A New Stable Local Radial Basis Function Approach for Option Pricing

Golbabai, A.; Mohebianfar, Ehsan

doi:10.1007/s10614-016-9561-8

Cited by 16 publications

(7 citation statements)

References 55 publications

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“…It is well known that stability analysis is only applied to partial differential equations with constant coefficients. Let the stencils with three uniform nodes be used [39]. For x = {x m-1 , x m , x m+1 }, we get…”

Section: Stability Analysismentioning

confidence: 99%

Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions

et al. 2018

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The exact solution of fractional combined Korteweg-de Vries and modified Korteweg-de Vries (KdV-mKdV) equation is obtained by using the (1/G ) expansion method. To investigate a geometrical surface of the exact solution, we choose γ = 1.The collocation method is applied to the fractional combined KdV-mKdV equation with the help of radial basis for 0 < γ < 1. L 2 and L ∞ error norms are computed with the Mathematica program. Stability is investigated by the Von-Neumann analysis. Instable numerical solutions are obtained as the number of node points increases. It is shown that the reason for this situation is that the exact solution contains some degenerate points in the Lorentz-Minkowski space.

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Section: Stability Analysismentioning

confidence: 99%

Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions

et al. 2018

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“…e most famous PDE in finance is the Black-Scholes (B-S) model, which is broadly adopted for option pricing. So far, studies have presented different approaches for finding the numerical solution of this model and its variation when the exact form does not exist [5][6][7][8][9][10][11][12][13]. By using tick-by-tick data, Cartea and del-Castillo-Negrete [14] found that the value of European-style options satisfies a FPDE containing a nonlocal operator in time-to-maturity known as the Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Novel ANN Method for Solving Ordinary and Time‐Fractional Black–Scholes Equation

Bajalan

2021

Complexity

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The main aim of this study is to introduce a 2-layered artificial neural network (ANN) for solving the Black–Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine-tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black–Scholes model are reported.

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“…The studies on the shape parameter can be categorized as modifying the shape parameter [13], using variable shape parameter [14][15][16][17], and finding optimal shape parameter [18][19][20][21][22][23][24][25]. Despite the success of identifying the optimal value of the shape parameter, the proposed methods are neither universal nor consistent.…”

Section: Introductionmentioning

confidence: 99%

New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter

2020

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Radial basis function (RBF) is gaining popularity in function interpolation as well as in solving partial differential equations thanks to its accuracy and simplicity. Besides, RBF methods have almost a spectral accuracy. Furthermore, the implementation of RBF-based methods is easy and does not depend on the location of the points and dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is primarily impacted by the basis function and the node distribution. At a small value of shape parameter, the RBF becomes more accurate, but unstable. Several approaches were followed in the open literature to overcome the instability issue. One of the approaches is optimizing the solver in order to improve the stability of ill-conditioned matrices. Another approach is based on searching for the optimal value of the shape parameter. Alternatively, modified bases are used to overcome instability. In the open literature, radial basis function using QR factorization (RBF-QR), stabilized expansion of Gaussian radial basis function (RBF-GA), rational radial basis function (RBF-RA), and Hermite-based RBFs are among the approaches used to change the basis. In this paper, the Taylor series is used to expand the RBF with respect to the shape parameter. Our analyses showed that the Taylor series alone is not sufficient to resolve the stability issue, especially away from the reference point of the expansion. Consequently, a new approach is proposed based on the partition of unity (PU) of RBF with respect to the shape parameter. The proposed approach is benchmarked. The method ensures that RBF has a weak dependency on the shape parameter, thereby providing a consistent accuracy for interpolation and derivative approximation. Several benchmarks are performed to assess the accuracy of the proposed approach. The novelty of the present approach is in providing a means to achieve a reasonable accuracy for RBF interpolation without the need to pinpoint a specific value for the shape parameter, which is the case for the original RBF interpolation.

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A New Stable Local Radial Basis Function Approach for Option Pricing

Cited by 16 publications

References 55 publications

Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions

Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions

Novel ANN Method for Solving Ordinary and Time‐Fractional Black–Scholes Equation

New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter

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