# Longstaff and Schwartz example in their paper

I was looking at the well known Longstaff and Schwartz paper "Valuing American Options by Simulation: A Simple Least-Squares Approach". There are a couple of examples where they applied the early exercise algorithm. Particularly, section 5 presents the valuation of a Cancelable Index Amortizing Swap.

For that case, the dynamics of the notional balance of the swap is needed and is given by:

$$dI(t) = -f(\text{CMS}_{10}(t)),$$

with $$\text{CMS}_{10}$$ a Constant Maturity Swap rate and $$f$$ a known deterministic function. So far, so good. The problem comes with the $$\text{CMS}_{10}$$ definition:

$$\text{CMS}_{10}(t) = 2 \frac{1 - P(t, 10)}{\sum_{i=1}^{20}P(t, i/2)}$$

with $$P(t, T)$$ a zero coupon bond that comes from a two factor short rate model of Vasicek type. Note that the paper uses a different notation for those zero coupon bonds, i.e. $$D(X, Y, T)$$ which, I believe, translates to

$$D(X, Y, T) = D(X(t), Y(t), T)$$

with $$X$$ and $$Y$$ the factor processes of the short rate model. They present the analytical solution of the zero coupon bond for this particular case. However, that expression is only applicable for $$t \leq T$$. The problem arises when computing $$\text{CMS}_{10}(t)$$ for $$t > 0.5$$ for example, since the summation in the denominator requires computing:

$$P(t, 0.5)$$

with $$t > 0.5$$, which does not make sense.

Is there something that I am missing? Should the summation start index be a function of time, such as:

$$\sum_{i=q(t)}^{10}P(t, i/2)$$

Thank you so much in advance!