10.1007/s40096-022-00474-0

Computational algorithm for financial mathematical model based on European option

  • Nikhil Srivastava1,
  • Aman Singh1,
  • Vineet Kumar Singh*1,
  1. Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, IN

Published in Issue 2022-05-30

How to Cite

Srivastava, N., Singh, A., & Singh, V. K. (2022). Computational algorithm for financial mathematical model based on European option. Mathematical Sciences, 17(4 (December 2023). https://doi.org/10.1007/s40096-022-00474-0
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Abstract

Abstract In this manuscript, a computational approach based on the combination of finite difference with the operational matrix approach is constructed for the time-fractional Black–Scholes model (TFBSM) arising in the financial market. We have applied the L1-2 scheme to approximate the Caputo derivative and the operational matrix method (OMM) by using shifted Legendre polynomials (SLP) and shifted Chebyshev polynomials (SCP) to approximate the space derivatives. The proposed algorithm easily transforms the TFBSM into a system of algebraic equations, which can be solved easily to get the numerical solution. Furthermore, theoretical unconditional stability and convergence of the numerical scheme are established for α∈(0,α¯]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,{\bar{\alpha }}]$$\end{document} and the stability of the proposed algorithm is also verified numerically. Finally, the scheme is tested on five numerical problems, including the European double barrier call option and the European call and put option. It is observed that this computational approach gives almost the same accuracy with both the basis functions, but the CPU time taken by scheme with SLP basis is less than the SCP basis function. The effect of different parameters like volatility, interest rate, fractional order, etc., on the option pricing, is also investigated. A comparative study of the numerical results by the proposed algorithm with the schemes given in Zhang et al. (Comput Math Appl 71(9):1772–1783, 2016) and De Staelen and Hendy (Comput Math Applic 74(6):1166–1175, 2017), is also provided to show its effectiveness and accuracy.

Keywords

  • Time-fractional Black–Scholes model,
  • L1-2 approximation,
  • Shifted Legendre polynomial,
  • Shifted Chebyshev polynomial,
  • Stability analysis,
  • Convergence analysis

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