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

• Werry Febrianti Departmen Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung
• Kuntjoro Adji Sidarto Departmen Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung
• Novriana Sumarti Departmen Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung
Keywords: Adaptive differential evolution, Approximation solution, Black-Scholes, Metaheuristic optimization, Partial differential equations

### Abstract

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.

### References

Babaei, M. A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization. Applied Soft Computing 13, 7 (2013), 3354–3365.

Babasola, O., Ngare, P., and Owoloko, E. Crank nicolson approach for the valuation of the barrier options. International of Journal of Applied Mathematical Sciences (JAMS) 11 (2018), 7–21.

Black, F., and Scholes, M. The pricing of options and corporate liabilities. Journal of political economy 81, 3 (1973), 637–654.

Boyle, P. P., and Tian, Y. An explicit finite difference approach to the pricing of barrier options. Applied Mathematical Finance 5, 1 (1998), 17–43.

Chaquet, J. M., and Carmona, E. J. Solving differential equations with fourier series and evolution strategies. Applied Soft Computing 12, 9 (2012), 3051–3062.

Chen, S.-H., and Lee, W.-C. Option pricing with genetic algorithms: The case of europeanstyle options. In ICGA (1997), pp. 704–711.

Eskiizmirliler, S., Gunel, K., and Polat, R. On the solution of the black–scholes equation using feed-forward neural networks. Computational Economics 58, 3 (2021), 915–941.

Febrianti, W., Sidarto, K. A., and Sumarti, N. Solving some ordinary differential equations numerically using differential evolution algorithm with a simple adaptive mutation scheme. In AIP Conference Proceedings (2021), vol. 2329, AIP Publishing LLC, p. 040003.

Febrianti, W., Sidarto, K. A., and Sumarti, N. Solving systems of ordinary differential equations using differential evolution algorithm with the best base vector of mutation scheme. In AIP Conference Proceedings (2021), vol. 2423, AIP Publishing LLC, p. 020006.

Gulen, S., Popescu, C., and Sari, M. A new approach for the black–scholes model with linear and nonlinear volatilities. Mathematics 7, 8 (2019), 760.

He, J., and Zhang, A. Finite difference/fourier spectral for a time fractional black–scholes model with option pricing. Mathematical Problems in Engineering 2020 (2020).

Higham, D. J. An introduction to financial option valuation: mathematics, stochastics and computation.

Ioffe, G., and Ioffe, M. Application of finite difference method for pricing barrier options. Paper available at www.egartech.com (2003).

Jeong, D., Yoo, M., and Kim, J. Finite difference method for the black–scholes equation without boundary conditions. Computational Economics 51, 4 (2018), 961–972.

Kazikova, A., Pluhacek, M., and Senkerik, R. Why tuning the control parameters of metaheuristic algorithms is so important for fair comparison? Mendel 26, 2 (2020), 9–16.

Khan, N. A., Razzaq, O. A., and Hameed, T. A metaheuristic approach to optimize european call function with boundary conditions. Journal of Systems Science and Information 6, 3 (2018), 260–268.

Kim, S., Lee, C., Lee, W., Kwak, S., Jeong, D., and Kim, J. Nonuniform finite difference scheme for the three-dimensional time-fractional black–scholes equation. Journal of Function Spaces 2021 (2021).

Matousek, R., Dobrovsky, L., and Kudela, J. How to start a heuristic? utilizing lower bounds for solving the quadratic assignment problem. International Journal of Industrial Engineering Computations 13, 2 (2022), 151–164.

Merton, R. C. Theory of rational option pricing. The Bell Journal of economics and management science (1973), 141–183.

Panagant, N., and Bureerat, S. Differential evolution algorithm for solving a nonlinear single pendulum problem. In Advanced Materials Research (2014), vol. 931, Trans Tech Publ, pp. 1129–1133.

Panagant, N., and Bureerat, S. Solving partial differential equations using a new differential evolution algorithm. Mathematical Problems in Engineering 2014 (2014).

Sadollah, A., Eskandar, H., Kim, J. H., et al. Approximate solving of nonlinear ordinary differential equations using least square weight function and metaheuristic algorithms. Engineering Applications of Artificial Intelligence 40 (2015), 117–132.

Sadollah, A., Sayyaadi, H., and Yadav, A. A dynamic metaheuristic optimization model inspired by biological nervous systems: Neural network algorithm. Applied Soft Computing 71 (2018), 747–782.

Storn, R., and Price, K. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization 11, 4 (1997), 341–359.

Umeorah, N., and Mashele, P. A cranknicolson finite difference approach on the numericalestimation of rebate barrier option prices. Cogent Economics & Finance (2019).

Published
2022-12-20
How to Cite

Febrianti, W., Sidarto, K. and Sumarti, N. 2022. Approximate Solution for Barrier Option Pricing Using Adaptive Differential Evolution With Learning Parameter. MENDEL. 28, 2 (Dec. 2022), 76-82. DOI:https://doi.org/10.13164/mendel.2022.2.076.
Section
Research articles