A Numerical Analysis of Comparing Several Practical Approaches for Pricing both American and European Put Options
Main Article Content
Abstract
The present paper focuses on pricing with European or American put option that has yet to be studied. In solving option pricing problems, numerical methods form an essential part. Five numerical methods: Black-Scholes-Merton, Monte Carlo, Binomial, Trinomial, and Finite Difference have been implemented in the present work. Here Computer Algebra System (CAS) Python is used for the simulation. A comparison of these methods for both European and American put options has been shown by a graphical representation. Results show that Crank Nicolson Finite Difference Methods (FDM) gives us more accurate results than the other methods.
Article Details
How to Cite
[1]
A. Akter, A. B. M. S. Hossain, and S. Parvin, “A Numerical Analysis of Comparing Several Practical Approaches for Pricing both American and European Put Options”, AJSE, vol. 23, no. 2, pp. 158 - 167, Aug. 2024.
Section
Articles
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
AJSE contents are under the terms of the Creative Commons Attribution License. This permits anyone to copy, distribute, transmit and adapt the worknon-commercially provided the original work and source is appropriately cited.
References
[1] F. Black and M. Scholes, “The pricing of options and corporate liabilities,” J. Polit. Econ., vol. 81, no. 3, pp. 637–654, 1973.
[2] L. Jódar, P. Sevilla-Peris, J. C. Cortes, and R. Sala, “A new direct method for solving the Black-Scholes equation,” Appl. Math. Lett., vol. 18, no. 1, pp. 29–32, 2005.
[3] R. Company, A. L. González, and L. Jódar, “Numerical solution of modified Black-Scholes equation pricing stock options with discrete dividend,” Math. Comput. Model., vol. 44, no. 11–12, pp. 1058–1068, 2006.
[4] A. K. Karagozoglu, “Option Pricing Models: From Black-Scholes-Merton to Present.,” J. Deriv., vol. 29, no. 4, 2022.
[5] P. Morales-Bañuelos, N. Muriel, and G. Fernández-Anaya, “A modified Black-Scholes-Merton model for option pricing,” Mathematics, vol. 10, no. 9, p. 1492, 2022.
[6] S. K. Parameswaran and S. Basu, “The black-scholes merton model—implications for the option delta and the probability of exercise,” Theor. Econ. Lett., vol. 10, no. 6, pp. 1307–1313, 2020.
[7] M. Aalaei and M. Manteqipour, “An adaptive Monte Carlo algorithm for European and American options,” Comput. Methods Differ. Equations, vol. 10, no. 2, pp. 489–501, 2022.
[8] K. M. Ondieki, “Swaption pricing under libor market model using Monte-Carlo method with simulated annealing optimization,” Strathmore University, 2021.
[9] A. Barbu, S.-C. Zhu, and others, Monte carlo methods, vol. 35. Springer, 2020.
[10] J. Zhang, “Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey,” Wiley Interdiscip. Rev. Comput. Stat., vol. 13, no. 5, p. e1539, 2021.
[11] Y. Zhang, “The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives,” PLoS One, vol. 15, no. 2, p. e0229737, 2020.
[12] D. An, N. Linden, J.-P. Liu, A. Montanaro, C. Shao, and J. Wang, “Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance,” Quantum, vol. 5, p. 481, 2021.
[13] Y. Mou, M. Li, H. Li, and B. Shi, “Investigation on the Mechanism of American Put Option Pricing under Binomial Consideration,” in 2022 2nd International Conference on Enterprise Management and Economic Development (ICEMED 2022), 2022, pp. 255–261.
[14] H. Xiaoping and C. Jie, “Randomized Binomial Tree and Pricing of American-Style Options,” Math. Probl. Eng., vol. 2014, no. 1, p. 291737, 2014.
[15] G. Leduc and M. Nurkanovic Hot, “Joshi’s split tree for option pricing,” Risks, vol. 8, no. 3, p. 81, 2020.
[16] Y. Muroi and S. Suda, “Binomial tree method for option pricing: Discrete cosine transform approach,” Math. Comput. Simul., vol. 198, pp. 312–331, 2022.
[17] H. Xiaoping, G. Jiafeng, D. Tao, C. Lihua, and C. Jie, “Pricing options based on trinomial markov tree,” Discret. Dyn. Nat. Soc., vol. 2014, no. 1, p. 624360, 2014.
[18] C. Lin, Y. Bao, and Y. He, “Non-arbitrage Pricing and Hedging Strategy for European Put Option Based on Trinomial Model,” in Proceedings of the 2021 5th International Conference on Software and e-Business, 2021, pp. 81–85.
[19] G. Leduc, “The Boyle--Romberg trinomial tree, a highly efficient method for double barrier option pricing,” Mathematics, vol. 12, no. 7, p. 964, 2024.
[20] D. Josheski and M. Apostolov, “A review of the binomial and trinomial models for option pricing and their convergence to the Black-Scholes model determined option prices,” Econom. Ekonom. Adv. Appl. Data Anal., vol. 24, no. 2, pp. 53–85, 2020.
[21] H. Nohrouzian, A. Malyarenko, and Y. Ni, “Constructing trinomial models based on cubature method on Wiener space: Applications to pricing financial derivatives,” arXiv Prepr. arXiv2204.10692, 2022.
[22] S. R. Chakravarty and P. Sarkar, “Option Pricing Using Finite Difference Method,” in An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain, Emerald Publishing Limited, 2020, pp. 49–56.
[23] E. S. Schwartz, “The valuation of warrants: Implementing a new approach,” J. financ. econ., vol. 4, no. 1, pp. 79–93, 1977.
[24] R. C. Merton, “Theory of rational option pricing,” Bell J. Econ. Manag. Sci., pp. 141–183, 1973.
[25] S. E. Fadugba and C. R. Nwozo, “Crank Nicolson finite difference method for the valuation of options,” Pacific J. Sci. Technol., vol. 14, pp. 136–146, 2013.
[26] M. S. Kumar, S. P. Das, and M. Reza, “A comparison between analytic and numerical solution of linear Black-Scholes equation governing option pricing: Using BANKNIFTY,” in 2012 World Congress on Information and Communication Technologies, 2012, pp. 437–441.
[27] G. Courtadon, “A more accurate finite difference approximation for the valuation of options,” J. Financ. Quant. Anal., vol. 17, no. 5, pp. 697–703, 1982.
[28] F. N. Nwobi, M. N. Annorzie, and I. U. Amadi, “The Impact of Crank-Nicolson Finite Difference Method in Valuation of Options,” Commun. Math. Financ., vol. 8, no. 1, pp. 93–122, 2019.
[29] M. N. Anwar and L. S. Andallah, “A Study on Numerical Solution of Black-Scholes Model,” J. Math. Financ., vol. 8, no. 2, 2018.
[30] S. Fadugba, C. Nwozo, and T. Babalola, “The comparative study of finite difference method and Monte Carlo method for pricing European option,” Math. Theory Model., vol. 2, no. 4, pp. 60–67, 2012.
[31] M. Mehdizadeh Khalsaraei, A. Shokri, Z. Mohammadnia, and H. M. Sedighi, “Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black-Scholes Equation,” J. Math., vol. 2021, 2021.
[32] M. Mehdizadeh Khalsaraei, M. M. Rashidi, A. Shokri, H. Ramos, and P. Khakzad, “A Nonstandard Finite Difference Method for a Generalized Black-Scholes Equation,” Symmetry (Basel)., vol. 14, no. 1, p. 141, 2022.
[33] T. E. Copeland, J. F. Weston, K. Shastri, and others, Financial theory and corporate policy, vol. 4. Pearson Addison Wesley Boston, 2005.
[34] S. C. Sutradhar and A. B. M. S. Hossain, “A Comparative Study of Pricing Option with Efficient Methods,” GUB J. Sci. Eng., pp. 1–7, 2020.
[35] J. C. Hull, Options futures and other derivatives. Pearson Education India, 2003.
[36] J. C. Cox, S. A. Ross, and M. Rubinstein, “Option pricing: A simplified approach,” J. financ. econ., vol. 7, no. 3, pp. 229–263, 1979, doi: https://doi.org/10.1016/0304-405X(79)90015-1.
[37] S. Shreve, Stochastic calculus for finance I: the binomial asset pricing model. Springer Science \& Business Media, 2005.
[38] H. Föllmer, A. Schied, and T. Lyons, “Stochastic finance. An introduction in discrete time,” Math. Intell., vol. 26, no. 4, pp. 67–68, 2004.
[39] T. K. Debnath and A. B. M. S. Hossain, A Comparative Analysis of some numerical techniques of option pricing. University of Dhaka, 2015.
[40] T. K. Debnath and A. B. M. S. Hossain, “A Comparative Study between Implicit and Crank-Nicolson Finite Difference Method for Option Pricing,” GANIT J. Bangladesh Math. Soc., vol. 40, no. 1, pp. 13–27, 2020.
[41] A. Akter, S. Sutradhar, A. B. M. Hossain, and S. Parvin, “A Numerical Study of Different Convenient Methods for Pricing Put Option,” SSRN Electron. J., 2022.
[2] L. Jódar, P. Sevilla-Peris, J. C. Cortes, and R. Sala, “A new direct method for solving the Black-Scholes equation,” Appl. Math. Lett., vol. 18, no. 1, pp. 29–32, 2005.
[3] R. Company, A. L. González, and L. Jódar, “Numerical solution of modified Black-Scholes equation pricing stock options with discrete dividend,” Math. Comput. Model., vol. 44, no. 11–12, pp. 1058–1068, 2006.
[4] A. K. Karagozoglu, “Option Pricing Models: From Black-Scholes-Merton to Present.,” J. Deriv., vol. 29, no. 4, 2022.
[5] P. Morales-Bañuelos, N. Muriel, and G. Fernández-Anaya, “A modified Black-Scholes-Merton model for option pricing,” Mathematics, vol. 10, no. 9, p. 1492, 2022.
[6] S. K. Parameswaran and S. Basu, “The black-scholes merton model—implications for the option delta and the probability of exercise,” Theor. Econ. Lett., vol. 10, no. 6, pp. 1307–1313, 2020.
[7] M. Aalaei and M. Manteqipour, “An adaptive Monte Carlo algorithm for European and American options,” Comput. Methods Differ. Equations, vol. 10, no. 2, pp. 489–501, 2022.
[8] K. M. Ondieki, “Swaption pricing under libor market model using Monte-Carlo method with simulated annealing optimization,” Strathmore University, 2021.
[9] A. Barbu, S.-C. Zhu, and others, Monte carlo methods, vol. 35. Springer, 2020.
[10] J. Zhang, “Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey,” Wiley Interdiscip. Rev. Comput. Stat., vol. 13, no. 5, p. e1539, 2021.
[11] Y. Zhang, “The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives,” PLoS One, vol. 15, no. 2, p. e0229737, 2020.
[12] D. An, N. Linden, J.-P. Liu, A. Montanaro, C. Shao, and J. Wang, “Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance,” Quantum, vol. 5, p. 481, 2021.
[13] Y. Mou, M. Li, H. Li, and B. Shi, “Investigation on the Mechanism of American Put Option Pricing under Binomial Consideration,” in 2022 2nd International Conference on Enterprise Management and Economic Development (ICEMED 2022), 2022, pp. 255–261.
[14] H. Xiaoping and C. Jie, “Randomized Binomial Tree and Pricing of American-Style Options,” Math. Probl. Eng., vol. 2014, no. 1, p. 291737, 2014.
[15] G. Leduc and M. Nurkanovic Hot, “Joshi’s split tree for option pricing,” Risks, vol. 8, no. 3, p. 81, 2020.
[16] Y. Muroi and S. Suda, “Binomial tree method for option pricing: Discrete cosine transform approach,” Math. Comput. Simul., vol. 198, pp. 312–331, 2022.
[17] H. Xiaoping, G. Jiafeng, D. Tao, C. Lihua, and C. Jie, “Pricing options based on trinomial markov tree,” Discret. Dyn. Nat. Soc., vol. 2014, no. 1, p. 624360, 2014.
[18] C. Lin, Y. Bao, and Y. He, “Non-arbitrage Pricing and Hedging Strategy for European Put Option Based on Trinomial Model,” in Proceedings of the 2021 5th International Conference on Software and e-Business, 2021, pp. 81–85.
[19] G. Leduc, “The Boyle--Romberg trinomial tree, a highly efficient method for double barrier option pricing,” Mathematics, vol. 12, no. 7, p. 964, 2024.
[20] D. Josheski and M. Apostolov, “A review of the binomial and trinomial models for option pricing and their convergence to the Black-Scholes model determined option prices,” Econom. Ekonom. Adv. Appl. Data Anal., vol. 24, no. 2, pp. 53–85, 2020.
[21] H. Nohrouzian, A. Malyarenko, and Y. Ni, “Constructing trinomial models based on cubature method on Wiener space: Applications to pricing financial derivatives,” arXiv Prepr. arXiv2204.10692, 2022.
[22] S. R. Chakravarty and P. Sarkar, “Option Pricing Using Finite Difference Method,” in An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain, Emerald Publishing Limited, 2020, pp. 49–56.
[23] E. S. Schwartz, “The valuation of warrants: Implementing a new approach,” J. financ. econ., vol. 4, no. 1, pp. 79–93, 1977.
[24] R. C. Merton, “Theory of rational option pricing,” Bell J. Econ. Manag. Sci., pp. 141–183, 1973.
[25] S. E. Fadugba and C. R. Nwozo, “Crank Nicolson finite difference method for the valuation of options,” Pacific J. Sci. Technol., vol. 14, pp. 136–146, 2013.
[26] M. S. Kumar, S. P. Das, and M. Reza, “A comparison between analytic and numerical solution of linear Black-Scholes equation governing option pricing: Using BANKNIFTY,” in 2012 World Congress on Information and Communication Technologies, 2012, pp. 437–441.
[27] G. Courtadon, “A more accurate finite difference approximation for the valuation of options,” J. Financ. Quant. Anal., vol. 17, no. 5, pp. 697–703, 1982.
[28] F. N. Nwobi, M. N. Annorzie, and I. U. Amadi, “The Impact of Crank-Nicolson Finite Difference Method in Valuation of Options,” Commun. Math. Financ., vol. 8, no. 1, pp. 93–122, 2019.
[29] M. N. Anwar and L. S. Andallah, “A Study on Numerical Solution of Black-Scholes Model,” J. Math. Financ., vol. 8, no. 2, 2018.
[30] S. Fadugba, C. Nwozo, and T. Babalola, “The comparative study of finite difference method and Monte Carlo method for pricing European option,” Math. Theory Model., vol. 2, no. 4, pp. 60–67, 2012.
[31] M. Mehdizadeh Khalsaraei, A. Shokri, Z. Mohammadnia, and H. M. Sedighi, “Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black-Scholes Equation,” J. Math., vol. 2021, 2021.
[32] M. Mehdizadeh Khalsaraei, M. M. Rashidi, A. Shokri, H. Ramos, and P. Khakzad, “A Nonstandard Finite Difference Method for a Generalized Black-Scholes Equation,” Symmetry (Basel)., vol. 14, no. 1, p. 141, 2022.
[33] T. E. Copeland, J. F. Weston, K. Shastri, and others, Financial theory and corporate policy, vol. 4. Pearson Addison Wesley Boston, 2005.
[34] S. C. Sutradhar and A. B. M. S. Hossain, “A Comparative Study of Pricing Option with Efficient Methods,” GUB J. Sci. Eng., pp. 1–7, 2020.
[35] J. C. Hull, Options futures and other derivatives. Pearson Education India, 2003.
[36] J. C. Cox, S. A. Ross, and M. Rubinstein, “Option pricing: A simplified approach,” J. financ. econ., vol. 7, no. 3, pp. 229–263, 1979, doi: https://doi.org/10.1016/0304-405X(79)90015-1.
[37] S. Shreve, Stochastic calculus for finance I: the binomial asset pricing model. Springer Science \& Business Media, 2005.
[38] H. Föllmer, A. Schied, and T. Lyons, “Stochastic finance. An introduction in discrete time,” Math. Intell., vol. 26, no. 4, pp. 67–68, 2004.
[39] T. K. Debnath and A. B. M. S. Hossain, A Comparative Analysis of some numerical techniques of option pricing. University of Dhaka, 2015.
[40] T. K. Debnath and A. B. M. S. Hossain, “A Comparative Study between Implicit and Crank-Nicolson Finite Difference Method for Option Pricing,” GANIT J. Bangladesh Math. Soc., vol. 40, no. 1, pp. 13–27, 2020.
[41] A. Akter, S. Sutradhar, A. B. M. Hossain, and S. Parvin, “A Numerical Study of Different Convenient Methods for Pricing Put Option,” SSRN Electron. J., 2022.