Geometric Brownian Motion Based on Stochastic Differential Equation Modeling Considering the Change Point Estimation for the Fluctuation of the Turkish Lira Against the US Dollar

Year 2025, Volume: 8 Issue: 2, 75 - 85, 28.06.2025

Sevda Ozdemır Calikusu , Fevzi Erdoğan , Nihal İnce

https://doi.org/10.33187/jmsm.1643023

https://izlik.org/JA36RX79HD

Abstract

In this study, the data showing the fluctuation of the Turkish Lira (TL) against the US Dollar (USD) between 19.06.2017 and 19.06.2022 were examined with Geometric Brownian Motion Stochastic Differential Equation Modeling (GBM SDEM). The study aims to get the GBM stochastic differential equation that best fits USD/TL data by considering the change point estimation (CP). Considering CP when working with abruptly changing datasets has a positive effect on the performance of the constructed model. In addition, there may be more than one CP in the data set, and as the number of CP increases, more suitable models can be obtained for the dataset. The results are supported by graphs that show the proposed SDE model fits the dataset.

Keywords

Change point estimation , Euler-Maruyama approximation method , Geometric Brownian Motion , Itô stochastic differential equation , Maximum likelihood estimation method

References

• [1] E. Kostrista, D. Çibuku, Introduction to stochastic differential equations, Journal of Natural Sciences and Mathematics of UT, 3(5-6) (2018), 189-195.
• [2] R. Rezaeyan, R. Farnoosh, Stochastic differential equations and application of the Kalman-Bucy filter in the modeling of RC Circuit, Appl. Math. Sci., 4(23) (2010), 1119-1127.
• [3] S. M. Iacus, N. Yoshida, Simulation and Inference for Stochastic Processes with YUIMA: A Comprehensive R Framework for SDEs and Other Stochastic Processes, Springer, 2018.
• [4] Z. Yang, D. Aldous, Geometric Brownian Motion Model in Financial Market, University of California, Berkeley, 2015.
• [5] K. Reddy, V. Clinton, Simulating stock prices using geometric Brownian motion: Evidence from Australian companies, Australas. Account. Bus. Finance J., 10(3) (2016), 23-47. http://dx.doi.org/10.14453/aabfj.v10i3.3
• [6] C. Lausberg, F. Brandt, Forecasting risk and return of listed real estate: A simulation approach with geometric Brownian motion for the German stock market, Zeitschrift für Immobilienökonomie, 10 (2024), 1–38. https://doi.org/10.1365/s41056-024-00070-4
• [7] S. N. Z. Abidin, M. M. Jaffar, A review on geometric Brownian motion in forecasting the share prices in Bursa Malaysia, World Appl. Sci. J., 17(1) (2012), 82-93.
• [8] S. E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004.
• [9] Y. Aıt-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach, Econometrica, 70(1) (2002), 223–262. https://doi.org/10.1111/1468-0262.00274
• [10] S. M. Iacus, Simulation and Inference for Stochastic Differential Equations: With R Examples, Springer, New York, 2008. http://dx.doi.org/10.1007/ 978-0-387-75839-8
• [11] S. M. Iacus, N. Yoshida, Estimation for the change point of volatility in a stochastic differential equation, Stochastic Process. Appl., 122(3) (2012), 1068-1092. https://doi.org/10.1016/j.spa.2011.11.005
• [12] L. H. Sun, Z. Y. Huang, C. Y. Chiu, et al., Detecting structural shifts and estimating single change-points in interval-based time series, Statist. Comput., 35(127) (2025), Article ID 127. https://doi.org/10.1007/s11222-025-10666-y
• [13] M. T. Malinowski, Financial models and well-posedness properties for symmetric set-valued stochastic differential equations with relaxed Lipschitz condition, Nonlinear Anal. Real World Appl., 84 (2025), Article ID 104323. https://doi.org/10.1016/j.nonrwa.2025.104323
• [14] J. Chen, A. K. Gupta, Parametric Statistical Change Point Analysis: With Applications to Genetics, Medicine, and Finance, Birkhäuser, Boston, 2012. http://dx.doi.org/10.1007/978-0-8176-4801-5
• [15] G. Chen, Y. Choi, Y. Zhou, Nonparametric estimation of structural change points in volatility models for time series, J. Econometrics, 126(1) (2005), 79-114. https://doi.org/10.1016/j.jeconom.2004.02.008
• [16] S. Lee, Y. Nishiyama, N. Yoshida, Test for parameter change in diffusion processes by cusum statistics based on one-step estimators, Ann. Inst. Statist. Math., 58 (2006), 211-222. https://doi.org/10.1007/s10463-006-0037-9
• [17] A. D. Gregorio, S. M. Iacus, Least squares volatility change point estimation for partially observed diffusion processes, Commun. Statist. Theory Methods, 37(15) (2008), 2342-2357. https://doi.org/10.1080/03610920801919692
• [18] A. M. B. de Oliveira, A. Mandal, G. J. Power, A primer on the pricing of electric energy options in Brazil via mean-reverting stochastic processes, Energy Rep., 5 (2019), 594-601. https://doi.org/10.1016/j.egyr.2019.03.010
• [19] Y. Tonaki, Y. Kaino, M. Uchida, Estimation for change point of discretely observed ergodic diffusion processes, (2021), arXiv:2102.06871 [math.ST]. http://dx.doi.org/10.48550/arXiv.2102.06871
• [20] O. Buberkoku, An empirical application of an asymmetric stochastic volatility model to ISE100 index,Igdir Univ. J. Soc. Sci., 18 (2019), 503-525.
• [21] O. Buberkoku, An Empirical Application of an Asymmetric Stochastic Volatility Model under the Generalized Hyperbolic Skew Student’s t-Distribution Assumption to the Turkish Exchange Rate Market, Conference: The Journal of Accounting and Finance, 89 (2021), 185-202.
• [22] O. Buberkoku, Modeling Interest Rates, and Forecasting the Yield Curve with Stochastic Interest Rate Models (CIR and Vasicek), Izmir J. Econ., 36(4) (2021), 893-911. https://doi.org/10.24988/ije.807286
• [23] E. Allen, Modeling with Ito Stochastic Differential Equations, Springer, 2007. https://doi.org/10.1007/978-1-4020-5953-7
• [24] N. Ince, Generalized entropy optimization methods in stochastic differential equation modeling, Ph.D. Thesis, Eskisehir Technical University, 2021.
• [25] F. Su, K. S. Chan, Quasi-likelihood estimation of a threshold diffusion process, Journal of Econometrics, 189(2) (2015), 473-484. https://doi.org/10.1016/j.jeconom.2015.03.038
• [26] S. M. Iacus, Option Pricing and Estimation of Financial Models with R, John Wiley and Sons, 2011. http://dx.doi.org/10.1002/9781119990079
• [27] S. Karamolegkos, D. E. Koulouriotis, Advancing short-term load forecasting with decomposed Fourier ARIMA: A case study on the Greek energy market, Energy, 325 (2025), Article ID 135854. https://doi.org/10.1016/j.energy.2025.135854
• [28] Yahoo Finance, USD/TRY historical exchange rates (June 19, 2017 – June 19, 2022), Available at: https://finance.yahoo.com. Accessed May 18, 2025.