UWCScholar :: Browsing by Author "Patidar, Kailash C."

Browsing by Author "Patidar, Kailash C."

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An efficient numerical scheme for a time-fractional black–scholes partial differential equation derived from the fractal market hypothesis
(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Nuugulu, Samuel M.; Patidar, Kailash C.; Gideon, Frednard
Since the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in realtime market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems.
An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection
(De Gruyter Open, 2013) Obaid, Hasim; Ouifki, Rachid; Patidar, Kailash C.
We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.
Construction and analysis of exponential time differencing methods for the robust simulation of ecological models
(University of Western Cape, 2021) Farah, Gassan Ali Mohamed Osman; Patidar, Kailash C.
In this thesis, we consider some interesting mathematical models arising in ecology. Our focus is on the exploration of robust numerical solvers which are applicable to models arising in mathematical ecology. To begin with, we consider a simple but nonlinear second-order time-dependent partial differential equation, namely, the Allen-Cahn equation. We discuss the construction of a class of exponential time differencing methods to solve this particular problem. This is then followed by a discussion on the extension of this approach to solve a more difficult fourth-order time-dependent partial differential equation, namely, Kuramoto-Sivashinsky equation. This equation is nonlinear. Further applications include the extension of this approach to solve a complex predator-prey system which is a system of fourth-order time-dependent non-linear partial differential equations. For each of these differential equation models, we presented numerical simulation results.
Contour integral method for European options with jumps
(Elsevier, 2013) Ngounda, Edgard; Patidar, Kailash C.; Pindza, Edson
We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points.
Effect of spatial configuration of an extended nonlinear Kierstead-Slobodkin reaction-transport model with adaptive numerical scheme
(SpringerOpen, 2016) Owolabi, Kolade M.; Patidar, Kailash C.
In this paper, we consider the numerical simulations of an extended nonlinear form of Kierstead-Slobodkin reaction-transport system in one and two dimensions. We employ the popular fourth-order exponential time differencing Runge-Kutta (ETDRK4) schemes proposed by Cox and Matthew (J Comput Phys 176:430-455, 2002), that was modified by Kassam and Trefethen (SIAM J Sci Comput 26:1214-1233, 2005), for the time integration of spatially discretized partial differential equations. We demonstrate the supremacy of ETDRK4 over the existing exponential time differencing integrators that are of standard approaches and provide timings and error comparison. Numerical results obtained in this paper have granted further insight to the question "What is the minimal size of the spatial domain so that the population persists?" posed by Kierstead and Slobodkin (J Mar Res 12:141-147, 1953 ), with a conclusive remark that the popula- tion size increases with the size of the domain. In attempt to examine the biological wave phenomena of the solutions, we present the numerical results in both one- and two-dimensional space, which have interesting ecological implications. Initial data and parameter values were chosen to mimic some existing patterns
A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer
(Wiley, 2019) Munyakazi, Justin B.; Patidar, Kailash C.; Sayi, Mbani T.
The objective of this paper is to construct and analyzea fitted operator finite difference method (FOFDM) forthe family of time-dependent singularly perturbed parabolicconvection–diffusion problems. The solution to the problemswe consider exhibits an interior layer due to the presence ofa turning point. We first establish sharp bounds on the solu-tion and its derivatives. Then, we discretize the time variableusing the classical Euler method. This results in a system ofsingularly perturbed interior layer two-point boundary valueproblems. We propose a FOFDM to solve the system above.
A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems
(Spring Verlag, 2013) Munyakazi, Justin B.; Patidar, Kailash C.
This paper treats a time-dependent singularly perturbed reaction-diffusion problem. We semidiscretize the problem in time by means of the classical backward Euler method. We develop a fitted operator finite difference method (FOFDM) to solve the resulting set of linear problems (one at each time level). We prove that the underlying fitted operator satisfies the maximum principle. This result is then used in the error analysis of the FOFDM. The method is shown to be first order convergent in time and second order convergent in space, uniformly with respect to the perturbation parameter. We test the method on several numerical examples to confirm our theoretical findings.
Fitted numerical methods for delay differential equations arising in biology
(2009) Bashier, Eihab Bashier Mohammed; Patidar, Kailash C.
Fitted Numerical Methods for Delay Di erential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics,Faculty of Natural Sciences, University of the Western Cape. This thesis deals with the design and analysis of tted numerical methods for some delay di erential models that arise in biology. Very often such di erential equations are very complex in nature and hence the well-known standard numerical methods seldom produce reliable numerical solutions to these problems. Ine ciencies of these methods are mostly accumulated due to their dependence on crude step sizes and unrealistic stability conditions.This usually happens because standard numerical methods are initially designed to solve a class of general problems without considering the structure of any individual problems. In this thesis, issues like these are resolved for a set of delay di erential equations. Though the developed approaches are very simplistic in nature, they could solve very complex problems as is shown in di erent chapters.The underlying idea behind the construction of most of the numerical methods in this thesis is to incorporate some of the qualitative features of the solution of the problems into the discrete models. Resulting methods are termed as tted numerical methods. These methods have high stability properties, acceptable (better in many cases) orders of convergence, less computational complexities and they provide reliable solutions with less CPU times as compared to most of the other conventional solvers. The results obtained by these methods are comparable to those found in the literature. The other salient feature of the proposed tted methods is that they are unconditionally stable for most of the problems under consideration.We have compared the performances of our tted numerical methods with well-known software packages, for example, the classical fourth-order Runge-Kutta method, standard nite di erence methods, dde23 (a MATLAB routine) and found that our methods perform much better. Finally, wherever appropriate, we have indicated possible extensions of our approaches to cater for other classes of problems. May 2009.
Fitted numerical methods to solve differential models describing unsteady magneto-hydrodynamic flow
(2011) Buzuzi, George; Patidar, Kailash C.
In this thesis, we consider some nonlinear differential models that govern unsteady magneto-hydrodynamic convective flow and mass transfer of viscous, incompressible,electrically conducting fluid past a porous plate with/without heat sources. The study focusses on the effect of a combination of a number of physical parameters (e.g., chemical reaction, suction, radiation, soret effect,thermophoresis and radiation absorption) which play vital role in these models.Non dimensionalization of these models gives us sets of differential equations. Reliable solutions of such differential equations can-not be obtained by standard numerical techniques. We therefore resorted to the use of the singular perturbation approaches. To proceed, each of these model problems is discretized in time by using a suitable time-stepping method and then by using a fitted operator finite difference method in spatial direction. The combined methods are then analyzed for stability and convergence. Aiming to study the robustness of the proposed numerical schemes with respect to change in the values of the key parame- ters, we present extensive numerical simulations for each of these models. Finally, we confirm theoretical results through a set of specificc numerical experiments.
Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options
(Kent State University, 2013) Pindza, Edson; Patidar, Kailash C.; Ngounda, Edgard
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.
Mathematical analysis and numerical simulation of a tumor-host model with chemotherapy application
(Communications in Mathematical Biology and Neuroscience, 2018) Owolab, Kolade M.; Patidar, Kailash C.; Shikongo, Albert
In this paper, a system of non-linear quasi-parabolic partial differential system, modeling the chemotherapy application of spatial tumor-host interaction is considered. At some certain parameters, we derive the steady state of the anti-angiogenic therapy, baseline therapy and anti-cytotoxic therapy models as well as their local stability condition. We use the method of upper and lower solutions to show that the steady states are globally stable. Since the system of non-linear quasi-parabolic partial differential cannot be solved analytically, we formulate a robust numerical scheme based on the semi-fitted finite difference operator. Analysis of the basic properties of the method shows that it is consistent, stable and convergent. Our numerical results are in agreement with our theoretical findings.
Numerical Methods for Mathematical Models on Warrant Pricing
(University of the Western Cape, 2010) Londani, Mukhethwa; Patidar, Kailash C.
Warrant pricing has become very crucial in the present market scenario. See, for example, M. Hanke and K. Potzelberger, Consistent pricing of warrants and traded options, Review Financial Economics 11(1) (2002) 63-77 where the authors indicate that warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing company's stocks. Another notable work is W.G. Zhang, W.L. Xiao and C.X. He, Equity warrant pricing model under Fractional Brownian motion and an empirical study, Expert System with Applications 36(2) (2009) 3056-3065 where the authors construct equity warrants pricing model under Fractional Brownian motion and deduce the European options pricing formula with a simple method. We study this paper in details in this mini-thesis. We also study some of the mathematical models on warrant pricing using the Black-Scholes framework. The relationship between the price of the warrants and the price of the call accounts for the dilution effect is also studied mathematically. Finally we do some numerical simulations to derive the value of warrants.
Performance of Richardson extrapolation on some numerical methods for a singularly perturbed turning point problem whose solution has boundary layers
(KOREAN MATHEMATICAL SOC, 2014) Munyakazi, Justin B.; Patidar, Kailash C.
Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used.
A robust spectral method for pricing of American put options on zero-coupon bonds
(Global-Science Press, 2018) Pindza, Edson; Patidar, Kailash C.
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.
A robust spectral method for solving Heston’s model
(Springer Verlag, 2014) Ngounda, E.; Patidar, Kailash C.; Pindza, E.
In this paper, we consider the Heston’s volatility model (Heston in Rev. Financ. Stud. 6: 327–343, 1993]. We simulate this model using a combination of the spectral collocation method and the Laplace transforms method. To approximate the two dimensional PDE, we construct a grid which is the tensor product of the two grids, each of which is based on the Chebyshev points in the two spacial directions. The resulting semi-discrete problem is then solved by applying the Laplace transform method based on Talbot’s idea of deformation of the contour integral (Talbot in IMA J. Appl. Math. 23(1): 97–120, 1979).

Browsing by Author "Patidar, Kailash C."

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