Question

Consider the optimal investment problem with a strictly concave utility U. Suppose that pu+qd≤(1+r) in the binomial model, and suppose that short-selling the stock is prohibited. Then the problem is to maximize E[U(X N )] over all strategies Δ 0 ,…,Δ N−1 satisfying Δ n ≥0 for all n=0,…,N−1, subject to the wealth equation X n+1 =Δ n S n+1 +(X n −Δ n S n )(1+r),n=0,…,N−1. Show that the optimal solution for this problem is Δ 0∗ =…=Δ N−1* =0. Hint: Start by showing that under the above assumptions, the discounted value process of any self-financing trading strategy is a supermartingale under the physical probability P

Answer 1

The optimal investment strategy in a binomial model with constraints and strictly concave utility function is to not invest at all, as shown by the supermartingale property of the discounted value process under these conditions.

Step-by-step explanation:

The student is considering an optimal investment problem within the context of a binomial model under certain constraints including no short-selling and a strictly concave utility function. To maximize their expected utility, they are looking at investment strategies subject to wealth equations. Given that the discounted value process of any self-financing strategy is a supermartingale under the given conditions, the optimal solution would be to not invest in any stocks (i.e., Δ0* = ... = ΔN-1* = 0). The key to understanding this problem lies in the properties of a supermartingale and the assumption that pu+qd ≤ (1+r), which indicate that the expected increase in price of the stock is not enough to outperform the risk-free rate, r.

An intuitive way of understanding this maximization problem involves comparing the marginal utility gained from one good to the marginal utility lost from another, whereby the optimal choice is reached when the ratio of marginal utility to the price of both goods is equal. This concept, while not a direct answer to the student's question, helps illustrate the broader principle at play: that decisions are made to balance the incremental gains and losses of utility.