On the Solution of the Black–Scholes Equation Using Feed-Forward Neural Networks

Eskiizmirliler, Saadet; Günel, Korhan; Polat, Refet

doi:10.1007/s10614-020-10070-w

Cited by 8 publications

(9 citation statements)

References 28 publications

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“…, 2019; Mehrkanoon et al. , 2012; Mehrkanoon and Suykens, 2015), neural networks (Eskiizmirliler et al. , 2021; Famelis and Kaloutsa, 2021; Günel and Gör, 2021; Li and Wang, 2021; Schiassi et al.…”

Section: Introductionmentioning

confidence: 99%

“…With the rapid development of computer hardware and various new theories, neural networks and other machine learning algorithms are gradually applied to various areas (Weng et al, 2021(Weng et al, , 2022. In fact, various machine learning algorithms such as least squares support vector machines (Lu et al, 2019;Mehrkanoon et al, 2012;Mehrkanoon and Suykens, 2015), neural networks (Eskiizmirliler et al, 2021;Famelis and Kaloutsa, 2021;G€ unel and G€ or, 2021;Li and Wang, 2021;Schiassi et al, 2021) and deep learning (Wang et al, 2021;Weinan et al, 2017) have been utilized to solve these differential equations. In particular, artificial neural networks (ANNs) are often superior to the traditional numerical methods on the field of solving differential equation Chakraverty, 2014, 2016;Yazdi et al, 2011Yazdi et al, , 2012Yazdi and Pourreza, 2010).…”

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confidence: 99%

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Numerical solution for high-order ordinary differential equations using H-ELM algorithm

Weng

Sun

2022

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PurposeThis paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).Design/methodology/approachThe proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.FindingsTo demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.Originality/valueThe H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Numerical solution for high-order ordinary differential equations using H-ELM algorithm

Weng

Sun

2022

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“…This is based on Chen and Lee [6], who introduced solving the BS equation with a metaheuristic approach using genetic algorithms. Further, we present the solution of BS equations with a trial solution, building on the concept from Khan et al [16] and Eskiizmirliler et al [7].…”

Section: Introductionmentioning

confidence: 99%

Approximate Solution for Barrier Option Pricing Using Adaptive Differential Evolution With Learning Parameter

Febrianti

Sidarto

Sumarti

2022

mendel

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Black-Scholes (BS) equations, which are in the form of stochastic partial differential equations, are fundamental equations in mathematical finance, especially in option pricing. Even though there exists an analytical solution to the standard form, the equations are not straightforward to be solved numerically. The effective and efficient numerical method will be useful to solve advanced and non-standard forms of BS equations in the future. In this paper, we propose a method to solve BS equations using an approach of optimization problems, where a metaheuristic optimization algorithm is utilized to find the best-approximated solutions of the equations. Here we use the Adaptive Differential Evolution with Learning Parameter (ADELP) algorithm. The BS equations being solved are meant to find values of European option pricing that is equipped with Barrier option pricing. The result of our approximation method fits well to the analytical approximation solutions.

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“…Andreou et al (2006) show that the artificial neural network models with the use of the Huber function outperform the ones optimized with least squares. Eskiizmirliler et al (2020) approximate the unknown function of the option value using a trial function, which depends on a neural network solution and satisfies the given boundary conditions of the Black-Scholes equation. Arin and Ozbayoglu (2020) develop hybrid deep learning based options pricing models to achieve better pricing compared to Black-Scholes.…”

Section: Introductionmentioning

confidence: 99%

A Neural Network Approach to Value R&D Compound American Exchange Option

Villani

2021

Comput Econ

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In this paper we show as the neural network methodology, coupled with the Least Squares Monte Carlo approach, can be very helpful in valuing R&D investment opportunities. As it is well known, R&D projects are made in a phased manner, with the commencement of subsequent phase being dependent on the successful completion of the preceding phase. This is known as a sequential investment and therefore R&D projects can be considered as compound options. In addition, R&D investments often involve considerable cost uncertainty so that they can be viewed as an exchange option, i.e. a swap of an uncertain investment cost for an uncertain gross project value. Finally, the production investment can be realized at any time before the maturity date, after that the effects of R&D disappear. Consequently, an R&D project can be considered as a compound American exchange option. In this context, the Least Squares Monte Carlo method is a powerful and flexible tool for capital budgeting decisions and for valuing American-type options. But, using the simulated values as “targets”, the implementation of a neural network allows to extend the results for any R&D valuation and to abate the waiting time of Least Squares Monte Carlo simulation.

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On the Solution of the Black–Scholes Equation Using Feed-Forward Neural Networks

Cited by 8 publications

References 28 publications

Numerical solution for high-order ordinary differential equations using H-ELM algorithm

Numerical solution for high-order ordinary differential equations using H-ELM algorithm

Approximate Solution for Barrier Option Pricing Using Adaptive Differential Evolution With Learning Parameter

A Neural Network Approach to Value R&D Compound American Exchange Option

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