I am trying to determine the pricing formula for a given claim inspired in prepayment obligations backed by mortgage portfolios $-$ I believe these were popular in the eighties.
The product mechanism is the following: consider an underlying mortgage of principal $N$ which is contracted at $t=0$ and which must be reimbursed at $t=T$, from which the principal payment has been stripped from the interest payments $-$ hence we are essentially considering a zero-coupon bond paying $N$ at $T$. However, the borrower has also the option to prepay the full amount at any time $t$ between $0$ and $T$. The buyer of the product will then get the amount $N$ when the borrower decides to pay.
Now, let's define the stopping time $\tau$ as the time at which the borrower decides to prepay. For example, you could assume that the borrower will prepay and subsequently refinance its mortgage if the mortgage's reference interest rate $r(t)$ decreases below a certain level $L$. In such a case, we would have:
$$\tau = \min\{t: r(t) \leq L, \: 0\leq t\leq T\}$$
My question is: is the price of this claim at $0$, $P_0$, given by the following risk-neutral expectation?
$$P_0 = \mathbb{E}^{\mathbb{Q}}\left[N\left(\mathbb{I}_{\{\tau<T\}}e^{-\int_0^{\tau}r(t)dt}+\mathbb{I}_{\{\tau\geq T\}}e^{-\int_0^{T}r(t)dt}\right)\right]$$
My doubt is mainly related to the first discount factor, which goes from $0$ to $\tau$.
