4
$\begingroup$

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 $\tau$.

$\endgroup$
  • 2
    $\begingroup$ Looks fine to me. $\endgroup$ – Quantuple Mar 9 '17 at 23:05
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Thank you for pointing that out @dm63. The stopping time defined above is for illustrative purposes mainly as I am on the preliminary phase of this work, I am also considering other modelling choices $-$ e.g. make it dependent on a Poisson process $N(t)$ of intensity $\lambda$ such that: $\tau = \min \{t : N(t) > 0, 0 < t < T\}$. I'll be careful in separating both rates if I carry on with this approach! $\endgroup$ – Daneel Olivaw Mar 11 '17 at 10:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.