# Mathematical Modelling of an Investor’s Wealth with different Stochastic Volatility Models

## Keywords:

Optimal portfolio distribution, stochastic volatility, Ito’s lemma, Hamilton Jacobi Bellman equation, financial market## Abstract

**Communication in Physical Sciences, 2024, 11(2): 355-372**

**Authors: Promise. A. Azor* and ****Amadi Ugwulo Chinyere**

*Received: 12 February 2024/Accepted: 04 May2024 *

This paper investigates the application of various stochastic volatility models in determining optimal investment strategies in the stock market. The study explores the geometric Brownian motion (GBM), constant elasticity of variance (CEV), modified CEV (M-CEV), and Heston volatility models. Each model offers a unique perspective on volatility dynamics and option pricing. The research formulates the Hamilton-Jacobi-Bellman (HJB) equations for each model and employs the Legendre transformation method to convert them into linear partial differential equations (PDEs). The quadratic utility function is utilized to derive optimal portfolio distributions under each model. Numerical simulations are conducted to analyze the impact of market parameters such as appreciation rate, volatility, interest rate, elasticity parameter, tax, and investor's wealth on the optimal portfolio distribution. The results indicate that optimal investment strategies vary significantly based on market conditions and investor preferences. Overall, this study provides valuable insights into the dynamic nature of financial markets and offers practical guidance for portfolio optimization and risk management strategies.

### Downloads

## References

Adedoyin, I. L, & Babajide, A. A. (2019). Estimation of parameters of MCEV model using implicit and explicit methods. International Journal of Mathematical, Engineering and Management Sciences, 4(3), pp. 62-70.

Adedoyin, I. L, & Babajide, A. A. (2019). Option pricing and Greeks under MCEV model with stochastic volatility. Journal of King Saud University - Science, 31(3), pp. 572-576.

Adedoyin, I. L, & Babajide, A. A. (2020). Portfolio optimization under MCEV model with stochastic volatility. Applied Mathematics and Computation, 377, pp. 125-137.

Adeleke, W.A., Akinwande, N.I. & Adeyemo, K.A. (2019). Option pricing under the constant elasticity of variance model: evidence from the Nigerian stock market. Investment Management and Financial Innovations, 16(3), pp. 103-112.

Agbam, S. A. & Azubuike, J. T., (2021). Stochastic forecasting of stock prices in Nigeria: Application of geometric Brownian motion model. International Journal of finance,

Akpanibah, E. E., & Catherine, O. D. (2023). Mathematical Analysis of Stock Prices Prediction in a Financial Market Using Geometric Brownian Motion Model. International Journal of Mathematical and Computational Sciences, 17(7), pp. 78-84.

Akpanibah E. E. & Ini U. O. (2020). Portfolio strategy for an investor with logarithm utility and stochastic interest rate under constant elasticity of variance model, Journal of the Nigerian Society of Physical Sciences, 2 (3), pp. 186-196.

Akpanibah E. E. &Ini U. O. (2021). An investor's investment plan with stochastic interest rate under the CEV model and the Ornstein-Uhlenbeck process, Journal of the Nigerian Society of Physical Sciences, 3 (3), pp. 186-196.

Akpanibah, E. E. &Ogheneoro, O. (2018). Optimal Portfolio Selection in a DC Pension with Multiple Contributors and the Impact of Stochastic Additional Voluntary Contribution on the Optimal Investment Strategy, International Journal of Mathematical and Computational sciences: 12, pp. 14 -19.

Akpanibah, E. E., Osu, B. O. &Ihedioha, S. A. (2020). On the optimal asset allocation strategy for a defined contribution pension system with refund clause of premium with predetermined interest under Heston’s volatility model. Journal of. Nonlinear Science and. Application. 13(1), pp. 53–64.

Akpanibah, E. E. & Samaila, S. K. (2017). Stochastic strategies for optimal investment in a defined contribution (DC) pension fund. International Journal of Applied Science and Mathematical Theory, 3, 3, pp. 48-55.

Akpanibah E. E & Samaila S. K. (2019). Optimal Investment Policy in a Pension Fund System with Return Clause and Multiple Assets under Volatility Risks. General Letters in Mathematics. 7(1), pp. 1 – 12.

Alina C., Adrian L., & George P. (2018). The Heston Model: A Review and its Applications. Numerical Methods for Partial Differential Equations, 34(3), pp. 918-942.

Aliyu, M. S. (2018). Pricing European Options with the Heston Model and the Black-Scholes Model: A Comparative Study. Journal of Mathematics, 2018, pp. 1-6.

Amadi, U. C., Gbarayorks, L. D. &Inamete, E. N.. H. (2022). Modelling an investment Portfolio with Mandatory and Voluntary contributions under MCEV model. International Journal of Mathematical and Mathematical Sciences 16(12), pp. 120-125.

Aremo, B., Adepoju, A.A. & Oladejo, B.A. (2019). Empirical analysis of the constant elasticity of variance (CEV) model: evidence from the Nigerian stock exchange. Journal of Finance and Investment Analysis, 8(1), pp. 1-12.

Ballotta, L. Hammouda, C. B. & Monteil, A. (2019). Pricing European options under the CEV model with stochastic volatility and interest rate: a Fourier spectral method. Applied Mathematics and Computation.

Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, pp. 637-654

Chan F & McAleer, M. (2020). Pricing European options under the CEV model with stochastic volatility and interest rate using Fourier transform methods. International Journal of Computational Economics and Econometrics

Chen, Z., Duan, H. & Zhang, Q. (2018). Heston volatility model with stochastic interest rate: A closed-form solution. Journal of Applied Mathematics, 2018, pp. 1-10.

Cox, J. (1975). Notes on option pricing I: Constant elasticity of variance diffusions. Unpublished note, Stanford University, Graduate School of Business.

Cui, Z. & Xu, W. (2019). Pricing European options under the CEV model with stochastic volatility and interest rate. Applied Mathematics and Computation.

Elliott , R. J. &Wagalath,L (2018). Option pricing under CEV model with stochastic volatility and interest rate. Journal of Derivatives and Quantitative Studies.

Gao J., (2009) . Optimal portfolios for DC pension plan under a CEV model. Insurance Mathematics and Economics, 44, 3, pp. 479-490.

Hamzah, S. R., Halul, H., Jeng, A., &Sha’ari, U. A. S. (2021). Forecasting Nestle Stock Price by using Brownian Motion Model during Pandemic Covid-19. Malaysian Journal of Science Health & Technology, 7(2), pp. 58-64.

Hu, W. & Huang, C. (2020). Pricing European options under the CEV model with stochastic volatility and interest rate using the finite element method. Journal of Computational and Applied Mathematics

Ini, U.O. and, Akpanibah, E. E. (2021). "Return of contribution clause in a DC plan under MCEV model" Abacus (Mathematics Science Series) 48(2), pp. 76-89.

Jianjun M, Hui H. &Xiaolei S. (2018). Option pricing and risk measurement under modified constant elasticity of variance model with stochastic interest rate. Journal of Applied Mathematics, 2018, pp. 1-16.

Jonsson M, Sircar R. (2002). Optimal investment problems and volatility homogenization approximations. Modern Methods in Scientific Computing and Applications NATO Science Series II, Springer, Germany, 75, pp. 255-281.

Kim Y. C. & Lee, J. W. (2019). Pricing European options under the Heston model with transaction costs and liquidity effects. Journal of Applied Mathematics, 2019, 1-15.

Li, D., Rong, X. and Zhao, H. (2013). Optimal investment problem with taxes, dividends and transaction costs under the constant elasticity of variance model, Transaction on Mathematics:12,pp. 243-255..

Li, S. & Lee, Y. (2019). Pricing European options under the CEV model with stochastic volatility and interest rate using finite difference method. Journal of Computational Analysis and Applications

Li, S. & Zhou, X. (2018). Pricing European options under the CEV model with stochastic volatility and interest rate, Applied Mathematics and Computation.

Li, Z & Cheng, T. (2020). Pricing European options under the CEV model with stochastic volatility and interest rate using the Legendre wavelet method. Journal of Computational and Applied Mathematics.

Liu, J., Yanxia S., &Yongjin W. (2020). Pricing options under the modified constant elasticity of variance model with stochastic interest rate. Journal of Risk and Financial Management, 13(8), pp. 159-168.

Liu, M. & Liu, H. (2020). Pricing European options under the CEV model with stochastic volatility and interest rate using the radial basis function method. Journal of Computational and Applied Mathematics.

Madalena, G. Sandra M, & Rui M. (2019). Option pricing under MCEV model with stochastic interest rates. Journal of Computational and Applied Mathematics, 347, pp. 190-201.

Merton, R.C. (1976). Option Pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3, pp. 125–144

Muhamad F. L. &, Ani, S. (2022). Forecasting The Crude Oil Price in Malaysia Using Geometric Brownian Motion. 9, pp. 185-194

Ogunjo, S.T. &Adegbaju, A.A. (2020). Stochastic volatility modeling of stock returns using the constant elasticity of variance (CEV) model: evidence from the Nigerian stock exchange, Journal of Applied Economic Sciences, 15(2), pp. 183-194

Olorunfemi, J. A. A &Oyewunmi, O. O. (2018). An Option Pricing Model under MCEV Process. Journal of Mathematics Research, 10(5), pp. 73-88.

Osu,B.O., Akpanibah,E.E, &Olunkwa, O.(2018). Mean-Variance Optimization of portfolios with returns of premium clauses in DC pension plan with multiple contributors under constant elasticity of variance model. International journal of mathematical and computational, 12(5), pp. 85-90

Sheng, D. &Rong, X. (2014). Optimal time consistent investment strategy for a DC pension with the return of premiums clauses and annuity contracts, Hindawi Publishing Corporation2014, pp. 1-13, http://dx.doi.org/10.1155/2014/862694

Song X. & Liang, H. (2021). Pricing European options under the CEV model with stochastic volatility and interest rate using the meshless method. Journal of Computational Finance

Tikhomirov, V & Nikitina, M V. (2019). Pricing of European options under CEV model with stochastic volatility. International Journal of Mathematical Education in Science and Technology.

Tiwari, S. K., Kumar, A. & Ahmad, S. A. (2019). Option pricing under Heston's stochastic volatility model with time dependent parameters. Journal of Computational and Applied Mathematics, 347, pp. 47-61.

Wang X & Meng, X. (2019). Pricing European options under the CEV model with stochastic volatility and interest rate using the fractional step method. Journal of Computational and Applied Mathematics

Wang, X & Zhu, Z. (2018). The constant elasticity of variance model with stochastic interest rate and jumps for pricing European options. Journal of Applied Mathematics and Computing

Yang, H & Yi Zhang (2019). Pricing options under the CEV model with stochastic volatility and interest rate, Journal of Computational and Applied Mathematics.

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2024 Journal and Author

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.