Interpreting CMS Swap Rates in a Simulated Amortizing Swap
Summary
The document examines the Constant Maturity Swap rate used to drive the notional balance of a cancelable index amortizing swap in a Longstaff–Schwartz American-option example. It describes a two-factor Vasicek-type short-rate model and a CMS definition built from a ten-year zero-coupon bond and a sum of bond prices at half-year maturities. The question is how to interpret that sum as simulation time advances.
The author observes that, after the first half-year, some maturities in the written sum would lie before the current time, while the cited analytical zero-coupon bond expression is stated only for times at or before maturity. They ask whether the summation’s lower index should move with time. No answer or resolution is included, so the document identifies a modeling and notation issue rather than establishing a convention. Readers should verify the swap’s remaining payment dates and the original paper’s definition before implementing the rate.
Key ideas
- The example uses a CMS rate to determine notional changes in a cancelable amortizing swap.
- The CMS expression combines a ten-year discount bond with a sum of half-year maturity bond prices.
- As simulation time advances, fixed payment dates may be in the past and cannot be treated as future cash flows.
- The document raises, but does not answer, whether the CMS summation should include only remaining payment dates.
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# Longstaff and Schwartz example in their paper
# 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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.