Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black–Scholes Equation

Khalsaraei, Mohammad Mehdizadeh; Shokri, Ali; Mohammadnia, Zahra; Sedighi, Hamid M.

doi:10.1155/2021/6679484

Cited by 7 publications

(6 citation statements)

References 32 publications

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“…As a mathematical model, certain assumptions, such as the log-normality of underlying prices, constant volatility, frictionless market, continuous trading without dividends applied to stocks, etc., are made for the Black-Scholes model to hold [13]. Though the Black-Scholes model has been criticized over the years due to some underlying assumptions, which are not applicable in the realworld scenario, certain recent works are associated with the model [33][34][35][36]. Additionally, Eskiizmirliler et al [37] numerically solved the Black-Scholes equation for the European call options using feed-forward neural networks.…”

Section: Extended Black-scholes Model For Barrier Optionsmentioning

confidence: 99%

Barrier Options and Greeks: Modeling with Neural Networks

Umeorah,

Mashele,

Agbaeze

et al. 2023

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This paper proposes a non-parametric technique of option valuation and hedging. Here, we replicate the extended Black–Scholes pricing model for the exotic barrier options and their corresponding Greeks using the fully connected feed-forward neural network. Our methodology involves some benchmarking experiments, which result in an optimal neural network hyperparameter that effectively prices the barrier options and facilitates their option Greeks extraction. We compare the results from the optimal NN model to those produced by other machine learning models, such as the random forest and the polynomial regression; the output highlights the accuracy and the efficiency of our proposed methodology in this option pricing problem. The results equally show that the artificial neural network can effectively and accurately learn the extended Black–Scholes model from a given simulated dataset, and this concept can similarly be applied in the valuation of complex financial derivatives without analytical solutions.

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Section: Extended Black-scholes Model For Barrier Optionsmentioning

confidence: 99%

Barrier Options and Greeks: Modeling with Neural Networks

Umeorah,

Mashele,

Agbaeze

et al. 2023

Axioms

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“…In this section, we will briefly introduce nonstandard finite difference methods (NSFDs) (for more details one can see [4,5,7,8,28,[33][34][35][36][37][38][39][40][41][42]). Numerical methods based on naive finite difference approximations to solve ODEs and PDEs problems may not work well, and properties such as positivity of the solution cannot be transferred to the numerical solutions.…”

Section: Nonstandard Finite-difference Strategymentioning

confidence: 99%

“…• Generally, the nonlinear terms can be variously approximated non-locally on the computational grid. For example, reaction terms can be modeled as follows: (see [4,8,17,28,[34][35][36][37][38][39][40][41][42])…”

Section: Nonstandard Finite-difference Strategymentioning

confidence: 99%

“…In [10], Anwar and Andallah introduced an explicit method for the numerical solution of the Black-Scholes equation in which the stability condition of the method is obtained by a convex combination. In addition to the above methods, there are other methods for numerically solving PDEs, ODEs, IEs, and Fractional equations [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

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A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation

et al. 2022

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In this paper, we evaluate and discuss different numerical methods to solve the Black–Scholes equation, including the θ-method, the mixed method, the Richardson method, the Du Fort and Frankel method, and the MADE (modified alternating directional explicit) method. These methods produce numerical drawbacks such as spurious oscillations and negative values in the solution when the volatility is much smaller than the interest rate. The MADE method sacrifices accuracy to obtain stability for the numerical solution of the Black–Scholes equation. In the present work, we improve the MADE scheme by using non-standard finite difference discretization techniques in which we use a non-local approximation for the reaction term (we call it the MMADE method). We will discuss the sufficient conditions to be positive of the new scheme. Also, we show that the proposed method is free of spurious oscillations even in the presence of discontinuous initial conditions. To demonstrate how efficient the new scheme is, some numerical experiments are performed at the end.

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“…In addition to the usual properties of consistency, stability and convergence, the NSDF methods give rise to numerical solutions that also maintain essential qualitative properties of the solutions [14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

et al. 2022

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An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.

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Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black–Scholes Equation

Cited by 7 publications

References 32 publications

Barrier Options and Greeks: Modeling with Neural Networks

Barrier Options and Greeks: Modeling with Neural Networks

A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation

A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

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