Forum Gurus,I am stuck with below question and not able to solve it. This question is not a Home Work, it's from previous year final term. So I am not looking for exact solution. Can someone be kind enough to provide some pointers, how to approach this? and where I can read more about this and practice some questions like this? I want to learn this. Suppose you own a vacant plot of land in a retirement community at the beach.You can either leave the land vacant or build a house to rent to other people. The community has strict zoning laws that permit you to build only one kind of house. As a result, your choice is always simple : either leave the land vacant or build on it. The value of house follows geometric Brownian motion, rising at a rate of 2% per year on average but also fluctuating with market conditions with a volatility of 0.2. The value of a new house today is V= $130,000. The rent that can be charged on a house is proportional to the market value of the house and equals 7% of the value. A new house costs $100,000 to build. The risk-free interest rate is 0.01.Explain whether you should build a house now or wait?This is some kind of option and I can calculate the price of this option using BS where I will consider S=130,000 and K = 100,000. But this rent income , how to handle this? Can I assume like this R = F(V) . where R -s rent Income and V=Value of house.and I can use Ito's lemma to solve dR?

Assume the rent income is like a Dividend payment(r-d+.5*sig^2)(T-t) I would just use basic black-scholes options pricing formula

Since there is no fixed maturity date, you cannot use the Black-Scholes formula. But the lack of maturity date along with time homogeneous parameters turn the usual B-S PDE into an ODE. The exercise boundary will be flat, so there is a single critical level such that it is optimal to build the house if the market price of the house exceeds this level. This is all with the proviso that the usual valuation framework applies, whether or not you can actually hedge the house price risk.