Modeling the Optimal Strategy in an Incomplete Market

We examine the optimal portfolio selection problem for a single agent who receives a unhedgeable endowment. The agent wishes to optimize his/her log-utility derived from his/her terminal wealth. We do not solve this problem analytically but rigorously prove that there exists a unique optimal portfolio strategy. We present a recursive computational algorithm which produces a sequence of portfolios converging to the optimal one. We present an ”intelligent” initial portfolio which requires, numerically, about 25% fewer corrective steps in the algorithm than a random initial portfolio, and outperforms the portfolio which ignores the unhedgeable risk of the endowment.