Robust Numerical Approach to CRR Model under Self-financing Assumption
Keywords:
Option Pricing, Metaheuristic Algorithms, Cox-Ross-Rubinstein Model, American Options
Abstract
The accurate pricing of options is crucial for minimizing financial risks and making informed investment decisions in dynamic markets. Traditional models like the Black-Scholes often fail to account for the early exercise feature of American options and the self-financing replicating portfolio concept, leading to less realistic pricing. This study address these gaps by employing various metaheuristic algorithms, including Particle Swarm Optimization, Differential Evolution, Grey Wolf Optimization, and Simulated Annealing Algorithm, to estimate the parameters for a modified Cox-Ross-Rubinstein model. We derive a Brownian motion model incorporating upward and downward factors and use the Euler-Maruyama method to simulate stock price paths. By comparing these simulated paths with real stock data, we evaluate the effectiveness of the estimated parameters. Additionally, we improve the numerical method for estimating American option prices via the CRR model by integrating the self-financing replicating portfolio concept. The results demonstrate that Particle Swarm and Grey Wolf optimization algorithms provide parameter estimates that yield simulated paths closely matching the real stock data, thereby offering computationally realistic prices for American options. This study highlights the potential of integrating metaheuristic algorithms with traditional models to enhance the accuracy and reliability of option pricing.References
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merton interest rate model. Communications in Statistics-Theory and Methods,
pages 1–10.
6g wireless communications: Recent advances and applications. Ad Hoc Networks, page
103474.
Aydınhan, A. O., Kolm, P. N., Mulvey, J. M., and Shu, Y. (2024). Identifying patterns in
financial markets: Extending the statistical jump model for regime identification. Annals
of Operations Research, pages 1–37.
Balb´as, A., Balb´as, B., and Balb´as, R. (2019). Golden options in financial mathematics.
Mathematics and Financial Economics, 13:637–659.
Beliavsky, G., Danilova, N., and Logunov, A. (2021). Robust estimation of european and
asian options. In Operator Theory and Harmonic Analysis: OTHA 2020, Part I–New General
Trends and Advances of the Theory 10, pages 101–117. Springer.
Cutland, N. J., Roux, A., Cutland, N. J., and Roux, A. (2013). American options. Derivative
Pricing in Discrete Time, pages 211–267.
Dembele, A., Mwangi, E., Bouchair, A., Ronoh, K. K., and Ataro, E. O. (2023). Automated
modified grey wolf optimizer for identification of unauthorized requests in softwaredefined
networks. International Journal of Advanced Computer Science and Applications,
14(7).
Ding, K., Cui, Z., and Yang, X. (2023). Pricing arithmetic asian and amerasian options: A
diffusion operator integral expansion approach. Journal of Futures Markets, 43(2):217–
241.
Dwivedi, S., Vardhan, M., and Tripathi, S. (2022). Defense against distributed dos attack
detection by using intelligent evolutionary algorithm. International Journal of Computers
and Applications, 44(3):219–229.
Estember, R. D. and Marana, M. J. R. (2016). Forecasting of stock prices using brownian
motion–monte carlo simulation. In International conference on industrial engineering and
operations management, pages 8–10.
Ferriani, F. and Zoi, P. (2022). The dynamics of price jumps in the stock market: an empirical
study on europe and us. The European Journal of Finance, 28(7):718–742.
10
Hoencamp, J., Jain, S., and Kandhai, B. (2024). A static replication approach for callable
interest rate derivatives: mathematical foundations and efficient estimation of simm–mva.
Quantitative Finance, pages 1–24.
Hussain, J., Soomro, M. A., Dahri, S. A., Memon, K., Bano, M., Awwad, F. A., Ismail, E. A.,
and Ahmad, H. (2024). A study of maximizing skew brownian motion with applications
to option pricing. Journal of Radiation Research and Applied Sciences, 17(1):7–32.
Idzorek, T. M. (2023). Personalized multiple account portfolio optimization. Financial Analysts
Journal, 79(3):155–170.
London, J. (2005). Modeling derivatives in C++, volume 263. John Wiley & Sons.
Mahmoodi, A., Hashemi, L., Jasemi, M., Mehraban, S., Lalibert´e, J., and Millar, R. C. (2023).
A developed stock price forecasting model using support vector machine combined with
metaheuristic algorithms. Opsearch, 60(1):59–86.
Mensah, E. T., Boateng, A., Frempong, N. K., and Maposa, D. (2023). Simulating stock prices
using geometric brownian motion model under normal and convoluted distributional assumptions.
Scientific African, 19:e01556.
Nafidi, A., Bahij, M., Achchab, B., and Guti´errez-S´anchez, R. (2019). The stochastic weibull
diffusion process: Computational aspects and simulation. Applied Mathematics and Computation,
348:575–587.
Ogundile, O. and Edeki, S. (2020). Karhunen-lo´eve expansion of brownian motion for approximate
solutions of linear stochastic differential models using picard iteration. J. Math.
Comput. Sci., 10(5):1712–1723.
Pelsser, A. and Vorst, T. (1996). Transaction costs and efficiency of portfolio strategies.
European journal of operational research, 91(2):250–263.
Phan, H. and Kim, S. (2022). Numerical approaches of pricing european options in the
cox-ross-rubinstein models. Universal Journal of Applied Mathematics, 10(3):43–48.
R¨oman, J. R. (2017). Analytical finance. Springer.
R¨uschendorf, L. (2023). Option pricing in discrete time models. In Stochastic Processes and
Financial Mathematics, pages 1–10. Springer.
Sarkissian, J. (2020). Quantum coupled-wave theory of price formation in financial markets:
Price measurement, dynamics and ergodicity. Physica A: Statistical Mechanics and its
Applications, 554:124300.
Shahvaroughi, M. and Farrokhi, H. (2023). Improved market prediction using meta-heuristic
algorithms and time series model and testing market efficiency. Iran Journal of Computer
Science, 6(1):29–61.
Yi, Z., Cao, X., Chen, Z., and Li, S. (2023). Artificial intelligence in accounting and finance:
Challenges and opportunities. IEEE Access, 11:129100–129123.
Zhang, H., Zhang, M., Liu, F., and Shen, M. (2024). Review of the fractional black-scholes
equations and their solution techniques. Fractal and Fractional, 8(2):101.
Zhao, P. and Guo, Z. (2024). Pricing of geometric average asian option under the subdiffusion
merton interest rate model. Communications in Statistics-Theory and Methods,
pages 1–10.
Published
2025-09-06
How to Cite
Abonongo, J., & Chidzalo, P. (2025). Robust Numerical Approach to CRR Model under Self-financing Assumption. Statistics, Optimization & Information Computing. https://doi.org/10.19139/soic-2310-5070-2331
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Research Articles
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