Abstracts

We study an investor maximizing the expected power utility of his terminal wealth. Some closed form solutions for such a problem have been found in the literature in a framework characterized a follows: (i) there is one state variable (typically the riskless interest rate or the risk premium) following a mean-reverting process either with constant or with linear volatility, (ii) there is one risky asset, (iii) a bond may exist.
The existence (and uniqueness) of the optimal portfolio is usually approached by establishing the existence (and uniqueness) of a viscosity solution to the Hamilton-Jacobi-Bellman equation deriving from the stochastic optimal control problem. Here, instead, we find sufficient conditions for easily checking whether there exists a closed form solution to an optimal portfolio problem when there exist both a set of (stochastic) state variables and a set of risky assets. Furthermore, we provide the algebraic form of this exact solution.
These sufficient conditions must hold on some combinations of drift and diffusion coefficients of state variables and risky assets. More precisely, we are able to demonstrate that all the exact solutions available in the literature for incomplete markets can be traced back to a framework where both state variables and risky assets follow mean-reverting processes with either constant or linear volatility. In such a framework, the value function solving the optimization problem has either a log-quadratic or a log-linear form whose exponent solves a Riccati differential equation. When the coefficients of this differential equation are constant (which is the most common case in the literature), then we are able to find a closed for solution for the optimal portfolio.
Furthermore, we derive the properties of the “linear” solution and we show that the absolute values of portfolio composition are monotonic functions of time. The direction of this monotonicity can be easily checked looking at the sign of a parameter. Finally, we present the structure of an incomplete financial market with one risky asset and two state variables which allows for a closed form solution according to our previous results. This is something new in the literature, since all the existing papers which present optimal portfolios in closed form, take into account only one state variable (at least to our knowledge).