Stabilized explicit Runge–Kutta methods for multi-asset American options

Martín-Vaquero, J.; Kleefeld, B.

doi:10.1016/j.camwa.2014.01.018

Cited by 35 publications

(19 citation statements)

References 43 publications

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“…Some stabilized explicit Runge-Kutta algorithms (SERK) were derived in [25,26,29,30]. They are also based on three-term recurrence formulae.…”

Section: Extrapolated Explicit Methodsmentioning

confidence: 99%

“…These types of problems are common in a large amount of applications such as atmospheric phenomena, biology, chemistry, combustion problems, financial mathematics, fluid mechanics, industrial engineering, laser modelling, malware propagation, medicine, molecular dynamics, nuclear kinetics, etc., see [3,6,12,13,14,15,17,20,30,43], to mention a few.…”

Section: Introductionmentioning

confidence: 99%

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ESERK5: A fifth-order extrapolated stabilized explicit Runge–Kutta method

Martín-Vaquero

Kleefeld

2019

Journal of Computational and Applied Mathematics

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A new algorithm is developed and analyzed for multi-dimensional non-linear parabolic partial differential equations (PDEs) which are semi-discretized in the spatial variables leading to a system of ordinary differential equations (ODEs). It is based on fifth-order extrapolated stabilized explicit Runge-Kutta schemes (ESERK). They are explicit methods, and therefore it is not necessary to employ complicated software for linear or non-linear system of equations. Additionally, they have extended stability regions along the negative real semi-axis, hence they can be considered to solve stiff problems coming from very common diffusion or reaction-diffusion problems.Previously, only lower-order codes (up to fourth-order) have been constructed and made available in the scientific literature. However, at the same time, higher-order codes were demonstrated to be very efficient to solve equations where it is necessary to have a high precision or they have transient zones that are very severe, and where functions change very fast. The new schemes allow changing the step length very easily and with a very small computational cost. Thus, a variable step length, with variable number of stages algorithm is constructed and compared with good numerical results in relation to other well-known ODE solvers.

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“…Some stabilized explicit Runge-Kutta algorithms (SERK) were derived in [25,26,29,30]. They are also based on three-term recurrence formulae.…”

Section: Extrapolated Explicit Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

ESERK5: A fifth-order extrapolated stabilized explicit Runge–Kutta method

Martín-Vaquero

Kleefeld

2019

Journal of Computational and Applied Mathematics

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“…We note that the SERK scheme is also limited in stage number, requiring 600 digits of precision for 320 stages, albeit principally due to severely ill-conditioned matrix systems used in calculating the stability polynomials by means of the Remez algorithm [18]. A subsequent revision of the SERK methodology has demonstrated a stability range which is four times larger [36].…”

Section: Internal Stabilitymentioning

confidence: 99%

A class of high-order Runge–Kutta–Chebyshev stability polynomials

O’Sullivan¹

2015

Journal of Computational Physics

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The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order N is presented. Roots of FRKC stability polynomials of degree L = M N are used to construct explicit schemes comprising L forward Euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to ∼ L 2 . The associated stability domain scales as M 2 along the real axis. Marginally stable real-valued points on the interior of the stability domain are removed via a prescribed damping procedure.By construction, FRKC schemes meet all linear order conditions; for nonlinear problems at orders above 2, complex splitting or Butcher series composition methods are required. Linear order conditions of the FRKC stability polynomials are verified at orders 2, 4, and 6 in numerical experiments.Comparative studies with existing methods show the second-order unsplit FRKC2 scheme and higher order (4 and 6) split FRKCs schemes are efficient for large moderately stiff problems.

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“…The three asset American-put option under the Black-Scholes model [26] is considered as a free boundary-value problem and can be written as [27] ∂U ∂t…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Partial Differential Equations via Local Meshless Method

et al. 2019

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In this paper, numerical simulation of one, two and three dimensional partial differential equations (PDEs) are obtained by local meshless method using radial basis functions (RBFs). Both local and global meshless collocation procedures are used for spatial discretization, which convert the given PDEs into a system of ODEs. Multiquadric, Gaussian and inverse quadratic RBFs are used for spatial discretization. The obtained system of ODEs has been solved by different time integrators. The salient feature of the local meshless method (LMM) is that it does not require mesh in the problem domain and also far less sensitive to the variation of shape parameter as compared to the global meshless method (GMM). Both rectangular and non rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy and ease implementation of the proposed method are shown via test problems.

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Stabilized explicit Runge–Kutta methods for multi-asset American options

Cited by 35 publications

References 43 publications

ESERK5: A fifth-order extrapolated stabilized explicit Runge–Kutta method

ESERK5: A fifth-order extrapolated stabilized explicit Runge–Kutta method

A class of high-order Runge–Kutta–Chebyshev stability polynomials

Numerical Simulation of Partial Differential Equations via Local Meshless Method

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