Optimal acquisition of a partially hedgeable house

We characterize the optimal time to purchase a house by a risk-averse investor who has access to complete financial markets and whose objective is to maximize expected utility from wealth at some fixed horizon. The house purchase is financially attractive (due to high expected returns or tax deductions, for example), which provides an incentive to buy as soon as possible; however, its value is only partially correlated with financial markets and, therefore, its price risk cannot be perfectly hedged, which provides an incentive to delay purchase. Using martingale duality method, we fully characterize the optimal terminal wealth in the case of CARA utility, so that we can generate simple numerical solutions for different parameter values, and study the trade-off between the two conflicting incentives. We derive more explicit results when the price follows a Gaussian process. In particular, we identify conditions under which an optimal stopping time is deterministic.