Theses and Dissertations (Applied Mathematics)
http://hdl.handle.net/10386/65
Sun, 04 Jun 2023 10:53:28 GMT2023-06-04T10:53:28ZMalliavin calculus and its applications to mathematical finance
http://hdl.handle.net/10386/3432
Malliavin calculus and its applications to mathematical finance
Kgomo, Shadrack Makwena
In this study,we consider two problems.The first one is the problem of computing hedging
portfolios for options that may have discontinuous payoff functions.For this problem we use the Malliavin property called the Clark-Ocone formula and give some examples for diferent types of pay of functions of the options of European type.The second problem is based on the
computation of price sensitivities (derivatives of the probabilistic representation of the pay off
functions with respect to the underlying parameters of the model) also known as`Greeks'
of discontinuous payoff functions and also give some examples.We restrict ourselves to the
computation of Delta, Gamma and Vega.For this problem we make use of the properties
of the Malliavin calculus like the integration by parts formula and the chain rule.We find
the representations of the price sensitivities in terms of the expected value of the random
variables that do not involve the actual direct differentiation of the payout function,that is,
E[g(XT ) ] where g is a payoff function which depend on the stochasticdic differential equation
XT at maturity time T and is the Malliavin weigh tfunction. In general, we study the
regularity of the solutions of the stochastic differentia lequations in the sense of Malliavin
calculus and explore its applications to Mathematical finance.
Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10386/34322020-01-01T00:00:00ZFinite element solution of the reaction-diffusion equation
http://hdl.handle.net/10386/3421
Finite element solution of the reaction-diffusion equation
Mahlakwana, Richard Kagisho
In this study we present the numerical solution o fboundary value problems for
the reaction-diffusion equations in 1-d and 2-d that model phenomena such as
kinetics and population dynamics.These differential equations are solved nu-
merically using the finite element method (FEM).The FEM was chosen due to
several desirable properties it possesses and the many advantages it has over
other numerical methods.Some of its advantages include its ability to handle
complex geometries very well and that it is built on well established Mathemat-
ical theory,and that this method solves a wider class of problems than most
numerical methods.The Lax-Milgram lemma will be used to prove the existence
and uniqueness of the finite element solutions.These solutions are compared
with the exact solutions,whenever they exist,in order to examine the accuracy
of this method.The adaptive finite element method will be used as a tool for
validating the accuracy of theFEM.The convergence of the FEM will be proven
only on the real line.
Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10386/34212020-01-01T00:00:00ZPathwise functional lto calculus and its applications to the mathematical finance
http://hdl.handle.net/10386/3410
Pathwise functional lto calculus and its applications to the mathematical finance
Nkosi, Siboniso Confrence
Functional Itˆo calculus is based on an extension of the classical Itˆo calculus to functionals depending
on the entire past evolution of the underlying paths and not only on its current value. The
calculus builds on F¨ollmer’s deterministic proof of the Itˆo formula Föllmer (1981) and a notion
of pathwise functional derivative recently proposed by Dupire (2019). There are no smoothness
assumptions required on the functionals, however, they are required to possess certain directional
derivatives which may be computed pathwise, see Cont and Fournié (2013); Schied and
Voloshchenko (2016a); Cont (2012).
In this project we revise the functional Itô calculus together with the notion of quadratic variation.
We compute the pathwise change of variable formula utilizing the functional Itô calculus and the
quadratic variation notion. We study the martingale representation for the case of weak derivatives,
we allow the vertical operator, rX, to operate on continuous functionals on the space of
square-integrable Ft-martingales with zero initial value. We approximate the hedging strategy,
H, for the case of path-dependent functionals, with Lipschitz continuous coefficients. We study
some hedging strategies on the class of discounted market models satisfying the quadratic variation
and the non-degeneracy properties. In the classical case of the Black-Scholes, Greeks are an
important part of risk-management so we compute Greeks of the price given by path-dependent
functionals. Lastly we show that they relate to the classical case in the form of examples.
Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2019
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10386/34102019-01-01T00:00:00ZProperties and calculus on price paths in the model-free approach to the mathematical finance
http://hdl.handle.net/10386/3354
Properties and calculus on price paths in the model-free approach to the mathematical finance
Galane, Lesiba Charles
Vovk and Shafer, [41], introduced game-theoretic framework for probability in
mathematical finance. This is a new trend in financial mathematics in which no
probabilistic assumptions on the space of price paths are made. The only assumption
considered is the no-arbitrage opportunity widely accepted by the financial
mathematics community. This approach rests on game theory rather than measure
theory. We deal with various properties and constructions of quadratic variation
for model-free càdlàg price paths and integrals driven by such paths. Quadratic
variation plays an important role in the analysis of price paths of financial securities
which are modelled by Brownian motion and it is sometimes used as the measure of
volatility (i.e. risk). This work considers mainly càdlàg price paths rather than just
continuous paths. It turns out that this is a natural settings for processes with jumps.
We prove the existence of partition independent quadratic variation. In addition,
following assumptions as in Revuz and Yor’s book, the existence and uniqueness of
the solutions of SDEs with Lipschitz coefficients, driven by model-free price paths
is proven.
Thesis (Ph.D. (Applied Mathematics)) -- University of Limpopo, 2021
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10386/33542021-01-01T00:00:00Z