Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization

Silas Abahia Ihedioha; Nanle Tanko Danat; Audu Buba

doi:doi:10.11648/j.pamj.20200903.12

| Peer-Reviewed

Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization

Silas Abahia Ihedioha, Nanle Tanko Danat, Audu Buba

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 3)

Received: 5 June 2020 Accepted: 20 June 2020 Published: 4 July 2020

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this work, we studied the optimal investment problem of an investor who had exponential utility preference and traded two assets; (1) a risky asset which price dynamics was governed by the Constant Elasticity of variance (CEV) model and (2) a risk-free asset which price system followed the Ornstein-Uhlenbeck model. We employed the maximum principle of dynamic programming to obtain the Hamilton-Jacobi-Bellman (H-J-B) equation on which the first principle and the elimination of variable dependency were applied to get the closed-form of the investor’s optimal strategies. Two scenarios where the Brownian motions correlated and where they did not correlate were investigated. Also considered were the cases of when transaction cost was involved and when transaction cost was not involved. This lead to six cases that among the results obtained was that the investor has an optimal investment strategy that requires more amount of money for investment when the Brownian motions do not correlate and there is transaction cost than when the Brownian motions correlate and there is no transaction.

Published in	Pure and Applied Mathematics Journal (Volume 9, Issue 3)
DOI	10.11648/j.pamj.20200903.12
Page(s)	55-63
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Investor, Optimal Strategy, Transaction Cost, Ornstein-Uhlenbeck Model, Constant of Elasticity of Variance (CEV) Model, Exponential Utility Maximization

References

[1]	Merton R. C. (1971). Optimum Consumption and Portfolio Rules in a Continuous-time model, Journal of Economic Theory, Vol. 3, No. 4, 373-413.
[2]	Gu A., Guo X., and Li Z. (2012). Optimal Control of excess-of-loss Reinsurance an Investment for Insurers under Constant Elasticity of variance (CEV) model, Insurance Mathematics and Economics, Vol. 51, No. 3, 674 – 684.
[3]	Liu J., Bai L., and Yiu K. C. (2012). Optimal Investment with a value-at-risk constraint. Journal of Industrial and Management Optimization [JIMO], 8 (3): 531-457
[4]	Zhao H., Rong X. (2012): Portfolio selection problem with multiple risky assets under the constant elasticity of variance model. Elsevier, vol. 50 (1), pages 179-190.
[5]	Chang H., Chang K. and Lu J. M. (2014): Portfolio selection with liability and affine interest rate in the HARA utility framework, Abstract and Applied Analysis. 2014, pp. 1-12
[6]	Ihedioha S. A, Oruh B. I. & Osu B. O. (2017). Effects of Correlation of Brownian Motions on an Investor’s Optimal Investment and Consumption Decision under Ornstein-Uhlenbeck Model, Academic Journal of Applied Mathematical Sciences, Academic Research Publishing Group, Vol 3 (6), pages 52-61, 06-2017. Handle: RePEc: arp: ajoams: 2017: 52-61.
[7]	Okonkwo, C. U., Osu B. O., Ihedioha S. A. and Chibuisi C. (2018). Optimal Investment Strategy for Defined Contribution Pension Scheme under the Heston Volatility Model, Journal of Mathematical Finance, 8, 613-622
[8]	Osu B. O., Njoku K. N. C and Basimanebotlhe O. S. (2019). Fund Management Strategies for a Defined Contribution (DC) Pension Scheme under the Default Fund Phase IV, Communications in Mathematical Finance, vol. 8, no. 1, 2019, 169-185 ISSN: 2241-195X (print), 2241- 1968 (online) Scientific Press International Limited.
[9]	Akpanibah E. E., Osu B. O., Njoku K. N. C and Eyo O. A (2017). Optimization of Wealth Investment Strategies for DCPension Fund with Stochastic Salary and Extra Contributions, International Journal of Partial Differential Equations and Applications, vol. 5. no. 1, 33-41.
[10]	Njoku K. N. C., Osu B. O., Akpanibah E. E. and Ujumadu R. N. (2017). Effect of Extra Contribution on Stochastic Optimal Investment Strategies for DC Pension with Stochastic Salary under the Affine Interest Rate Model, Journal of Mathematical Finance, 7, 821-833.
[11]	Wang L. and Chen, Z. (2018). Nash Equilibrium Strategy for a DC Pension Plan with State Dependent Risk Aversion: A Multiperiod Mean-Variance Framework. Hindawi Discrete Dynamics in Nature and Society Article ID 7581231 https://doi.org/10.1155/2018/7581231.
[12]	Osu B. O., Akpanibah E. E. and Olunkwa C. (2018). Mean-Variance Optimization of portfolios with return of premium clauses in a DC pension plan with multiple contributors under constant elasticity of variance model, Int. J. Math. Comput. Sci., 12 (2018), 85–90. 1
[13]	Osu B. O., Akpanibah E. E. and Oruh B. I. (2017). Optimal investment strategies for Deﬁned Contribution (DC) pension fund with multiple contributors via Legendre transform and dual theory, Int. J. Pure Appl. Res., 2 97–105. 1
[14]	Osu B. O., Njoku K. N. C. and Oruh B. I. (2019). On the Effect of Inflation and Impact of Hedging on Pension Wealth Generation Strategies under the Geometric Brownian motion Mode, Earthline Journal of Mathematical Sciences, Volume 1, Number 2, 2019, Pages 119-142. http://www.earthlinepublishers.com; https://doi.org/10.34198/ejms.1219.119142
[15]	Njoku K. N. C. and Osu B. O. (2019). On the Modified Optimal Investment Strategy for Annuity Contracts under the Constant Elasticity of Variance (CEV) Model, Earthline Journal of Mathematical Sciences, Volume 1, Number 1, 63-90. http://www.earthlinepublishers.com, https://doi.org/10.34198/ejms.1119.6390

Cite This Article

Plain Text BibTeX RIS

APA Style

Silas Abahia Ihedioha, Nanle Tanko Danat, Audu Buba. (2020). Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization. Pure and Applied Mathematics Journal, 9(3), 55-63. https://doi.org/10.11648/j.pamj.20200903.12

Copy | Download

ACS Style

Silas Abahia Ihedioha; Nanle Tanko Danat; Audu Buba. Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization. Pure Appl. Math. J. 2020, 9(3), 55-63. doi: 10.11648/j.pamj.20200903.12

Copy | Download

AMA Style

Silas Abahia Ihedioha, Nanle Tanko Danat, Audu Buba. Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization. Pure Appl Math J. 2020;9(3):55-63. doi: 10.11648/j.pamj.20200903.12

Copy | Download

@article{10.11648/j.pamj.20200903.12,
  author = {Silas Abahia Ihedioha and Nanle Tanko Danat and Audu Buba},
  title = {Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {3},
  pages = {55-63},
  doi = {10.11648/j.pamj.20200903.12},
  url = {https://doi.org/10.11648/j.pamj.20200903.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200903.12},
  abstract = {In this work, we studied the optimal investment problem of an investor who had exponential utility preference and traded two assets; (1) a risky asset which price dynamics was governed by the Constant Elasticity of variance (CEV) model and (2) a risk-free asset which price system followed the Ornstein-Uhlenbeck model. We employed the maximum principle of dynamic programming to obtain the Hamilton-Jacobi-Bellman (H-J-B) equation on which the first principle and the elimination of variable dependency were applied to get the closed-form of the investor’s optimal strategies. Two scenarios where the Brownian motions correlated and where they did not correlate were investigated. Also considered were the cases of when transaction cost was involved and when transaction cost was not involved. This lead to six cases that among the results obtained was that the investor has an optimal investment strategy that requires more amount of money for investment when the Brownian motions do not correlate and there is transaction cost than when the Brownian motions correlate and there is no transaction.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization
AU  - Silas Abahia Ihedioha
AU  - Nanle Tanko Danat
AU  - Audu Buba
Y1  - 2020/07/04
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200903.12
DO  - 10.11648/j.pamj.20200903.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 55
EP  - 63
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200903.12
AB  - In this work, we studied the optimal investment problem of an investor who had exponential utility preference and traded two assets; (1) a risky asset which price dynamics was governed by the Constant Elasticity of variance (CEV) model and (2) a risk-free asset which price system followed the Ornstein-Uhlenbeck model. We employed the maximum principle of dynamic programming to obtain the Hamilton-Jacobi-Bellman (H-J-B) equation on which the first principle and the elimination of variable dependency were applied to get the closed-form of the investor’s optimal strategies. Two scenarios where the Brownian motions correlated and where they did not correlate were investigated. Also considered were the cases of when transaction cost was involved and when transaction cost was not involved. This lead to six cases that among the results obtained was that the investor has an optimal investment strategy that requires more amount of money for investment when the Brownian motions do not correlate and there is transaction cost than when the Brownian motions correlate and there is no transaction.
VL  - 9
IS  - 3
ER  -

Copy | Download

Author Information

Silas Abahia Ihedioha

Department of Mathematics, Plateau State University Bokkos, Jos, Plateau State, Nigeria
Nanle Tanko Danat

Department of Mathematics, Plateau State University Bokkos, Jos, Plateau State, Nigeria
Audu Buba

Department of Actuarial Science, University of Jos, Jos, Plateau State, Nigeria

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Silas Abahia Ihedioha, Nanle Tanko Danat, Audu Buba. (2020). Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization. Pure and Applied Mathematics Journal, 9(3), 55-63. https://doi.org/10.11648/j.pamj.20200903.12

Copy | Download

ACS Style

Silas Abahia Ihedioha; Nanle Tanko Danat; Audu Buba. Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization. Pure Appl. Math. J. 2020, 9(3), 55-63. doi: 10.11648/j.pamj.20200903.12

Copy | Download

AMA Style

Silas Abahia Ihedioha, Nanle Tanko Danat, Audu Buba. Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization. Pure Appl Math J. 2020;9(3):55-63. doi: 10.11648/j.pamj.20200903.12

Copy | Download

@article{10.11648/j.pamj.20200903.12,
  author = {Silas Abahia Ihedioha and Nanle Tanko Danat and Audu Buba},
  title = {Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {3},
  pages = {55-63},
  doi = {10.11648/j.pamj.20200903.12},
  url = {https://doi.org/10.11648/j.pamj.20200903.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200903.12},
  abstract = {In this work, we studied the optimal investment problem of an investor who had exponential utility preference and traded two assets; (1) a risky asset which price dynamics was governed by the Constant Elasticity of variance (CEV) model and (2) a risk-free asset which price system followed the Ornstein-Uhlenbeck model. We employed the maximum principle of dynamic programming to obtain the Hamilton-Jacobi-Bellman (H-J-B) equation on which the first principle and the elimination of variable dependency were applied to get the closed-form of the investor’s optimal strategies. Two scenarios where the Brownian motions correlated and where they did not correlate were investigated. Also considered were the cases of when transaction cost was involved and when transaction cost was not involved. This lead to six cases that among the results obtained was that the investor has an optimal investment strategy that requires more amount of money for investment when the Brownian motions do not correlate and there is transaction cost than when the Brownian motions correlate and there is no transaction.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Investor’s Optimal Strategy with and Without Transaction Cost Under Ornstein-Uhlenbeck and Constant Elasticity of Variance (CEV) Models Via Exponential Utility Maximization
AU  - Silas Abahia Ihedioha
AU  - Nanle Tanko Danat
AU  - Audu Buba
Y1  - 2020/07/04
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200903.12
DO  - 10.11648/j.pamj.20200903.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 55
EP  - 63
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200903.12
AB  - In this work, we studied the optimal investment problem of an investor who had exponential utility preference and traded two assets; (1) a risky asset which price dynamics was governed by the Constant Elasticity of variance (CEV) model and (2) a risk-free asset which price system followed the Ornstein-Uhlenbeck model. We employed the maximum principle of dynamic programming to obtain the Hamilton-Jacobi-Bellman (H-J-B) equation on which the first principle and the elimination of variable dependency were applied to get the closed-form of the investor’s optimal strategies. Two scenarios where the Brownian motions correlated and where they did not correlate were investigated. Also considered were the cases of when transaction cost was involved and when transaction cost was not involved. This lead to six cases that among the results obtained was that the investor has an optimal investment strategy that requires more amount of money for investment when the Brownian motions do not correlate and there is transaction cost than when the Brownian motions correlate and there is no transaction.
VL  - 9
IS  - 3
ER  -

Copy | Download