**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1635

# World Academy of Science, Engineering and Technology

## [Mathematical and Computational Sciences]

### Online ISSN : 1307-6892

##### 1635 The Martingale Options Price Valuation for European Puts Using Stochastic Differential Equation Models

**Authors:**
H. C. Chinwenyi,
H. D. Ibrahim,
F. A. Ahmed

**Abstract:**

In modern financial mathematics, valuing derivatives such as options is often a tedious task. This is simply because their fair and correct prices in the future are often probabilistic. This paper examines three different Stochastic Differential Equation (SDE) models in finance; the Constant Elasticity of Variance (CEV) model, the Balck-Karasinski model, and the Heston model. The various Martingales option price valuation formulas for these three models were obtained using the replicating portfolio method. Also, the numerical solution of the derived Martingales options price valuation equations for the SDEs models was carried out using the Monte Carlo method which was implemented using MATLAB. Furthermore, results from the numerical examples using published data from the Nigeria Stock Exchange (NSE), all share index data show the effect of increase in the underlying asset value (stock price) on the value of the European Put Option for these models. From the results obtained, we see that an increase in the stock price yields a decrease in the value of the European put option price. Hence, this guides the option holder in making a quality decision by not exercising his right on the option.

**Keywords:**
monte carlo method,
martingales,
equivalent martingale measure,
European put option,
girsanov theorem,
option price valuation formula,
option price valuation

##### 1634 Several Spectrally Non-Arbitrary Ray Patterns of Order 4

**Authors:**
Ling Zhang,
Feng Liu

**Abstract:**

A matrix is called a ray pattern matrix if its entries are either 0 or a ray in complex plane which originates from 0. A ray pattern *A *of order *n *is called spectrally arbitrary if the complex matrices in the ray pattern class of *A* give rise to all possible *n*th degree complex polynomial. Otherwise, it is said to be spectrally non-arbitrary ray pattern*.* We call that a spectrally arbitrary ray pattern *A *of order *n *is minimally spectrally arbitrary if any nonzero entry of *A* is replaced, then *A *is not spectrally arbitrary. In this paper, we find that is not spectrally arbitrary when n equals to 4 for any θ which is greater than or equal to 0 and less than or equal to n. In this article, we give several ray patterns A(θ) of order n that are not spectrally arbitrary for some θ which is greater than or equal to 0 and less than or equal to n. by using the nilpotent-Jacobi method. One example is given in our paper.

**Keywords:**
spectrally arbitrary,
nilpotent matrix,
ray patterns,
sign patterns

##### 1633 Bidirectional Discriminant Supervised Locality Preserving Projection for Face Recognition

**Abstract:**

**Keywords:**
Face Recognition,
Dimension Reduction,
locality
preserving projection,
discriminant information,
bidirectional
projection

##### 1632 Metric Dimension on Line Graph of Honeycomb Networks

**Authors:**
M. Hussain,
Aqsa Farooq

**Abstract:**

**Keywords:**
Resolving set,
Metric dimension,
Honeycomb network,
Line graph

##### 1631 The Non-Stationary BINARMA(1,1) Process with Poisson Innovations: An Application on Accident Data

**Authors:**
N. Mamode Khan,
V. Jowaheer,
Y. Sunecher

**Abstract:**

**Keywords:**
non-stationary,
CML,
BINARMA(1,
Poisson
Innovations

##### 1630 Analysis of One Dimensional Advection Diffusion Model Using Finite Difference Method

**Authors:**
Ravneet Kaur,
Vijay Kumar Kukreja

**Abstract:**

**Keywords:**
Stability,
Consistency,
Crank-Nicolson scheme,
Lax-Richtmyer theorem,
Peclet number,
Gerschgorin
circle

##### 1629 An Automated Stock Investment System Using Machine Learning Techniques: An Application in Australia

**Authors:**
Carol Anne Hargreaves

**Abstract:**

A key issue in stock investment is how to select representative features for stock selection. The objective of this paper is to firstly determine whether an automated stock investment system, using machine learning techniques, may be used to identify a portfolio of growth stocks that are highly likely to provide returns better than the stock market index. The second objective is to identify the technical features that best characterize whether a stock’s price is likely to go up and to identify the most important factors and their contribution to predicting the likelihood of the stock price going up. Unsupervised machine learning techniques, such as cluster analysis, were applied to the stock data to identify a cluster of stocks that was likely to go up in price – portfolio 1. Next, the principal component analysis technique was used to select stocks that were rated high on component one and component two – portfolio 2. Thirdly, a supervised machine learning technique, the logistic regression method, was used to select stocks with a high probability of their price going up – portfolio 3. The predictive models were validated with metrics such as, sensitivity (recall), specificity and overall accuracy for all models. All accuracy measures were above 70%. All portfolios outperformed the market by more than eight times. The top three stocks were selected for each of the three stock portfolios and traded in the market for one month. After one month the return for each stock portfolio was computed and compared with the stock market index returns. The returns for all three stock portfolios was 23.87% for the principal component analysis stock portfolio, 11.65% for the logistic regression portfolio and 8.88% for the K-means cluster portfolio while the stock market performance was 0.38%. This study confirms that an automated stock investment system using machine learning techniques can identify top performing stock portfolios that outperform the stock market.

**Keywords:**
Machine Learning,
stock market trading,
automated stock investment system,
logistic principal component analysis

##### 1628 Relation between Roots and Tangent Lines of Function in Fractional Dimensions: A Method for Optimization Problems

**Authors:**
Ali Dorostkar

**Abstract:**

In this paper, a basic schematic of fractional dimensional optimization problem is presented. As will be shown, a method is performed based on a relation between roots and tangent lines of function in fractional dimensions for an arbitrary initial point. It is shown that for each polynomial function with order N at least N tangent lines must be existed in fractional dimensions of 0 < α < N+1 which pass exactly through the all roots of the proposed function. Geometrical analysis of tangent lines in fractional dimensions is also presented to clarify more intuitively the proposed method. Results show that with an appropriate selection of fractional dimensions, we can directly find the roots. Method is presented for giving a different direction of optimization problems by the use of fractional dimensions.

**Keywords:**
optimization problem,
root,
tangent line,
fractional dimension

##### 1627 Multi-Objective Optimization of Combined System Reliability and Redundancy Allocation Problem

**Authors:**
Vijaya K. Srivastava,
Davide Spinello

**Abstract:**

This paper presents established 3** ^{n}** enumeration procedure for mixed integer optimization problems for solving multi-objective reliability and redundancy allocation problem subject to design constraints. The formulated problem is to find the optimum level of unit reliability and the number of units for each subsystem. A number of illustrative examples are provided and compared to indicate the application of the superiority of the proposed method.

**Keywords:**
Multi-objective optimization,
Integer Programming,
mixed integer programming,
Reliability Redundancy Allocation

##### 1626 Characterizations of Γ-Semirings by Their Cubic Ideals

**Authors:**
Debabrata Mandal

**Abstract:**

**Keywords:**
intra-regular,
Cartesian product,
Γ-semiring,
cubic ideal,
normal cubic ideal,
cubic
bi-ideal,
cubic quasi-ideal,
regular

##### 1625 Monte Carlo Estimation of Heteroscedasticity and Periodicity Effects in a Panel Data Regression Model

**Authors:**
Nureni O. Adeboye,
Dawud A. Agunbiade

**Abstract:**

This research attempts to investigate the effects of heteroscedasticity and periodicity in a Panel Data Regression Model (PDRM) by extending previous works on balanced panel data estimation within the context of fitting PDRM for Banks audit fee. The estimation of such model was achieved through the derivation of Joint Lagrange Multiplier (LM) test for homoscedasticity and zero-serial correlation, a conditional LM test for zero serial correlation given heteroscedasticity of varying degrees as well as conditional LM test for homoscedasticity given first order positive serial correlation via a two-way error component model. Monte Carlo simulations were carried out for 81 different variations, of which its design assumed a uniform distribution under a linear heteroscedasticity function. Each of the variation was iterated 1000 times and the assessment of the three estimators considered are based on Variance, Absolute bias (ABIAS), Mean square error (MSE) and the Root Mean Square (RMSE) of parameters estimates. Eighteen different models at different specified conditions were fitted, and the best-fitted model is that of within estimator when heteroscedasticity is severe at either zero or positive serial correlation value. LM test results showed that the tests have good size and power as all the three tests are significant at 5% for the specified linear form of heteroscedasticity function which established the facts that Banks operations are severely heteroscedastic in nature with little or no periodicity effects.

**Keywords:**
Periodicity,
heteroscedasticity,
audit fee,
lagrange multiplier test

##### 1624 Three-Dimensional Generalized Thermoelasticity with Variable Thermal Conductivity

**Authors:**
Hamdy M. Youssef,
Mowffaq Oreijah,
Hunaydi S. Alsharif

**Abstract:**

In this paper, a three-dimensional model of the generalized thermoelasticity with one relaxation time and variable thermal conductivity has been constructed. The resulting non-dimensional governing equations together with the Laplace and double Fourier transforms techniques have been applied to a three-dimensional half-space subjected to thermal loading with rectangular pulse and traction free in the directions of the principle co-ordinates. The inverses of double Fourier transforms, and Laplace transforms have been obtained numerically. Numerical results for the temperature increment, the invariant stress, the invariant strain, and the displacement are represented graphically. The variability of the thermal conductivity has significant effects on the thermal and the mechanical waves.

**Keywords:**
Fourier Transforms,
Thermal Conductivity,
thermoelasticity,
three-dimensional,
Laplace transforms

##### 1623 On the Efficiency of Five Step Approximation Method for the Solution of General Third Order Ordinary Differential Equations

**Authors:**
N. M. Kamoh,
M. C. Soomiyol

**Abstract:**

In this work, a five step continuous method for the solution of third order ordinary differential equations was developed in block form using collocation and interpolation techniques of the shifted Legendre polynomial basis function. The method was found to be zero-stable, consistent and convergent. The application of the method in solving third order initial value problem of ordinary differential equations revealed that the method compared favorably with existing methods.

**Keywords:**
convergent,
shifted legendre polynomials,
third order block method,
discrete method

##### 1622 Jeffrey's Prior for Unknown Sinusoidal Noise Model via Cramer-Rao Lower Bound

**Authors:**
Rasaki O. Olanrewaju,
Samuel A. Phillips,
Emmanuel A. Ayanlowo,
Olayode Fatoki

**Abstract:**

This paper employs the Jeffrey's prior technique in the process of estimating the periodograms and frequency of sinusoidal model for unknown noisy time variants or oscillating events (data) in a Bayesian setting. The non-informative Jeffrey's prior was adopted for the posterior trigonometric function of the sinusoidal model such that Cramer-Rao Lower Bound (CRLB) inference was used in carving-out the minimum variance needed to curb the invariance structure effect for unknown noisy time observational and repeated circular patterns. An average monthly oscillating temperature series measured in degree Celsius (0C) from 1901 to 2014 was subjected to the posterior solution of the unknown noisy events of the sinusoidal model via Markov Chain Monte Carlo (MCMC). It was not only deduced that two minutes period is required before completing a cycle of changing temperature from one particular degree Celsius to another but also that the sinusoidal model via the CRLB-Jeffrey's prior for unknown noisy events produced a miniature posterior Maximum A Posteriori (MAP) compare to a known noisy events.

**Keywords:**
sinusoidal,
Maximum A Posteriori (MAP),
Cramer-Rao lower bound (CRLB),
Jeffrey's prior,
Markov Chain Monte
Carlo (MCMC),
Periodograms

##### 1621 Comparing the Efficiency of Simpson’s 1/3 and 3/8 Rules for the Numerical Solution of First Order Volterra Integro-Differential Equations

**Authors:**
N. M. Kamoh,
D. G. Gyemang,
M. C. Soomiyol

**Abstract:**

This paper compared the efficiency of Simpson’s 1/3 and 3/8 rules for the numerical solution of first order Volterra integro-differential equations. In developing the solution, collocation approximation method was adopted using the shifted Legendre polynomial as basis function. A block method approach is preferred to the predictor corrector method for being self-starting. Experimental results confirmed that the Simpson’s 3/8 rule is more efficient than the Simpson’s 1/3 rule.

**Keywords:**
Collocation shifted Legendre polynomials,
Simpson’s rule and Volterra integro-differential equations

##### 1620 A Study of Two Disease Models: With and Without Incubation Period

**Authors:**
H. C. Chinwenyi,
H. D. Ibrahim,
J. O. Adekunle

**Abstract:**

The incubation period is defined as the time from infection with a microorganism to development of symptoms. In this research, two disease models: one with incubation period and another without incubation period were studied. The study involves the use of a mathematical model with a single incubation period. The test for the existence and stability of the disease free and the endemic equilibrium states for both models were carried out. The fourth order Runge-Kutta method was used to solve both models numerically. Finally, a computer program in MATLAB was developed to run the numerical experiments. From the results, we are able to show that the endemic equilibrium state of the model with incubation period is locally asymptotically stable whereas the endemic equilibrium state of the model without incubation period is unstable under certain conditions on the given model parameters. It was also established that the disease free equilibrium states of the model with and without incubation period are locally asymptotically stable. Furthermore, results from numerical experiments using empirical data obtained from Nigeria Centre for Disease Control (NCDC) showed that the overall population of the infected people for the model with incubation period is higher than that without incubation period. We also established from the results obtained that as the transmission rate from susceptible to infected population increases, the peak values of the infected population for the model with incubation period decrease and are always less than those for the model without incubation period.

**Keywords:**
asymptotic stability,
incubation period,
Routh-Hurwitz criterion,
Runge Kutta method

##### 1619 Comparison of Methods of Estimation for Use in Goodness of Fit Tests for Binary Multilevel Models

**Authors:**
I. V. Pinto,
M. R. Sooriyarachchi

**Abstract:**

It can be frequently observed that the data arising in our environment have a hierarchical or a nested structure attached with the data. Multilevel modelling is a modern approach to handle this kind of data. When multilevel modelling is combined with a binary response, the estimation methods get complex in nature and the usual techniques are derived from quasi-likelihood method. The estimation methods which are compared in this study are, marginal quasi-likelihood (order 1 & order 2) (MQL1, MQL2) and penalized quasi-likelihood (order 1 & order 2) (PQL1, PQL2). A statistical model is of no use if it does not reflect the given dataset. Therefore, checking the adequacy of the fitted model through a goodness-of-fit (GOF) test is an essential stage in any modelling procedure. However, prior to usage, it is also equally important to confirm that the GOF test performs well and is suitable for the given model. This study assesses the suitability of the GOF test developed for binary response multilevel models with respect to the method used in model estimation. An extensive set of simulations was conducted using MLwiN (v 2.19) with varying number of clusters, cluster sizes and intra cluster correlations. The test maintained the desirable Type-I error for models estimated using PQL2 and it failed for almost all the combinations of MQL. Power of the test was adequate for most of the combinations in all estimation methods except MQL1. Moreover, models were fitted using the four methods to a real-life dataset and performance of the test was compared for each model.

**Keywords:**
Power,
multilevel modelling,
goodness-of-fit test,
marginal quasi-likelihood,
penalized quasi-likelihood,
quasi-likelihood,
type-I error

##### 1618 An Attack on the Lucas Based El-Gamal Cryptosystem in the Elliptic Curve Group Over Finite Field Using Greater Common Divisor

**Authors:**
Tze Jin Wong,
Nik Mohd Asri Nik Long,
Lee Feng Koo,
Pang Hung Yiu

**Abstract:**

Greater common divisor (GCD) attack is an attack that relies on the polynomial structure of the cryptosystem. This attack required two plaintexts differ from a fixed number and encrypted under same modulus. This paper reports a security reaction of Lucas Based El-Gamal Cryptosystem in the Elliptic Curve group over finite field under GCD attack. Lucas Based El-Gamal Cryptosystem in the Elliptic Curve group over finite field was exposed mathematically to the GCD attack using GCD and Dickson polynomial. The result shows that the cryptanalyst is able to get the plaintext without decryption by using GCD attack. Thus, the study concluded that it is highly perilous when two plaintexts have a slight difference from a fixed number in the same Elliptic curve group over finite field.

**Keywords:**
Encryption,
decryption,
elliptic curve,
greater common divisor

##### 1617 Cryptographic Attack on Lucas Based Cryptosystems Using Chinese Remainder Theorem

**Authors:**
Tze Jin Wong,
Lee Feng Koo,
Pang Hung Yiu

**Abstract:**

_{4,6}) cryptosystem under the Lenstra’s attack as compared to the other two Lucas based cryptosystems such as LUC and LUC

_{3}cryptosystems. All the Lucas based cryptosystems were exposed mathematically to the Lenstra’s attack using Chinese Remainder Theorem and Dickson polynomial. Result shows that the possibility for successful Lenstra’s attack is less against LUC

_{4,6}cryptosystem than LUC

_{3}and LUC cryptosystems. Current study concludes that LUC

_{4,6}cryptosystem is more secure than LUC and LUC

_{3}cryptosystems in sustaining against Lenstra’s attack.

**Keywords:**
congruence,
Lucas sequence,
Dickson polynomial,
faulty signature,
corresponding signature

##### 1616 Autonomous Vehicle Navigation Using Harmonic Functions via Modified Arithmetic Mean Iterative Method

**Authors:**
Jumat Sulaiman,
Azali Saudi

**Abstract:**

**Keywords:**
Path Planning,
Laplace’s equation,
Modified Arithmetic Mean method,
Harmonic
functions

##### 1615 The Influence of Beta Shape Parameters in Project Planning

**Authors:**
Dimitra Alexiou,
Stefanos Katsavounis,
Αlexios Kotsakis

**Abstract:**

Networks can be utilized to represent project planning problems, using nodes for activities and arcs to indicate precedence relationship between them. For fixed activity duration, a simple algorithm calculates the amount of time required to complete a project, followed by the activities that comprise the critical path. Program Evaluation and Review Technique (PERT) generalizes the above model by incorporating uncertainty, allowing activity durations to be random variables, producing nevertheless a relatively crude solution in planning problems. In this paper, based on the findings of the relevant literature, which strongly suggests that a Beta distribution can be employed to model earthmoving activities, we utilize Monte Carlo simulation, to estimate the project completion time distribution and measure the influence of skewness, an element inherent in activities of modern technical projects. We also extract the activity criticality index, with an ultimate goal to produce more accurate planning estimations.

**Keywords:**
Monte Carlo Simulation,
Skewness,
PERT,
beta distribution,
project completion time distribution

##### 1614 An IM-COH Algorithm Neural Network Optimization with Cuckoo Search Algorithm for Time Series Samples

**Authors:**
Wullapa Wongsinlatam

**Abstract:**

**Keywords:**
Artificial Neural Networks,
Time series,
local minima problem,
back propagation
algorithm,
metaheuristic
optimization

##### 1613 Natural Emergence of a Core Structure in Networks via Clique Percolation

**Authors:**
A. Melka,
N. Slater,
A. Mualem,
Y. Louzoun

**Abstract:**

**Keywords:**
Networks,
Cliques,
percolation,
core structure,
phase
transition

##### 1612 Forecasting Issues in Energy Markets within a Reg-ARIMA Framework

**Authors:**
Ilaria Lucrezia Amerise

**Abstract:**

**Keywords:**
Time series,
electricity prices,
interval forecasts,
Forecasting problem,
reg-plus-SARMA methods

##### 1611 Flood Modeling in Urban Area Using a Well-Balanced Discontinuous Galerkin Scheme on Unstructured Triangular Grids

**Authors:**
Rabih Ghostine,
Craig Kapfer,
Viswanathan Kannan,
Ibrahim Hoteit

**Abstract:**

**Keywords:**
Flood Modeling,
shallow water equations,
dam-break,
discontinuous Galerkin scheme,
MUSCL scheme

##### 1610 Optimal Location of the I/O Point in the Parking System

**Authors:**
Jing Zhang,
Jie Chen

**Abstract:**

In this paper, we deal with the optimal I/O point location in an automated parking system. In this system, the S/R machine (storage and retrieve machine) travels independently in vertical and horizontal directions. Based on the characteristics of the parking system and the basic principle of AS/RS system (Automated Storage and Retrieval System), we obtain the continuous model in units of time. For the single command cycle using the randomized storage policy, we calculate the probability density function for the system travel time and thus we develop the travel time model. And we confirm that the travel time model shows a good performance by comparing with discrete case. Finally in this part, we establish the optimal model by minimizing the expected travel time model and it is shown that the optimal location of the I/O point is located at the middle of the left-hand above corner.

**Keywords:**
Response Time,
optimal location,
parking system,
S/R machine

##### 1609 Box Counting Dimension of the Union L of Trinomial Curves When α ≥ 1

**Authors:**
Kaoutar Lamrini Uahabi,
Mohamed Atounti

**Abstract:**

**Keywords:**
fractal dimension,
feasible angles,
trinomial curves,
trinomial equation,
Minkowski
sausage

##### 1608 Adomian’s Decomposition Method to Generalized Magneto-Thermoelasticity

**Authors:**
Hamdy M. Youssef,
Eman A. Al-Lehaibi

**Abstract:**

Due to many applications and problems in the fields of plasma physics, geophysics, and other many topics, the interaction between the strain field and the magnetic field has to be considered. Adomian introduced the decomposition method for solving linear and nonlinear functional equations. This method leads to accurate, computable, approximately convergent solutions of linear and nonlinear partial and ordinary differential equations even the equations with variable coefficients. This paper is dealing with a mathematical model of generalized thermoelasticity of a half-space conducting medium. A magnetic field with constant intensity acts normal to the bounding plane has been assumed. Adomian’s decomposition method has been used to solve the model when the bounding plane is taken to be traction free and thermally loaded by harmonic heating. The numerical results for the temperature increment, the stress, the strain, the displacement, the induced magnetic, and the electric fields have been represented in figures. The magnetic field, the relaxation time, and the angular thermal load have significant effects on all the studied fields.

**Keywords:**
Adomian’s decomposition method,
magneto-thermoelasticity,
finite conductivity,
iteration method,
thermal load

##### 1607 An Efficient Collocation Method for Solving the Variable-Order Time-Fractional Partial Differential Equations Arising from the Physical Phenomenon

**Authors:**
Haniye Dehestani,
Yadollah Ordokhani

**Abstract:**

**Keywords:**
collocation method,
legendre-laguerre functions,
fractional partial differential
equations,
pseudo-operational matrix
of integration

##### 1606 Mathematical Expression for Machining Performance

**Authors:**
Md. Ashikur Rahman Khan,
M. M. Rahman

**Abstract:**

In electrical discharge machining (EDM), a complete and clear theory has not yet been established. The developed theory (physical models) yields results far from reality due to the complexity of the physics. It is difficult to select proper parameter settings in order to achieve better EDM performance. However, modelling can solve this critical problem concerning the parameter settings. Therefore, the purpose of the present work is to develop mathematical model to predict performance characteristics of EDM on Ti-5Al-2.5Sn titanium alloy. Response surface method (RSM) and artificial neural network (ANN) are employed to develop the mathematical models. The developed models are verified through analysis of variance (ANOVA). The ANN models are trained, tested, and validated utilizing a set of data. It is found that the developed ANN and mathematical model can predict performance of EDM effectively. Thus, the model has found a precise tool that turns EDM process cost-effective and more efficient.

**Keywords:**
Modelling,
Artificial Neural Network,
Surface Finish,
analysis of variance,
material removal rate,
response surface method