The Impact of a Mean-Reverting Stochastic Differential Equation on Optimal Portfolio Allocation

Abstract

<p>We consider the dilemma an investor contemplates when faced with the decision to allocate proportions of initial wealth within a multi-"risky" asset framework in order to maximize terminal wealth. It is assumed that Geometric Brownian Motion generates the "risky" asset price paths for this problem formulation. We consider the particular setting where the "risky" assets exhibit dependence on an Arithmetic Ornstein-Uhlenbeck stochastic differential equation (AOU) in the form of correlation and an embedded adjustment to the stochastic drift. We exemplify the motivation for this problem formulation by highlighting the empirical dependence that has occurred between the daily returns 1 of the TSX Composite Index (S1), the daily returns of the Scotia Capital Overall Bond Index (b1) and the Yield Ratio^2 (Rt). We then derive the optimal portfolio control for this investor, using the Hamilton-Jacobi-Bellman equation method. We then construct optimal portfolios using Mean-Variance-Optimization (MVO) and compare terminal wealth for two investors using Monte-Carlo Simulation. Investor A is incognizant of the above dependence whereas Investor B is cognizant. We vary the above dependence parameters and assess the overall impact on the probability distribution of terminal wealth</p>