Monday, September 25, 2017

pricing - 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 τ 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:



τ=min{t:r(t)L,0tT}


My question is: is the price of this claim at 0, P0, given by the following risk-neutral expectation?


$$P_0 = \mathbb{E}^{\mathbb{Q}}\left[N\left(\mathbb{I}_{\{\tau

My doubt is mainly related to the first discount factor, which goes from 0 to τ.




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