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
  IJETT-book-cover           
  
© 2023 by IJETT Journal
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.
[CrossRef] [Google Scholar] [Publisher Link]
[2] Dan Stefanica, A Primer for the Mathematics of Financial Engineering, 2nd ed., United States of America: Financial Engineering Press, 2011.
[Google Scholar] [Publisher Link]
[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.
[Google Scholar] [Publisher Link]
[4] S.D. Howison, “Applied Mathematics and Finance,” Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences, 1994.
[CrossRef] [Google Scholar] [Publisher Link]
[5] John C. Hull, Options, Futures, and Other Derivatives, 11th Ed., New Jersey, United States of America: Pearson, 2021.
[Google Scholar] [Publisher Link]
[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.
[CrossRef] [Google Scholar] [Publisher Link]
[7] Lee Tse Yueng, “Crank-Nicolson Scheme for Asian Option,” M.Sc. Thesis, Universiti Tunku Abdul Rahman, Malaysia, 2012.
[Google Scholar] [Publisher Link]
[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.
[CrossRef] [Publisher Link]
[9] Hélyette Geman, and Marc Yor, “Bessel Processes, Asian Options, and Perpetuities,” Mathematical Finance, vol. 3, no. 4, pp. 349-375, 1993.
[CrossRef] [Google Scholar] [Publisher Link]
[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.
[CrossRef] [Google Scholar] [Publisher Link]
[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.
[CrossRef] [Google Scholar] [Publisher Link]
[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.
[Publisher Link]
[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.
[CrossRef] [Google Scholar] [Publisher Link]
[14] Steven Chapra, and Raymond Canale, Numerical Methods for Engineers, 8th ed., United States of America: McGraw-Hill, 2021.
[Publisher Link]
[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.
[Google Scholar] [Publisher Link]
[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.
[Google Scholar] [Publisher Link]
[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.
[CrossRef] [Google Scholar] [Publisher Link]
[18] Stefan Kronawitter, “Automatic Performance Optimization of Stencil Codes,” Doctoral Dissertation, Universität Passau, Germany, 2019.
[Google Scholar] [Publisher Link]
[19] Jun Zhang, “Acceleration of Five-Point Red-Black Gauss-Seidel in Multigrid for Poisson Equation,” Applied Mathematics and Computation, vol. 80, no. 1, pp. 73-93, 1996.
[CrossRef] [Google Scholar] [Publisher Link]
[20] Irad Yavneh, “Multigrid Smoothing Factors for Red-Black Gauss–Seidel Relaxation Applied to a Class of Elliptic Operators,” SIAM Journal on Numerical Analysis, vol. 32, no. 4, pp. 1126-1138, 1996.
[CrossRef] [Google Scholar] [Publisher Link]
[21] Mahmoud El Maghrbay, Reda Ammar, and Sanguthevar Rajasekaran, “Fast GPU Algorithms for Implementing the Red-Black Gauss-Seidel Method for Solving Partial Differential Equations,” IEEE Symposium on Computers and Communications, pp. 269-274, 2013.
[CrossRef] [Google Scholar] [Publisher Link]
[22] Cosmo D. Santiago et al., “A Multigrid Waveform Relaxation Method for Solving the Pennes Bioheat Equation,” Numerical Heat Transfer, Part A: Applications, vol. 83, no. 9, pp. 976-990, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[23] Johanna Potyka et al., “Towards DNS of Droplet-Jet Collisions of Immiscible Liquids with FS3D,” arXiv, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[24] C.W.W. Haddon, and D.P. Hampshire, “Fast Multigrid Simulations of Pinning in REBCO with Highly Resistive Nanorods,” IEEE Transactions on Applied Superconductivity, vol. 33, no. 5, pp. 1-5, 2023.
[CrossRef] [Google Scholar] [Publisher Link]