Department of Mathematical Scienceshttp://hdl.handle.net/10500/30162014-05-22T16:57:22Z2014-05-22T16:57:22ZPricing European and American bond options under the Hull-White extended Vasicek ModelMpanda, Marc Mukendihttp://hdl.handle.net/10500/133462014-04-23T15:24:03Z2013-01-01T00:00:00ZPricing European and American bond options under the Hull-White extended Vasicek Model
Mpanda, Marc Mukendi
In this dissertation, we consider the Hull-White term structure problem with the boundary value condition given as the payoff of a European bond option. We restrict ourselves to the case where the parameters of the Hull-White model are strictly positive constants and from the risk neutral valuation formula, we first derive simple closedâ€“form expression for pricing European bond option in the Hull-White extended Vasicek model framework. As the European option can be exercised only on the maturity date, we then examine the case of early exercise opportunity commonly called American option. With the analytic representation of American bond option being very hard to handle, we are forced to resort to numerical experiments. To do it excellently, we transform the Hull-White term structure equation into the diffusion equation and we first solve it through implicit, explicit and Crank-Nicolson (CN) difference methods. As these standard finite difference methods (FDMs) require truncation of the domain from infinite to finite one, which may deteriorate the computational efficiency for American bond option, we try to build a CN method over an unbounded domain. We introduce an exact artificial boundary condition in the pricing boundary value problem to reduce the original to an initial boundary problem. Then, the CN method is used to solve the reduced problem. We compare our performance with standard FDMs and the results through illustration show that our method is more efficient and accurate than standard FDMs when we price American bond option.
2013-01-01T00:00:00ZConcerning ideals of pointfree function ringsIghedo, Oghenetegahttp://hdl.handle.net/10500/133422014-04-23T15:20:27Z2013-11-01T00:00:00ZConcerning ideals of pointfree function rings
Ighedo, Oghenetega
We study ideals of pointfree function rings. In particular, we study the lattices of z-ideals
and d-ideals of the ring RL of continuous real-valued functions on a completely regular
frame L. We show that the lattice of z-ideals is a coherently normal Yosida frame; and
the lattice of d-ideals is a coherently normal frame. The lattice of z-ideals is demonstrated
to be
atly projectable if and only if the ring RL is feebly Baer. On the other hand, the
frame of d-ideals is projectable precisely when the frame is cozero-complemented.
These ideals give rise to two functors as follows: Sending a frame to the lattice of
these ideals is a functorial assignment. We construct a natural transformation between the
functors that arise from these assignments. We show that, for a certain collection of frame
maps, the functor associated with z-ideals preserves and re
ects the property of having a
left adjoint.
A ring is called a UMP-ring if every maximal ideal in it is the union of the minimal
prime ideals it contains. In the penultimate chapter we give several characterisations for
the ring RL to be a UMP-ring. We observe, in passing, that if a UMP ring is a Q-algebra,
then each of its ideals when viewed as a ring in its own right is a UMP-ring. An example
is provided to show that the converse fails.
Finally, piggybacking on results in classical rings of continuous functions, we show that,
exactly as in C(X), nth roots exist in RL. This is a consequence of an earlier proposition
that every reduced f-ring with bounded inversion is the ring of fractions of its bounded
part relative to those elements in the bounded part which are units in the bigger ring. We
close with a result showing that the frame of open sets of the structure space of RL is isomorphic to L.
2013-11-01T00:00:00ZLogistic regression to determine significant factors associated with share price changeMuchabaiwa, Honesthttp://hdl.handle.net/10500/132292014-04-23T15:19:01Z2013-02-01T00:00:00ZLogistic regression to determine significant factors associated with share price change
Muchabaiwa, Honest
This thesis investigates the factors that are associated with annual changes in the share price of Johannesburg Stock Exchange (JSE) listed companies. In this study, an increase in value of a share is when the share price of a company goes up by the end of the financial year as compared to the previous year. Secondary data that was sourced from McGregor BFA website was used. The data was from 2004 up to 2011.
Deciding which share to buy is the biggest challenge faced by both investment companies and individuals when investing on the stock exchange. This thesis uses binary logistic regression to identify the variables that are associated with share price increase.
The dependent variable was annual change in share price (ACSP) and the independent variables were assets per capital employed ratio, debt per assets ratio, debt per equity ratio, dividend yield, earnings per share, earnings yield, operating profit margin, price earnings ratio, return on assets, return on equity and return on capital employed.
Different variable selection methods were used and it was established that the backward elimination method produced the best model. It was established that the probability of success of a share is higher if the shareholders are anticipating a higher return on capital employed, and high earnings/ share. It was however, noted that the share price is negatively impacted by dividend yield and earnings yield. Since the odds of an increase in share price is higher if there is a higher return on capital employed and high earning per share, investors and investment companies are encouraged to choose companies with high earnings per share and the best returns on capital employed.
The final model had a classification rate of 68.3% and the validation sample produced a classification rate of 65.2%
2013-02-01T00:00:00ZBydraes tot die oplossing van die veralgemeende knapsakprobleemVenter, Geertienhttp://hdl.handle.net/10500/86032014-04-23T18:25:50Z2013-02-06T00:00:00ZBydraes tot die oplossing van die veralgemeende knapsakprobleem
Venter, Geertien
In this thesis contributions to the solution of the generalised knapsack problem are given and discussed.
Attention is given to problems with functions that are calculable but not necessarily in a closed form.
Algorithms and test problems can be used for problems with closed-form functions as well.
The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be
investigated and good test problems must be designed. A measure of convexity for convex functions
is developed and adapted for concave functions. A test problem generator makes use of this measure
of convexity to create challenging test problems for the concave, convex and mixed knapsack problems.
Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped
as well as the generalised knapsack problem.
The in
uence of the size of the problem and the funding ratio on the speed and the accuracy of the
algorithms are investigated. When applicable, the in
uence of the interval length ratio and the ratio of
concave functions to the total number of functions is also investigated.
The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf-
cient conditions for optimality for the convex knapsack problem with xed interval lengths is given
and proved. For the general convex knapsack problem, the key theorem, which contains the stronger
necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the
adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well.
The exact search-lambda algorithm is developed for the concave knapsack problem with functions that
are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped
knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with
xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using
the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution
was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this
heuristic as well as on the S-shaped and generalised knapsack problems.
Text in Afikaans
2013-02-06T00:00:00Z