A minimum variance result in continuous trading portfolio optimization

The problem of minimizing the variance of discounted wealth at the end of a fixed period is solved when the expectation of terminal wealth is constrained to a specified investment goal. The results are obtained in a continuous trading framework under the assumption that the funds can be exchanged between a riskless bond and a stock whose discounted price is described by a geometric Brownian motion. Transaction costs are ignored (i.e., the market is ‘frictionless’) and unlimited borrowing is permitted at the same rate as the return on the bond. Typically the optimal trading policy under the above assumptions involves a highly leveraged investment in the stock in the early stages followed by an accumulation of the bond in the later stages. Numerical results are provided as an illustration of the theory.