If we associate the mortgage market rate with the 7-year index, then the short end of the curve will affect its forward value in 3 years, albeit as a 3:7 ratio.

When do we need a two-factor model?
Knowing, now, that the curve shape carries a pronounced valuation effect, one should wonder whether the traditional single-factor term structure modeling is a legitimate approach at all. Paradoxically enough, it still should be able to do the job it is meant for - valuation, not measuring risk.

Valuation aspect. Even some seasoned experts fall into the misperception that imperfect correlation between rates may significantly affect the prepay option's value. Their leading argument has been the fact that short rates are formally employed by valuation schemes for discounting. This is an artifact that is completely quantified by the observed market for European options. Namely, a simple consideration reminiscent of the Black-Scholes model proves that fixed-rate mortgages will be valued similarly, provided that:

   (A) The prepay option is modeled as a series European option, and
   (B) The entire volatility term structure is matched for the mortgage rate(s) that drives refinancing.

Indeed, correlations between rates don't enter the analysis of European options or any of their portfolios. In contrast, American options depend on joint distribution of many rates. Therefore, if a prepay option were modeled as an American option, the above conclusion would become theoretically invalid. ARMs, unlike fixed-rate mortgages, may be affected by inter-rate correlations since their index is typically not the mortgage rate and may have an imperfect correlation with it.

For many valuation exercises with fixed-rate mortgages, a two-factor model will not add or subtract considerable value extracted by a one-factor model - once volatility term structures are made identical (taken from the market) for both models. This is a good time to remind readers that AD&Co's models satisfy both conditions (A) and (B). We model the prepay option as a sequence of European payoffs (inefficiently exercised), whereas our suite of Term Structure Models contains three single-factor term structure models that can be tied to an arbitrary family of swap and swaptions.

Risk management aspect. Without a doubt, this is the most important application of the multi-factor yield curve modeling. Can one hedge a 5-year swap with a 1-year swap? A single-factor model suggests, "Yes, just take enough notional". Any practitioner knows that this method is impractical, as revealed by a two-factor consideration (strict or intuitive). >>>