购买和出售房屋的最佳时间
Optimal times to buy and sell a home
摘要：
我们考虑一个金融市场，其中无风险利率被建模为马尔可夫扩散。我们假设房屋价格是由一个有代表性的购房者决定的，他只能支付单位时间内固定的现金流。现金流只是代表性购房者工资的一小部分，而工资的增长率与无风险利率成正比。因此，从长远来看，更高的利率会导致房价更快地增长。代表性的购房者通过抵押贷款为购房提供资金。购房者支付的抵押贷款利率在购买时是固定的，等于无风险利率加上一个正常数。由于购房者只能支付单位时间内固定的现金流，较高的抵押贷款利率限制了购房者可以获得的贷款规模。因此，高利率的短期影响是降低房屋的价值。在这种情况下，我们考虑一个投资者谁希望购买，然后出售一个家，以最大限度地提高他的贴现预期利润。这导致了一个嵌套的最优停时问题。我们使用一个非负凹优方法来获得投资者的最优买卖策略。此外，我们提供了一个详细的分析和数值研究的情况下，无风险利率的Cox-Ingersoll-Ross（CIR）过程建模。我们还研究，在CIR利率的情况下，预期的时间，投资者等待之前购买，然后出售房屋时，遵循最优策略。
英文摘要：
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.
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