In part 3.2 of Lu and Neftci (2003) "Convexity Adjustments and Forward Libor Model: Case of Constant Maturity Swaps", the authors propose a new way of pricing CMS swaps, with Monte Carlo simulations. They show that the par CMS swap rate is a function of forward rates. So we just need a model of forward rate to do the Monte Carlo pricing (such as the Libor Market Model).

Specifically, the formula they find for the par CMS swap rate (equation 22 and 23 in their paper) is a function of these forward rates: $F(t_1,t_1,t_2)$, $F(t_2,t_2,t_3)$ and $F(t_3,t_3,t_4)$, with the following notation: $F(a,b,c)$ = forward rate viewed from time $a$ between time $b$ and time $c$.

These forward rates are then of the form : $F(a,a,c)$. I don't understand why they are said to be forward then. It seems to me that if a forward rate start at the date of observation, it is not a forward rate but a spot rate. Isn't it? Why would they say that then?

Many thanks for your help!


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