  We consider a financial market in which the risk-free rate of interest is
modeled as a Markov diffusion. We suppose that home prices are set by a
representative home-buyer, who can afford to pay only a fixed cash-flow per
unit time for housing. The cash-flow is a fraction of the representative
home-buyer's salary, which grows at a rate that is proportional to the
risk-free rate of interest. As a result, in the long-run, higher interest rates
lead to faster growth of home prices. The representative home-buyer finances
the purchase of a home by taking out a mortgage. The mortgage rate paid by the
home-buyer is fixed at the time of purchase and equal to the risk-free rate of
interest plus a positive constant. As the home-buyer can only afford to pay a
fixed cash-flow per unit time, a higher mortgage rate limits the size of the
loan the home-buyer can take out. As a result, the short-term effect of higher
interest rates is to lower the value of homes. In this setting, we consider an
investor who wishes to buy and then sell a home in order to maximize his
discounted expected profit. This leads to a nested optimal stopping problem. We
use a nonnegative concave majorant approach to derive the investor's optimal
buying and selling strategies. Additionally, we provide a detailed analytic and
numerical study of the case in which the risk-free rate of interest is modeled
by a Cox-Ingersoll-Ross (CIR) process. We also examine, in the case of CIR
interest rates, the expected time that the investor waits before buying and
then selling a home when following the optimal strategies.
