Stochastic Analysis of Asset Values in Financial Markets: Effects of Drift and Volatility

C. P.  Ogbogbo; I. A.  Amadi; M. O.  Kolawole; T. K.  Atsektor

C. P. Ogbogbo Department of Mathematics, University of Ghana, Legon, Accra, Ghana
I. A. Amadi Department of Mathematics & Statistics, Captain Elechi Amadi Polytechnics, Port Harcourt, Nigeria
M. O. Kolawole Department of Mathematics & Statistics, Captain Elechi Amadi Polytechnics, Port Harcourt, Nigeria
T. K. Atsektor Department of Mathematics, University of Ghana, Legon, Accra, Ghana

Keywords: Stock Prices, Financial Markets, Stochastic Analysis, Drift and Volatility

Abstract

The Stochastic Differential Equation (SDE) is well known prevailing mathematical tools used for the estimation of asset values over time. In particular, this paper considered stochastic systems with prominence on variations of stock parameters. These problems were solved analytically by implementing the Ito's method of solution where precise measures were given on the assessments of asset values. Therefore, the impressions on Tables for investors in financial markets were analyzed to demonstrate empirically the behavior of asset values when volatility increases. Also the expectations of each independent solution were obtained graphically. From the analysis we deduce that; increase in volatility decreases the value of assets, increase in volatility shows rise and fall in the assessments of asset values due to periodic parameter incorporated in the model, and finally describes the average value of the asset over time as it affects financial markets in time varying investments. This work presents a unique contribution and first-time approach that is, assessing asset values by modeling stock drift coefficients, with some constants and periodic event parameters.

References

[1] Afolabi, A. S. (2022). On the constrained optimal control problems with mixed constraints. International Journal of Mathematical Analysis and Modelling. Vol. 5: 25-35.
[2] Afolabi, A.S. and Olotu, O. (2020), Analytical Solutions of Optimal Control Problems with Mixed Constraints. Iconic Research and Engineering Journals. Vol. 3, Issue 12: 1–9.
[3] Akpanibah, C. E., and Ini, U. O. (2021). Refund Clause of Contributions with Predetermined Interest under CEV Model. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 7(1): 1-16.
[4] Belloti, P., Berthold,T. and Neves, K. (2020). Algorithms for discrete nonlinear optimization in FICO Xpress.Technische Universität Berlin. Berlin, Germany: 10-20.
[5] Brandenberg, R. and Silbernagl, M. (2014). Implementing a unit commitment power market model in FICO Xpress Mosel. FICO Community: 1-37.
[6] Betts, J. T. (2001). Practical Methods for Optimal Control Using Nonlinear Programming. Philadelphia. PA SIAM: 23-30.
[7] Belloti, P. (2010). Dash Optimization. Xpress-MP Essentials, 2nd ed. Birmingham. UK: Dash Associates: 1-10.
[8] Forsgren, A., Gill, P. E. and Wright, M. H. (2002). Interior-Point Methods for Nonlinear Optimization. SIAM Review. Vol. 44: 525-597.
[9] Hassan, M.K., Ahamed, M.S., Alam, M.S. and Hossain, M.B. (2013). An implicit method for numerical solution of singular and stiff initial value problems. Applied Mathematics and Nonlinear Sciences. Vol. 18: 23-45.
[10] FICO® Analytics. How Mosel Fits into FICO® Xpress Optimization. Analytic Cloud Youtube channel,https://www.youtube.com/watch?v=iGMhjLi09uw. Accessed: Nov. 26, 2019
[11] Kogut, P. I. and Leugering, G.(2022). Matrix-valued
L
−
1
L
−1
-optimal controls in the coefficients of linear elliptic problems. DFG-Cluster CE315: Engineering of Advanced Materials: 1-20.
[12] Kogut,P. I., Kupenko, O. P. and Leugering, G. (2020). Optimal Control in Matrix-Valued Coefficients for Nonlinear Monotone Problems: Optimality Conditions. Part I: 1-20.
[13] Lenhart, S. and Workman, J.T. (2007). Optimal Control Applied to Biological Models. Boca Raton. FL: Chapman & Hall/CRC: 25-33.
[14] Miquel, T. (2020). Introduction to Optimal Control. Master's course material. Journal of Optimization Theory and Applications. Vol. 187: 99-124.
[15] MIT OpenCourseWare. (2024).MIT Lecture Notes on Lebesgue Integration, 2024. [Online]. Available: https://math.mit.edu/ rbm/18-102-S14/Chapter2. pdf. Accessed: Nov. 26, 2024.
[16] Mattheij, R. and Molenaar, J. (2002). Ordinary Differential Equations in Theory and Practice. Philadelphia, SIAM: 231-237.
[17] Rao, S.S. (2009). Engineering Optimization: Theory and Practice 4th ed. John Wiley & Sons, Hoboken, NJ: 2-9.
[18] Naidu, D. S. (2003). Optimal Control Systems. Boca Raton FL: CRC Press: 25-65.
[19] Olaiya, O. O. (2024). One-step fourth derivative block integrator for the numerical solution of singularly perturbed third-order boundary value problems. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 10(1): 60-71.
[20] Olotu, O and Adekunle, A. I. (2010). Analytic and numeric solutions of discretized constrained optimal control problem with vector-matrix coefficients. Advanced Modeling and Optimization. Vol. 12, No. 1: 119-131.
[21] Olotu, O and Olorunsola, S. A. (2008). Convergence Ratio Profile for Optimal Control Problems Governed by Ordinary Differential Equations with Matrix Coefficients. American Journal of Applied Sciences. No. 2: 89-92.
[22] Orji, J.C., Miswato, M. and Afolabi, A.S. (2024). Optimal Control Problems Constrained by Non-Linear Ordinary Differential Equations. International Research and Innovation Journal. 1(1):134-149
[23] Oyewale, A.A. and Afolabi, A.S. (2024). On the Optimal Control Problems constrained by Ordinary Differential Equations Using Conjugate Gradient Method and FICO Xpress Mosel. Journal of Mathematical and Computational Science. ISSN:1927-5307, 14:18.
[24] Rahaman, M. H., Hasan, M. K., Li, M. A. and Alam, A. S. (2020). Implicit methods for numerical solution of singular initial value problems. Applied Mathematics and Nonlinear Sciences. Vol. 5: 18-30.
[25] Ross, I. M. and Fahroo, F. (2004). Pseudospectral Methods for Optimal Control. Journal of Guidance, Control, and Dynamics. Vol. 27, No. 3: 397-405.
[26] Walker, P. Worcester Polytechnic Institute, Handout on Lipschitz Continuity, [Online]. Available:https://users.wpi.edu/ walker/MA500/HANDOUTS/LipschitzContinuity.pdf. Accessed: Nov. 26, 2024.