Several methods have been described and implemented to employ backward valuation for ARMs. For example, ARMVAL, the joint Kalotay - AD&Co product, expanded a valuation grid to the coupon rate, albeit a very sparse one, see Howard (2001). Independently, Levin (2001) describes several correcting techniques, including using Brownian bridges; they are all ad-hoc and imprecise, but require no grid expansion and add little computational time.

Interestingly enough, a straightforward use of the backward induction (for decomposed collateral) leads to high accuracy for most ARMs despite their material reset dependence. Here are several plausible explanations for the luck:

• A linear dependence of cashflow on the last reset does not distort an ARM’s value. The dependence has to be convex (asymmetric) to cause trouble, which is often the case with prepayable loans. In practice, past its first reset, an ARM’s prepayment speed becomes weakly dependent on interest rates.

• The existence of a fixed-rate period reduces an ARM’s path-dependence. In practice, the embedded option is greater in hybrid ARMs having longer first roll.

• Frequent resets reduce path-dependence (e.g. a monthly floater is not path-dependent). In practice, most ARMs reset annually or more frequent.

We can easily conjure up an ARM that will be valued very inaccurately using backward induction. An example would be a bond resetting soon, and only once in its lifetime. Accurate valuation on the grid would be possible only with improvements to the straightforward method. For new actual hybrids that are often seen in dealers’ reports, we compared valuation results of the backward induction and 20,000 Monte-Carlo paths and found them to be only 1 bp apart in price (0.3-0.4 bp OAS difference). We have included the backward induction valuation method for ARMs into our OAS v7.1. It has truly empowered the risk-neutral calibrating process, as I explain later in the article.

 

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