I am reading a paper (prepayment risk and expected MBS return, Diep et all, 2017). And I am confused as to how the research derives the main pricing equation (see page 6 and 7 of the paper if you have access to it). In particular the below expression is of issue:
r is the discount rate, t is time, b is the mortgage balance (b naught is initial balance), c is the mortgage rate paid, p is the prepayment rate.
The change in balance is first defined as :
$$ db_t/dt = -pb_t $$ And the first order approximation of the price is then given as: $$ Price_o = \int_{0}^{inf} e^{-rt}(b_tc_t -db_t) dt=b_o + (c - r) \int_{0}^{inf} e^{-t(r+p)} dt= 1+ (c -r)/ (r+p) \,. $$
The first statement makes perfect sense to me, the rate of change of the MBS balance is simply the current balance times the prepayment rate per unit time. I think I understand the left hand side of the second equation. The price of a MBS is the discounted value of the mortgage payments less the change in the principal over time, but it seems like they are not including the present value of principal payments in that statement. I am not sure how they arrive with the two subsequent expressions. Mainly I am not sure where the b_0 come from and how they arrive at the new exponential equation within the integral.