43 research outputs found
Robust Spectral Methods for Solving Option Pricing Problems
Doctor 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
Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options
In this paper we present a robust numerical method to solve several types of European style option pricing problems. The governing equations are described by variants of Black-Scholes partial differential equations (BS-PDEs) of the reaction-diffusion-advection type. To discretise these BS-PDEs numerically, we use the spectral methods in the asset (spatial) direction and couple them with a third-order implicit-explicit predictor-corrector (IMEX-PC) method for the discretisation in the time direction. The use of this high-order time integration scheme sustains the better accuracy of the spectral methods for which they are well-known. Our spectral method consists of a pseudospectral formulation of the BS-PDEs by means of an improved Lagrange formula. On the other hand, in the IMEX-PC methods, we integrate the diffusion terms implicitly whereas the reaction and advection terms are integrated explicitly. Using this combined approach, we first solve the equations for standard European options and then extend this approach to digital options, butterfly spread options, and European calls in the Heston model. Numerical experiments illustrate that our approach is highly accurate and very efficient for pricing financial options such as those described above
Price discovery in the cryptocurrency option market : a univariate GARCH approach
Abstract:In this paper, two univariate generalised autoregressive conditional heteroskedasticity (GARCH) option pricing models are applied to Bitcoin and the Cryptocurrency Index (CRIX). The first model is symmetric and the other takes asymmetric effects into account. Furthermore, the accuracy of the GARCH option pricing model applied to Bitcoin is tested. Empirical results indicate that asymmetry is not an important factor to consider when pricing options on Bitcoin or CRIX, this is consistent with findings in the literature. In addition, the GARCH option pricing model provides realistic price discovery within the bid-ask spreads suggested by the market
Sinc collocation method for solving the Benjamin-Ono equation
We 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
Solving the generalized regularized long wave equation using a distributed approximating functional method
The generalized regularized long wave (GRLW) equation is solved numerically by using a distributed approximating functional
(DAF) method realized by the regularized Hermite local spectral kernel. Test problems including propagation of single solitons,
interaction of two and three solitons, and conservation properties of mass, energy, and momentum of the GRLW equation are
discussed to test the efficiency and accuracy of the method. Furthermore, using the Maxwellian initial condition, we show that
the number of solitons which are generated can be approximately determined. Comparisons are made between the results of
the proposed method, analytical solutions, and numerical methods. It is found that the method under consideration is a viable
alternative to existing numerical methods.This paper was supported by BradWelch, RidgeCape Capital,
Tokai, Cape Town.http://www.hindawi.com/journals/ijcm/am201
Sinc Collocation Method for Solving the Benjamin-Ono Equation
We 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 numerical method. The method is easy to implement and yields accurate results
A robust spectral method for pricing of American put options on zero-coupon bonds
American put options on a zero-coupon bond problem is reformulated as a
linear complementarity problem of the option value and approximated by a nonlinear
partial differential equation. The equation is solved by an exponential time differencing
method combined with a barycentric Legendre interpolation and the Krylov projection
algorithm. Numerical examples shows the stability and good accuracy of the method. A bond is a financial instrument which allows an investor to loan money to an entity
(a corporate or governmental) that borrows the funds for a period of time at a fixed interest rate (the coupon) and agrees to pay a fixed amount (the principal) to the investor
at maturity. A zero-coupon bond is a bond that makes no periodic interest payments
Fourier spectral method for higher order space fractional reaction-diffusion equations
Evolution equations containing fractional derivatives can provide suitable mathemati-
cal models for describing important physical phenomena. In this paper, we propose a
fast and accurate method for numerical solutions of space fractional reaction-diffusion
equations. The proposed method is based on a exponential integrator scheme in time
and the Fourier spectral method in space. The main advantages of this method are
that it yields a fully diagonal representation of the fractional operator, with increased
accuracy and efficiency, and a completely straightforward extension to high spatial di-
mensions. Although, in general, it is not obvious what role a high fractional derivative
can play and how to make use of arbitrarily high-order fractional derivatives, we in-
troduce them to describe fractional hyper-diffusions in reaction diffusion. The scheme
justified by a number of computational experiments, this includes two and three dimen-
sional partial differential equations. Numerical experiments are provided to validate
the effectiveness of the proposed approach.http://www.elsevier.com/locate/cnsns2017-11-30hb2016Mathematics and Applied Mathematic
Adaptive techniques for solving chaotic system of parabolic-type
Time-dependent partial differential equations of parabolic type are known to have a lot of applications in biology, mechanics, epidemiology and control processes. Despite the usefulness of this class of differential equations, the numerical approach to its solution, especially in high dimensions, is still poorly understood. Since the nature of reaction-diffusion problems permit the use of different methods in space and time, two important approximation schemes which are based on the spectral and barycentric interpolation collocation techniques are adopted in conjunction with the third-order exponential time-differencing Runge-Kutta method to advance in time. The accuracy of the method is tested by considering a number of time-dependent reaction-diffusion problems that are still of current and recurring interests in one and high dimensions.© 2022 The Authors. Published by Elsevier B.V. on behalf of African Institute of Mathematical Sciences / Next Einstein Initiative. This is an open access article under the CC BY license.http://www.elsevier.com/locate/sciafhj2023Mathematics and Applied Mathematic
Numerical simulation of chaotic maps with the new generalized Caputo-type fractional-order operator
This work considers a new generalized operator which is based on the application of Caputo-type fractional
derivative is applied to model a number of nonlinear chaotic phenomena, such as the Oiseau mythique
Bicéphale, Oiseau mythique and L’Oiseau du paradis maps. Numerical approximation of the generalized
Caputo-type fractional derivative using the novel predictor–corrector scheme, which indeed is regarded as
an extension of a well-known Adams–Bashforth–Moulton classical-order algorithm. A range of new strange
chaotic wave propagation was observed for various maps with varying fractional parameters.http://www.elsevier.com/locate/rinpdm2022Mathematics and Applied Mathematic