On Merton's Optimal Portfolio Problem with Sporadic Bankruptcy for Isoelastic Utility

Abstract

We consider a stock that follows a geometric Brownian motion (GBM) and a riskless asset continuously compounded at a constant rate. We assume that the stock can go bankrupt, i.e., lose all of its value, at some exogenous random time (independent of the stock price) modeled as the first arrival time of a homogeneous Poisson process. For this setup, we study Merton's optimal portfolio problem consisting in maximizing the expected isoelastic utility of the total wealth at a given finite maturity time. We obtain an analytical solution using coupled Hamilton-Jacobi-Bellman (HJB) equations. The optimal strategy bans borrowing and never allocates more wealth into the stock than the classical Merton ratio recommends. For non-logarithmic isoelastic utilities, the optimal weights are non-myopic. This is an example where a realistic problem, being merely a slight modification of the usual GBM model, leads to non-myopic weights. For logarithmic utility, we additionally present an alternative derivation using a stochastic integral and verify that the weights obtained are identical to our first approach. We also present an example for our strategy applied to a stock with non-zero bankruptcy probability.