galerkin method for finding approximate solutions of Fredholm Volterra Integral Equation (FVIE) of 2nd kind, and this method shows the result in respect of the linear combinations of basis polynomials. Here, BF (product of Bernstein and Fibonacci polynomials), CH (product of Chebyshev and Hermite polynomials), CL (product of Chebyshev and Laguerre polynomials), FL (product of Fibonacci and Laguerre polynomials) and LLE (product of Legendre and Laguerre polynomials) polynomials are established and considered as basis function in Galerkin method. Also, we have tried to observe the behavior of all these approximate solutions finding from Galerkin method for different problems and then a comparison is shown using some standard error estimations. In addition, we observe the error graphs of numerical solutions in Galerkin method for different problems of FVIE of second kind. GANITJ. Bangladesh Math. Soc.41.1 (2021) 1–14

A mathematical analysis of the dynamics of chikungunya virus transmission

In this paper, a deterministic model for the dynamics of chikungunya virus transmission is formulated and analyzed. It is shown that the model has a disease free equilibrium (DFE) and by using the basic reprodution number (R0) local stability of DFE is proved when R0 < 1. Also, the global stability of DFE is investigated by Lyapunov function and LaSalle Invariance Principle. We show that there exists a unique endemic equilibrium (EE) of the model which is locally asymptotically stable whenever R0 > 1 and establish the global stability of the EE when R0 > 1, by using Lyapunov function and LaSalle Invariance Principle for a special case. Numerical simulations and sensitivity analysis show that the destruction of breeding sites and reduction of average life spans of vector would be effective prevention to control the outbreak. Controlling of effective contact rates and reducing transmissions probabilities may reduce the disease prevalence. GANITJ. Bangladesh Math. Soc.41.1 (2021) 41-61

Arterial pharmacokinetics in a patient-specific atherosclerotic artery-a simulation study

Of concern in the paper is a numerical study of endovascular drug delivery in a patient-specific atherosclerotic artery through a mathematical model in which the luminal flow is governed by an incompressible vis- cous Newtonian fluid, and the transport of luminal as well as tissue concentration is modeled as an unsteady convection-diffusion process. An image processing technique has been successfully adopted to detect the edges of the computational domain extracted from an asymmetric (about the centerline of the artery) patient-specific atherosclerotic artery. Considering each pixel as a control volume, the Marker and Cell (MAC) method has been leveraged to get a quantitative insight of the model considered by exploiting physiologically realistic initial, boundary as well as interface conditions. Simulated results reveal that the number as well as the length of separation zone does increase with increasing Re, and the near-wall velocity contour might be important for estimating the near-wall residence time for the pool of drug molecules available for tissue uptake. Results also show that the more the tissue porosity and interface roughness do not necessarily imply the more the effective- ness of delivery, even though they enhance the averaged concentration in the tissue domains, and also the area under concentration diminishes with increasing Peclet number. Thus, the tissue porosity, the Peclet number and various geometrical shapes (interface roughness) play a pivotal role in the dispersion and the effectiveness of drug delivery. GANITJ. Bangladesh Math. Soc.41.1 (2021) 62-77

Study on Flow Behavior of Parallel Lumped-Model under Constant Flow for Bronchial Tree

Redistribution of flow in the bronchial tree is an important factor that enhances gas exchange in the lungs, especially, in diseased lungs. The bifurcated bronchial tree is like an electric network in series and parallel. A lumped-model of parallel system for constant flow rate is solved analytically to demonstrate the intrinsic characteristics and the dynamic behavior of the system. Inertial and capacitive time constants are calculated for 19th generation airways of human lung to control the solution. The investigation revealed that (i) higher inertial force takes more time to maximize the inertial flow to steady state and more time to minimize the resistive flow to steady state and (ii) the compliant effect is negligible for a relatively higher inertial time constant, on the same experimental conditions. GANIT J. Bangladesh Math. Soc. 40.2 (2020) 86–94

Lyapunov Stability Analysis of a Competition Model with Crowding Eects

The present study is connected to the analysis of a nonlinear system that covered a wide range of mathematical biology in terms of competition, cooperation, and symbiosis interactions between two species. We focus on how populations change their densities when two different species follow the non-symmetric logistic growth laws. We have investigated the stability of the corresponding densities of population, and to control the convergence of solutions by proper choice of interacting constant and periodic parameters. It shows the effect of crowding tolerance on both species. It will show that there exists an infinite number of coexistence solutions if the resource distributions are identical for both populations. If the carrying capacity of the first species exceeds the rest one, then eventually the second population drops down to extinction. The results are presented studying the Lyapunov functional, phase portraits, and in a series of numerical examples. GANIT J. Bangladesh Math. Soc. 40.2 (2020) 95-110

Performance of k-ω and k-ε Model for Blood Flow Simulation in Stenosed Artery

Blood flow through arterial stenosis can play a crucial role at the post stenotic flow regions. This produces a disturbance in the normal flow path. The intensity of the flow disturbance (i.e. laminar, transitional and turbulent flow characteristics) depends not only on the severity of the stenosis but also on the pattern of the geometrical model. In that case, the turbulence model plays vital role to measure these flow disturbances. However, it is very important to choose a proper flow simulation model that can predict the flow behavior of fluid accurately and efficiently with less computational cost and time. Thus, this study aims to analyze the results of two turbulence models i.e. k-ω and k-ε for blood flow simulation to compare their performance for the prediction of the flow behavior. Simulations have been performed with 75% area reductions in the arteries. The results of simulation show that, the flow parameters obtained from the k-ε model exhibits lack of fits with the experimental data. On the other hand, k-ω model can capture large scale gradient in the different parameters of blood flow and has a good agreement with the experimental data. This study suggests that, k-ω model has the better performance comparing to k-ε model to predict the behavior of blood flow in stenosed artery. GANIT J. Bangladesh Math. Soc. 40.2 (2020) 111-125

The Economic Order Quantity Repair and Waste Disposal Model: Solution Approaches

As an inventory management instrument, Economic Order Quantity (EOQ) is gaining increased attention. EOQ has become the focus point of everyone’s interest, especially scientists. There are inventory problems in the production and repair system where the used products are collected to repair, and after repairing, it is considered as a new product. A two-stage EOQ model for manufacturing, repairing and disposing of products is discussed in our analysis. The present model follows a former model introduced by Nahimas and Rivera, where the repair system is finite. We propose a new mathematical model in reverse logistics system where demand in production and repair items follows an exponential rate. Mathematical expressions for determining the production and repair quantities are also initiated here. In this model, it is assumed that the available space for supply and repair depots is limited, and we impose two constraints that turn the problem into a constrained optimization model. We also conduct extensive numerical experiments, and advantages are addressed. GANIT J. Bangladesh Math. Soc. 40.2 (2020) 134-144

Lagrangian Relaxation Method in Approximating the Pareto Front of Multiobjective Optimization Problems

In this paper, we propose that the Lagrangian relaxation approach can be used to approximate the Pareto front of the multiobjective optimization problems. We introduce Lagrangian relaxation approach to solve scalarized subproblems. The scalarization is a technique employed to transform multiple objectives optimization problems into single-objective optimization problems so that existing optimization techniques are used to solve the problems. The relaxation approach exploits transformation and creates a Lagrangian problem in which some of the constraints are replaced from the original problem to make the problem easier to solve. The method is very effective when the problem is large scale and difficult to solve; this means if the problem has nonconvex and nonsmooth structure, then our proposed method efficiently solves the problem. We succeed in establishing proper Karush Kuhn-Tucker type necessary conditions for our proposed approach. We establish the relation between our proposed approach and the well-known existing approach weighted-sum scalarization methods. We conduct extensive numerical experiments and demonstrated the advantages of the proposed method of adopting a test problem. GANIT J. Bangladesh Math. Soc. 40.2 (2020) 126-133