Development of Red-Black Gauss-Seidel Algorithm for Efficiently Pricing Fixed Strike Asian Options based on Arithmetic Average

## Development of Red-Black Gauss-Seidel Algorithm for Efficiently Pricing Fixed Strike Asian Options based on Arithmetic Average

Volume-71 Issue-11
Year of Publication : 2023
Author : Wei Sin Koh, Saiful Hafizah Jaaman, Rokiah Rozita Ahmad, Jumat Sulaiman
DOI : 10.14445/22315381/IJETT-V71I11P219

How to Cite?

Wei Sin Koh, Saiful Hafizah Jaaman, Rokiah Rozita Ahmad, Jumat Sulaiman, "Development of Red-Black Gauss-Seidel Algorithm for Efficiently Pricing Fixed Strike Asian Options based on Arithmetic Average," International Journal of Engineering Trends and Technology, vol. 71, no. 11, pp. 181-189, 2023. Crossref, https://doi.org/10.14445/22315381/IJETT-V71I11P219

Abstract
In this paper, the Red-Black Gauss-Seidel (RBGS) algorithm is developed to solve the arithmetic Asian option pricing. Developing such an algorithm is crucial for optimizing computational resources and reducing the processing time of the financial instrument. The pricing of arithmetic Asian options is formulated by approximating the Black-Scholes Partial Differential Equation (PDE) through the Crank-Nicolson finite difference method. Subsequently, the RBSG iterative algorithm is employed to solve the system of linear equations derived from the Crank-Nicolson approximation. Extensive computational experiments are conducted to measure the accuracy and efficiency of the RBGS algorithm to the conventional Gauss-Seidel (GS) iterative method. The evaluation criteria include the iteration count, computational time, and root mean squared error (RMSE). The results indicate that the RBSG iterative algorithm significantly reduces the number of iterations and computational time compared to the GS iterative method. Moreover, both iterations yield accurate numerical solutions that align closely. These findings demonstrate the effectiveness of the RBSG algorithm in efficiently pricing arithmetic Asian options while maintaining a high level of accuracy.

Keywords
Asian option, Black-Scholes PDE, Crank-Nicolson finite difference method, Red-Black Gauss-Seidel algorithm, Resource efficiency.

References
[1] Abdul Quadir Md et al., “Novel Optimization Approach for Stock Price Forecasting using Multi-layered Sequential LSTM,” Applied Soft Computing, vol. 134, 2023.
[2] Dan Stefanica, A Primer for the Mathematics of Financial Engineering, 2nd ed., United States of America: Financial Engineering Press, 2011.
[3] Ali Bolfake, Seyed Nourollah Mousavi, and Sima Mashayekhi, “Deep Learning Application in Rainbow Options,” Advances in Mathematical Finance & Applications, vol. 8, no. 3, pp. 951-963, 2023.
[4] S.D. Howison, “Applied Mathematics and Finance,” Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences, 1994.
[5] John C. Hull, Options, Futures, and Other Derivatives, 11th Ed., New Jersey, United States of America: Pearson, 2021.
[6] Christian-Oliver Ewald, Yuexiang Wu, and Aihua Zhang, “Pricing Asian Options with Stochastic Convenience Yield and Jumps,” Quantitative Finance, vol. 23, no. 4, pp. 677-692, 2023.
[7] Lee Tse Yueng, “Crank-Nicolson Scheme for Asian Option,” M.Sc. Thesis, Universiti Tunku Abdul Rahman, Malaysia, 2012.
[8] M. Kamalakkannun, and N.D. Sridhar, “Linear Programming Based Optimal Power Flow Optimization of DCOPF for an IEEE 5 and IEEE 14 Bus System,” SSRG International Journal of Electrical and Electronics Engineering, vol. 9, no. 11, pp. 95-102, 2022.
[9] Hélyette Geman, and Marc Yor, “Bessel Processes, Asian Options, and Perpetuities,” Mathematical Finance, vol. 3, no. 4, pp. 349-375, 1993.
[10] L.C.G. Rogers, and Z. Shi, “The Value of an Asian Option,” Journal of Applied Probability, vol. 32, no. 4, pp. 1077-1088, 1995.
[11] Jan Vecer, “A New PDE Approach for Pricing Arithmetic Average Asian Options,” Journal of Computational Finance, vol. 4, no. 4, pp. 105-113, 2001.
[12] Lee Tse Yueng, and Chin Seong Tah, “A Simple Crank-Nicolson Scheme for Asian Option,” The 6th IMT-GT Conference on Mathematics, Statistics and its Applications, pp. 381-394, 2010.
[13] Nordin Saad, Andang Sunarto, and Azali Saudi, “Accelerated Red-Black Strategy for Image Composition using Laplacian Operator,” International Journal of Computing and Digital Systems, vol. 10, no. 1, pp. 1085-1195, 2021.
[14] Steven Chapra, and Raymond Canale, Numerical Methods for Engineers, 8th ed., United States of America: McGraw-Hill, 2021.
[15] Okechukwu U. Solomon, “Application of Gauss-Seidel Method on Refined Financial Matrix for Solution to Partial Differential Equation (PDE) in Finance,” Asian Journal of Pure and Applied Mathematics, vol. 4, no. 1, pp. 666-667, 2022.
[16] Okechukwu U. Solomon, and Udoinyang I. Efiong, “A Comparative Study of Gauss-Seidel Method and Pseudo Inversion Method in Option Pricing,” Science Set Journal of Physics, vol. 2, no. 1, pp. 1-8, 2023.
[17] Stephen Kinsella, and Terence O'Shea, “Solution and Simulation of Large Stock Flow Consistent Monetary Production Models via the Gauss Seidel Algorithm,” Journal of Policy Modeling, Forthcoming, 2010.
[18] Stefan Kronawitter, “Automatic Performance Optimization of Stencil Codes,” Doctoral Dissertation, Universität Passau, Germany, 2019.