Discounting a Mortgage Prepayment Claim with Separate Reference and Short Rates

# Price of a prepayment-based claim


# Price of a prepayment-based claim












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 $\tau$.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/32946

What looks odd to me is that r(t) is used in two places. First, as the mortgage reference rate , and second as the short rate ( for discounting). I would have expected these rates to differ. In the US, the reference mortgage rate is more of a long dated rate, for example. So the formula should refer to r1(t) and r2(t) in my opinion.