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Robust Spectral Methods for Solving Option Pricing Problems
Full text linkDoctor Scientiae - DScRobust Spectral Methods for Solving Option Pricing Problems
by
Edson Pindza
PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of
Natural Sciences, University of the Western Cape
Ever since the invention of the classical Black-Scholes formula to price the financial
derivatives, a number of mathematical models have been proposed by numerous researchers
in this direction. Many of these models are in general very complex, thus
closed form analytical solutions are rarely obtainable. In view of this, we present a
class of efficient spectral methods to numerically solve several mathematical models of
pricing options. We begin with solving European options. Then we move to solve their
American counterparts which involve a free boundary and therefore normally difficult
to price by other conventional numerical methods. We obtain very promising results
for the above two types of options and therefore we extend this approach to solve
some more difficult problems for pricing options, viz., jump-diffusion models and local
volatility models. The numerical methods involve solving partial differential equations,
partial integro-differential equations and associated complementary problems which are
used to model the financial derivatives. In order to retain their exponential accuracy,
we discuss the necessary modification of the spectral methods. Finally, we present
several comparative numerical results showing the superiority of our spectral methods
Spectral difference methods for solving equations of the KdV hierarchy
Full text linkThesis (MSc (Applied Mathematics))--Stellenbosch University, 2008.The Korteweg-de Vries (KdV) hierarchy is an important class of nonlinear evolution equa-
tions with various applications in the physical sciences and in engineering.
In this thesis analytical solution methods were used to ¯nd exact solutions of the third and
¯fth order KdV equations, and numerical methods were used to compute numerical solutions
of these equations.
Analytical methods used include the Fan sub-equation method for constructing exact trav-
eling wave solutions, and the simpli¯ed Hirota method for constructing exact N-soliton
solutions. Some well known cases were considered.
The Fourier spectral method and the ¯nite di®erence method with Runge-Kutta time dis-
cretisation were employed to solve the third and the ¯fth order KdV equations with periodic
boundary conditions. The one soliton and the two soliton solutions were used as initial
conditions. The numerical solutions are obtained and compared with the exact solutions.
The propagation of a single soliton as well as the interaction of double soliton solutions is
modeled well by both numerical methods, although the Fourier spectral method performs
better.
The stability, consistency and convergence of these numerical methods were investigated.
Error propagation is studied. The theoretically predicted quadratic convergence of the ¯nite
di®erence method as well as the exponential convergence of the Fourier spectral method is
con¯rmed in numerical experiments
Pricing Options under Levy models using Spectral methods
Full text linkDissertation (MSc)--University of Pretoria, 2017.Spectral methods have been actively developed in the last decades. The
main advantage of these methods is to yield exponential order of accuracy
when the function is smooth. However, for discontinuous functions,
their accuracy deteriorates due to the Gibbs phenomenon. When functions
are contaminated with the Gibbs phenomenon, proper workarounds
can be applied to recover their accuracy. In this dissertation, we review the
spectral methods and their convergence remedies such as grid stretching,
discontinuity inclusion and domain decomposition methods in pricing options.
The basic functions of L´evy processes models are also reviewed.
The main purpose of this dissertation is to show that high order of accuracy
can be recovered from spectral approximations. We explored and
designed numerical methods for solving PDEs and PIDEs that arise in finance.
It is known that most standard numerical methods for solving financial
PDEs and PIDEs are reduced to low order accurate results due to
the discontinuity at strike prices in the initial condition.
Firstly the Black Scholes (BS) PDE was solved numerically. The computation
of the PDE is done by using barycentric spectral methods. Three different
payoffs call options are used as initial and boundaries conditions.
It appears that the grid stretching, the discontinuity inclusion and the domain
decomposition methods provide efficient ways to remove Gibbs phenomenon.
On the other hand, these methods restore the high accuracy of
spectral methods in pricing financial options. The spectral domain decomposition
method appears to be the most accurate workaround when
we solve a BS PDE in this dissertation.
Secondly, a financial PIDE was discretized and solved by using a barycen tric spectral domain decomposition method algorithm. The method is applied
to two different options pricing problems under a class of infinite activity
L´evy models. The use of barycentric spectral domain decomposition
methods allows the computation of ODEs obtained from the discretization
of the PIDE. The ODEs are solved by exponential time integration scheme.
Several numerical tests for the pricing of European and butterfly options
are given to illustrate the efficiency and accuracy of this algorithm. We
also show that the option Greeks such as the Delta and Gamma sensitivity
measures are computed with no spurious oscillation. The methods produce
accurate results.RidgeCape Capital companyMathematics and Applied MathematicsMScUnrestricte
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Analysis of cryptocurrencies adoption using fractional grey Lotka-Volterra models
Full text linkAbstract: Solving analytically nonlinear dynamical system in continuous time scale is often problematic. The accumulation generating operations provide a tool of formulating a discrete dynamical form whose properties are relatively close to that of corresponding nonlinear systems. The present study discusses threes versions of 2- and 3- dimensional discrete Lotka-Volterra dynamical system with application to cryptocurrencies adoption. The application is interested on 3 cryptocurrencies namely Bitcoin, Litecoin and Ripple. The 2-dimensional application is on Bitcoin and Litecoin while the 3-dimensional application is on Bitcoin, Litecoin and Ripple. The dataset include records from 28-April-2013 to 10-February-2018 which provide forecasting values for Bitcoin and Litecoin through 2-dimensional study, while records from 7-August-2013 to 10- February-2018 provide forecasting values of Bitcoin, Litecoin and Ripple through 3-dimensional study. The thesis has produced four papers that have been published and presented in international conferences.Ph.D. (Mathematics
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Sinc collocation method for solving the Benjamin-Ono equation
Full text linkWe propose a simple, though powerful, technique for numerical solutions of the Benjamin-Ono equation. This approach is based on
a global collocation method using Sinc basis functions. Some properties of the Sinc collocation method required for our subsequent
development are given and utilized to reduce the computation of the Benjamin-Ono equation to a system of ordinary differential
equations.The propagation of one soliton and the interaction of two solitons are used to validate our numericalmethod.Themethod
is easy to implement and yields accurate results.Edson Pindza is thankful to Brad Welch for the financial
support from RidgeCape Capital.http://www.hindawi.com/journals/jcmp/am201
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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