A nonstandard finite difference scheme for a nonlinear Black–Scholes equation

Arenas, Abraham J.; González-Parra, Gilberto; Caraballo, Blas M.

doi:10.1016/j.mcm.2011.11.009

Cited by 18 publications

(8 citation statements)

References 14 publications

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“…Many of these require numerical approaches to enable a solution to be quantified. Other numerical and analytic solutions can be found in Abe and Ishimura [21], Ankudinova and Ehrhardt [22], Anwar and Andallah [23], Arenas, Gonza-lez-Parra, and Caraballo [24], Cen and Le [25], Company, Jodar, and Pintos [26],…”

Section: Introductionmentioning

confidence: 99%

Using the Power Series Method to Evaluate Non-Linear Contingent Claim Partial Differential Equations

Buetow¹,

Hanke²

2022

JMF

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We use the Power Series Method (PSM) numerical framework for estimating nonlinear variations of the Black-Scholes partial differential equations (PDE). The PSM offers an alternative to using traditional finite difference methods. Traditional approaches often require complicated mathematical manipulations of the nonlinear PDE before they can be deployed; or non-linear relationships are approximated linearly usually involving an iterative solver. The PSM does not have these requirements to be implemented and avoids needing an iterative solver. The PSM allows both symbolic and numerical analysis with full choice of the order of the solver in time. This paper illustrates how the PSM offers an alternative and superior, framework to evaluate nonlinear PDE's. Several extensions for future research are also offered.

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Section: Introductionmentioning

confidence: 99%

Using the Power Series Method to Evaluate Non-Linear Contingent Claim Partial Differential Equations

Buetow¹,

Hanke²

2022

JMF

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“…Во многих математических моделях, в частности, динамики популяций, ценообразования опционов, химической кинетики, газовой динамики и других используются уравнения в частных производных с нелинейностью в операторе дифференцирования, смотри, например, [1,2] и ссылки в них. К тому же модели могут быть осложнены эффектами запаздывания, дробными производными, многомерностью пространственных переменных.…”

Section: Introductionunclassified

“…Наиболее простой -рассмотрение явных схем, которые, как правило, являются условно усточивыми, что накладывает жесткие ограничения на шаги по времени. При этом основное внимание при построениии разностной схемы уделяется монотонности численного решения решения или его неотрицательности [1].…”

Section: Introductionunclassified

Numerical solving of partial differential equations with heredity and nonlinearity in the differential operator

Gorbova¹,

Пименов²,

Solodushkin³

2019

Sib. Elektron. Mat. Izv.

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The problem to be considered is a numerical solving of nonlinear partial differential equations with heredity effect. Nonlinearity is contained in the operator of differentiation as well as in the inhomogeneity function. We propose a nonlinear implicit difference scheme, which implies the use of iterative methods to find the solution on each time layer. To take into account the heredity effect the interpolation and extrapolation of grid solution were used. Stability and convergence of the proposed difference scheme were proved. Numerical experiments were carried out and results coincides with the theoretical ones.

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“…Nonstandard finite difference (NSFD) techniques, developed by Mickens [28], to design elementary stable NSFD methods that preserve the local stability of equilibria of the approximated differential system for arbitrary time step sizes. It is important for constructing the positivity preserving schemes to avoid unrealistic negative values for the solution [4], [11], [15], [17], [18], [28], [33]. The explicit schemes are generally less expensive than other classical methods since larger step sizes can be taken without generating negative solutions [15], [28], [29].…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatments of the Transmission Dynamics of West Nile Virus and It’s Optimal Control N. H. Sweilam, O. M. Saad and D. G. Mohamed

2019

Electronic Journal of Mathematical Analysis and Applications

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In this paper, numerical studies for transmission dynamics of West Nile Virus mathematical model are presented. The nonstandard finite difference method is introduced to solve the posed model. Positivity, boundedness, and convergence of the nonstandard finite difference scheme are studied. Also, numerical stability analysis of fixed points is studied. An optimal control problem is formulated and studied theoretically using the Pontryagin's maximum principle. The obtained results by using nonstandard finite difference method are compared with standard finite difference method. It can be concluded that the nonstandard finite difference method is more efficient and preserves the stability and positivity of the solutions in large regions.

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A nonstandard finite difference scheme for a nonlinear Black–Scholes equation

Cited by 18 publications

References 14 publications

Using the Power Series Method to Evaluate Non-Linear Contingent Claim Partial Differential Equations

Using the Power Series Method to Evaluate Non-Linear Contingent Claim Partial Differential Equations

Numerical solving of partial differential equations with heredity and nonlinearity in the differential operator

Numerical Treatments of the Transmission Dynamics of West Nile Virus and It’s Optimal Control N. H. Sweilam, O. M. Saad and D. G. Mohamed

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