Peter J. Witbooi

A reliable numerical method to price arithmetic Asian options

Abstract We design and analyze two numerical methods for pricing Asian options. The first one is an explicit finite difference method and therefore, as usual, only conditionally stable. The second method is an implicit finite difference method and unconditionally stable. To explore the basic ideas of analysis, we discuss the explicit method in detail and then highlight the crucial steps in the analysis of the implicit method. Numerical results obtained by these two methods are compared with those obtained by an improved Monte Carlo method. The proposed implicit method is very robust as is evident from the comparative numerical results. We also provide additional numerical results that confirm theoretical investigations done by many other researchers.

Modelling the influence of temperature and rainfall on the population dynamics of Anopheles arabiensis.

BackgroundMalaria continues to be one of the most devastating diseases in the world, killing more humans than any other infectious disease. Malaria parasites are entirely dependent on Anopheles mosquitoes for transmission. For this reason, vector population dynamics is a crucial determinant of malaria risk. Consequently, it is important to understand the biology of malaria vector mosquitoes in the study of malaria transmission. Temperature and precipitation also play a significant role in both aquatic and adult stages of the Anopheles.MethodsIn this study, a climate-based, ordinary-differential-equation model is developed to analyse how temperature and the availability of water affect mosquito population size. In the model, the influence of ambient temperature on the development and the mortality rate of Anopheles arabiensis is considered over a region in KwaZulu-Natal Province, South Africa. In particular, the model is used to examine the impact of climatic factors on the gonotrophic cycle and the dynamics of mosquito population over the study region.ResultsThe results fairly accurately quantify the seasonality of the population of An. arabiensis over the region and also demonstrate the influence of climatic factors on the vector population dynamics. The model simulates the population dynamics of both immature and adult An. arabiensis. The simulated larval density produces a curve which is similar to observed data obtained from another study.ConclusionThe model is efficiently developed to predict An. arabiensis population dynamics, and to assess the efficiency of various control strategies. In addition, the model framework is built to accommodate human population dynamics with the ability to predict malaria incidence in future.

A model for control of HIV/AIDS with parental care

In this study we investigate the HIV/AIDS epidemic in a population which experiences a significant flow of immigrants. We derive and analyze a mathematical model that describes the dynamics of HIV infection among the immigrant youths and how parental care can minimize or prevent the spread of the disease in the population. We analyze the model with both screening control and parental care, then investigate its stability and sensitivity behavior. We also conduct both qualitative and quantitative analyses. It is observed that in the absence of infected youths, disease-free equilibrium is achievable and is globally asymptotically stable. We establish optimal strategies for the control of the disease with screening and parental care, and provide numerical simulations to illustrate the analytic results.

The Whitehead Square of the 6-point 2-sphere

We describe a model of the Whitehead square [ι 2, ι 2] ∈ π3(S 2) in the form of an order-preserving map from a 56-point model of S 3 into the minimal model of S 2 in the category of finite posets. The simplicity of the model enables the map to be visualised.

A nontrivial pairing of finite T0 spaces

Abstract There is a four-point space S 1 weakly homotopy equivalent to the circle. The restriction to S 1 of the complex number multiplication is not continuous, nevertheless a continuous model of the multiplication with values in S 1 can be defined on an eight-point circle. Applying an analogue of Hopfs construction we obtain a finite model of Hopfs famous map S3→S2.

Non-cancellation for certain classes of groups

For a certain class of groups, which are semidirect products arising from an action of a finite rank free abelian group on another group, we study cancellation of the infinite cyclic group in isomorphic direct products. As an application we obtain a sufficient condition for triviality of the genus of certain nilpotent groups.

Non-cancellation and Mislin genus of certain groups and H0-spaces

Abstract The non-cancellation set of a group G measures the extent to which the infinite cyclic group cannot be cancelled as a direct factor of G× Z . If G is a finitely generated group with finite commutator subgroup, then there is a group structure on its non-cancellation set, which coincides with the Hilton–Mislin genus group when G is nilpotent. Using a notion closely related to Nielsen equivalence classes of presentations of a finite abelian group, we give an alternative description of the group structure on the non-cancellation set of groups of a certain kind, and we include some computations. Analogously, we consider non-cancellation, up to homotopy, of the circle as a direct factor of a topological space. In particular, we show how the Mislin genera of certain H 0 -spaces with two non-vanishing homotopy groups can be identified with the genera of certain nilpotent groups.

An Optimal Portfolio and Capital Management Strategy for Basel III Compliant Commercial Banks

We model a Basel III compliant commercial bank that operates in a financial market consisting of a treasury security, a marketable security, and a loan and we regard the interest rate in the market as being stochastic. We find the investment strategy that maximizes an expected utility of the bank’s asset portfolio at a future date. This entails obtaining formulas for the optimal amounts of bank capital invested in different assets. Based on the optimal investment strategy, we derive a model for the Capital Adequacy Ratio (CAR), which the Basel Committee on Banking Supervision (BCBS) introduced as a measure against banks’ susceptibility to failure. Furthermore, we consider the optimal investment strategy subject to a constant CAR at the minimum prescribed level. We derive a formula for the bank’s asset portfolio at constant (minimum) CAR value and present numerical simulations on different scenarios. Under the optimal investment strategy, the CAR is above the minimum prescribed level. The value of the asset portfolio is improved if the CAR is at its (constant) minimum value.

Vaccination Control in a Stochastic SVIR Epidemic Model

For a stochastic differential equation SVIR epidemic model with vaccination, we prove almost sure exponential stability of the disease-free equilibrium for ℛ 0 < 1, where ℛ 0 denotes the basic reproduction number of the underlying deterministic model. We study an optimal control problem for the stochastic model as well as for the underlying deterministic model. In order to solve the stochastic problem numerically, we use an approximation based on the solution of the deterministic model.

A non-Hausdorff quaternion multiplication

We denote by (S3)? the barycentric subdivision of the minimal model S3 of the three-dimensional sphere in the category of finite posets and order-preserving functions, op(X) is the poset obtained by reversing the order relations in a poset X. We describe a finite model of a quaternion multiplication in the form of a morphism op(S3)?×(S3)??S3 that restricts to weak homotopy equivalences on the axes. For such multiplications a version of Hopfs construction can be defined that yields finite models of non-trivial homotopy classes.