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!



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