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.
Highlights
The problem of utility maximization is of great importance
This paper intends to find the optimal investment strategy for an investor who participates in a financial market, in which the interest rate of the risk-free asset is stochastic and governed by the Ornstein-Uhlenbeck model and the risky asset assumed to follow the constant elasticity of variance (CEV) model and look into the variation that would occur when the Brownian motions correlate and when the Brownian motions do not correlate as well as find what happens when transaction cost is charged
Njoku and Osu [15] worked on the optimal pension wealth investment strategy during the decumulation phase, in a defined contribution (DC) pension scheme where the pension plan member was allowed to invest in a risk free and a risky asset, under the constant elasticity of variance (CEV) model
Summary
The problem of utility maximization is of great importance. This has lead to many researchers in financial mathematics to greatly focus on solving optimal investment problem of utility maximization. Three different assets namely risk free asset (cash), zero coupon bonds and the risky asset (stock) were considered They obtained the optimal investment strategies for the three investments using Legendre transformation method and dual theory where exponential utility function for two of the cases was involved and found result that showed that the strategies for the respective investments when there was no extra contribution could be used when the extra contribution rate was constant but could not be used when it was stochastic. Njoku and Osu [15] worked on the optimal pension wealth investment strategy during the decumulation phase, in a defined contribution (DC) pension scheme where the pension plan member was allowed to invest in a risk free and a risky asset, under the constant elasticity of variance (CEV) model. It can be seen that most of the works reviewed did not discuss transaction cost our choice of this topic
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