Abstract
This work presented and solved the problem of portfolio optimization within the context of continuous-time stochastic model of financial variables. It has considered an investment problem of two assets, namely, risk-free assets and risky assets. The evolution of the risk-free asset is described deterministically while the dynamics of the risky asset is described by the geometric mean reversion (GMR) model. The controlled wealth stochastic differential Equation (SDE) and the optimal portfolio problem were successfully formulated and solved with the help of the theory of stochastic control technique where the dynamic programming principle (DPP) and the HJB theory were used. Two utility functions which are members of hyperbolic absolute risk aversion (HARA) family have been employed, and these are power utility and exponential utility. In both cases, the optimal control has explicit form and is wealth dependent Linearization of the logarithmic term in the portfolio problem was necessary for simplification of the problem.
Highlights
The concept of portfolio optimization is of fundamental importance in financial investment theory and practice
In similar situations of portfolio returns considered as uncertain variables, [7] proposed a semi variance technique to be used for handling the diversified portfolio selection problem
We focus on optimal strategies in the sense that portfolio depends on asset prices and no borrowing and short-selling
Summary
The concept of portfolio optimization is of fundamental importance in financial investment theory and practice. Merton [3] [4] used optimal stochastic control technique in continuous time to explicitly determine a closed-form solution of the optimal portfolio problem in the financial investment market comprising risk-less asset and a stock as investment alternatives [5]. The general abstraction behind this problem is the selection of the best strategies that could provide optimal results at times an investor is faced with huge varieties of investment decisions about his wealth. The study by [12] investigated the portfolio selection consisting of instruments whose logarithms are mean-reverting They assumed that portfolios are constant and short-selling and borrowing are allowed, and the optimal strategies were found in the sense of time-independent portfolios, i.e. portfolios which do not depend on asset prices, which is not the case in real life situation.
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