For example, we may assume that prepay decisions are based on a "remaining-maturity" rate rather than on a single tenor.

Since the majority of commonly used practical models simulate prepay speeds as the sequence of European options on the MBS current market rate, volatility selection effectively "freezes" the value of a prepay option. Perhaps, the most important point that many practitioners miss is that model selection should be studied after an option volatility matrix is matched. This leaves much less freedom to deviate from the value given by a single-factor model.

What Makes the Difference?
A) Asymmetric response to curve's twist. Note that so far we bound the prepay speed to a single rate. What if the prepay option is triggered by the difference between two points of the curve? If the dependence is linear, an MBS would lose and gain the same amount from steepening and flattening, thereby causing no need for a prepay option overhaul. It is only when the pay-off depends on the curve shape asymmetrically that a curve's random twist changes the value.

The Curve-At-Origination (CATO) prepay effect can serve a good example. If the yield curve steepens from origination, homeowners who originally elected a fixed-rate mortgage may review their choice and switch to an ARM. If the curve flattens, they will not change their selection. Hence, CATO causes an asymmetric prepayment signal in response to random twist.

Another example of asymmetric pricing can be found in floater and inverse floater IO tranches. Suppose that x denotes the random twist factor; without loss of generality, we can center it on zero by interpreting x as the deviation of the curve's slope from the historical average. Suppose further that the positive value of x corresponds to steepening. The value of a 1% fixed-rate IO will increase with x due to slower prepayments; let us write it down as a + bx (where b > 0), ignoring higher-order terms. The actual floating coupon will change with x as c + dx where d is positive for an inverse floater and negative for a direct floater. Hence, the total value of an IO will be (a + bx)(c + dx). Let us take mathematical expectation of this expression; keeping in mind that the expectation of x is zero, we ultimately get
ac + bds². Therefore, the slope's volatility (s) boosts the value of an inverse IO, which pays a higher rate longer and lower rate shorter: vice versa, it deflates the value of a direct IO floater.

Once the effect of using two-factor model is proven for IO floaters, it can be extrapolated to some other CMO tranches having embedded IO floating components. >>>