# A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model

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## Abstract

The mathematical modeling in trade and finance issues is the key purpose in the computation of the value and considering option during preferences in contract. This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. Due to the outstanding memory effect present in the fractional derivatives, approximating financial options with regards to their hereditary characteristics can be well interpreted and stated. Motivated by the reason mentioned, relatively reliable and also efficient numerical approaches have to be found while facing with fractional differential equations. The main objective of the current paper is to obtain the approximation solution of the time fractional Black–Scholes model of order \(0<\alpha \le 1\) governing European options based on the moving least-squares (MLS) method. In proposed method, firstly, the mentioned equation is discretized in the time sense based on finite difference scheme of order \({\mathcal {O}}(\delta t^{2-\alpha })\) and then approximated by using MLS approach in the space variable. Furthermore, the stability and convergence of the proposed method are discussed in detail throughout the paper. Numerical evidences and comparisons demonstrate that the proposed method is very accurate and efficient.

## Keywords

Time fractional Black–Scholes model Double barrier option MLS method Stability Convergence## Mathematics Subject Classification

34K37 97N50 91G80## Notes

### Acknowledgements

The authors are thankful to the referees for their valuable comments and constructive suggestions towards the improvement of the original paper. The authors are also very grateful to the Editor-in-Chief, Professor Hans Amman.

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