The Using Neural Network and Finite Difference Method for Option Pricing under Black-Scholes-Vasicek Model
الموضوعات :
Mahdiye Mohmmadi
1
,
Elham Dastranj
2
,
Abdolmajid Abdolbaghi Ataabadi
3
,
Hossein Sahebi fard
4
1 - Department of Mathematics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran
2 - Department of Mathematics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran
3 - Department of Management, Faculty of Industrial Engineering and Management, Shahrood University of Technolo-gy, Shahrood, Semnan, Iran
4 - Department of Mathematics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran
الکلمات المفتاحية: Option Pricing , Neural Networks , Finite Difference Method , Black-Scholes-Vasicek Model,
ملخص المقالة :
In this paper, the European option pricing is done using neural networks in the Black-Scholes-Vasicek market. The general purpose of this research is to compare the accuracy of neural network and Black-Scholes-Vasicek models for the pricing of call options. In the sequel, the finite difference method is applied to find approximate solutions of partial differential equation related to option pricing in the considered market. In the design of the artificial neural network required for this research, the parameters of the Black-Scholes-Vasicek model have been used as network inputs, as well as 720 data from the daily price of stock options available in the Tehran Stock Exchange market (in 1400) as the network output. The approximate solutions obtained in this article, which were carried out by two methods of neural networks and finite differences on the Tehran stock exchange based on the daily price of stock options, are shown that neural networks are more accurate method comparing with finite difference. The comparison of pricing results using neural networks with real prices in the assumed market is presented and shown via diagram, as well. In this paper, the European option pricing is done using neural networks in the Black-Scholes-Vasicek market. The general purpose of this research is to compare the accuracy of neural network and Black-Scholes-Vasicek models for the pricing of call options. In the sequel, the finite difference method is applied to find approximate solutions of partial differential equation related to option pricing in the considered market. In the design of the artificial neural network required for this research, the parameters of the Black-Scholes-Vasicek model have been used as network inputs, as well as 720 data from the daily price of stock options available in the Tehran Stock Exchange market (in 1400) as the network output. The approximate solutions obtained in this article, which were carried out by two methods of neural networks and finite differences on the Tehran stock exchange based on the daily price of stock options, are shown that neural networks are more accurate method comparing with finite difference. The comparison of pricing results using neural networks with real prices in the assumed market is presented and shown via diagram, as well.
[1] Bennell, J., Sutcliffe, C., Black-Scholes versus neural networks in pricing FTSE 100 options, Intelligent Systems in Accounting, Finance & Management: International Journal, 2000; 12(4), 243-260. Doi: 10.1002/isaf.254.
[2] Björk, T., Arbitrage theory in continuous time. Oxford university press, 2009.
[3] Dastranj, E., Sahebi Fard, H., Abdolbaghi, A., Hejazi, S, R., Power option pricing under the unstable conditions (Evidence of power option pricing under fractional Heston model in the Iran gold mar-ket), Physica A: Statistical Mechanics and its Applications, 2020; 537, 122690. Doi: 10.1016/j.physa.2019.122690
[4] Dastranj, E., Sahebi Fard, H., Exact solutions and numerical simulation for Bakstein-Howison model, Computational Methods for Differential Equations, 2022; 10(2), 461-474. Doi:10.22034/cmde.2021.42640.1834.
[5] Dura, G., Mosneagu, A. M., Numerical approximation of Black-Scholes equation, Annals of the Alexandru Ioan Cuza University-Mathematics, 2010; 56(1), 39-64. Doi:10.2478/v10157-010-0004-x.
[6] François-Lavet, V., Henderson, P., Islam, R., Bellemare, M. G., Pineau, J., An introduction to deep rein-forcement learning, Foundations and Trends® in Machine Learning, 2018;11(3-4): 219-354. Doi:10.1561/2200000071.
[7] Hull, J.c, Risk management and financial institutions, John Wiley & Sons, 2012.
[8] MacBeth, J. D., Merville, L, J., An empirical examination of the Black-Scholes call option pricing model, The journal of finance, 1979; 34(5), 1173-1186. Doi:10.2307/2327242
[9] Pagnottoni, P., Neural network models for Bitcoin option pricing, Frontiers in Artificial Intelligence, 2019; 2: 5. Doi:10.3389/frai.2019.00005
[10] Ren, P., Parametric estimation of the Heston model under the indirect observability framework, Doctoral dissertation, University of Houston. 2014.
[11] Sahebi Fard, H., Easapour, H., Dastranj, E., Stochastic analysis to find exact solutions of rainbow option under 2-D Black-Scholes model, Journal of Interdisciplinary Mathematics, accepted.
[12] Sahebi Fard, H., Dastranj, E., Abdolbaghi Ataabadi, A., Analytical and numerical solutions for the pricing of a combination of two financial derivatives in a market under Hull-White model, Advances in Mathematical Finance and Applications, 2022; 7(4):1013-1023. Doi: 10.22034/AMFA.2021.1902303.1447
[13] Kaustubh, Y., Formulation of a rational option pricing model using artificial neural networks, South-eastCon, IEEE, 2021, Doi:10.20944\preprints202012.0426.v1
[14] Shahvaroughi Farahani, M., Prediction of interest rate using artificial neural network and novel meta-heuristic algorithms, Iranian Journal of Accounting, Auditing and Finance, 2021; 5(1): 1-30. Doi:10.22067/ijaaf.2021.67957.1017
[15] Roul, P., Goura, V, P., A compact finite difference scheme for fractional Black-Scholes option pricing model, Applied Numerical Mathematics, 2021; 166: 40-60. Doi:10.1016/j.apnum.2021.03.017
[16] Mitra, S.K., An option pricing model that combines neural network approach and Black-Scholes formula, Global Journal of Computer Science and Technology, 2012; 12(4): 7-15.
[17] Chalasani, P., and Jha, S., Steven Shreve: Stochastic Calculus and Finance, Lecture notes, 1997;1-343.
[18] Wang, X., He, X., Bao, Y., Zhao, Y., Parameter estimates of Heston stochastic volatility model with MLE and consistent EKF algorithm, Science China Information Sciences, 2018; 6:1-17.
[19] Araghi, M., Dastranj, E., Abdolbaghi Ataabadi, A., Sahebi Fard, H., Option pricing with artificial neural network in a time dependent market, Advances in Mathematical Finance and Applications, 2024; 2(2): 723.
[20] Zohrabi, A., Tehrani, R., Souri, A., Sadeghi Sharif, S.J., Provide an improved factor pricing model using
neural networks and the gray wolf optimization algorithm, Advances in Mathematical Finance and
Applications, 2023; 8(1):23-45. Doi: 10.22034/AMFA.2022.1942547.1641.
[21] Nikumaram, H., Rahnamay Roodposhti, F., Zanjirdar, M., The explanation of risk and expected rate of return by using of Conditional Downside Capital Assets Pricing Model, Financial knowledge of securities analysis,2008;3(1):55-77
[22] Zamanpour,A., Zanjirdar, M., Davodi Nasr,M, Identify and rank the factors affecting stock portfolio optimization with fuzzy network analysis approach, Financial Engineering And Portfolio Management,2021;12(47): 210-236
