LMM. Calibration to swaptions by Brigo and Morini. Volatility of swaption that matures at T=0

Question

I'm reading Brigo D., Mercurio F. Interest Rate Models - Theory and Practice (Springer, 2006)(ISBN 3540221492) and also a source article on LMM cascade calibration to swaptions by Brigo and Morini.

I completely confused with notation. From what I understand it follows that they reference to volatility of swaptions that mature at time T=0. Either I miss something basic or there are no such swaptions.

Here are more details.

In the article in section 3 they introduce notation for black's swaption volatility $V_{a,b}$ - this is vol of swaptions with maturity at $T_a$ and underlying swap length $T_b - T_a$. But in next section they write:

where formula 8 is how to calculate swaption vol from cap vols:

My main question is - do $V_{0,1}$, $V_{0,2}$ have any meaning under these definitions (volatility of swaps that mature at T=0 with length 1 and 2 years)?

Also in their book on page 323 in section 7.4 they provide table that completely confuses me.

How do indices of $V$'s correlate with these maturities and lenthts? From my understanding table should look like this:

$V_{1,2} V_{1,3} V_{1,4}$

$V_{2,3} V_{2,4}$

$V_{3,4}$

Answer 1

Thanks to my research leader, I found what I missed. $V_{0,1}$ is vol of swaption that matures at $T_0$ which is not 0 (as I thought), rather it is maturity of the first libor. So $V_{0,1}$ is the closest available point on market. And now this is all clear with table on page 323 in section 7.4. $V_{0,2}$ is realy vol of swaption that matures at $T_0$=1y and has length = 2y.