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Interpreting Swaption Volatility Indices in LMM Calibration

Article Quant Q&A · Author: lkjldfkjhljk
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Summary

The note resolves a notation misunderstanding in a discussion of calibrating the Libor Market Model to swaptions. The indexed quantity V(a,b) refers to a swaption with option expiry at tenor date T(a) and an underlying swap running to T(b). The index zero does not mean a calendar expiry at time zero: T(0) is the date of the first Libor tenor, so V(0,1) represents the nearest available market point under the convention described.

With that interpretation, V(0,2) is the volatility for an option expiring at the first tenor date and referencing a swap of two-year length, as given in the source answer. The corrected reading also explains the triangular arrangement of entries in the cited table. This is a clarification of index conventions in a particular interest-rate modeling context; it does not describe the broader calibration procedure or address alternative tenor definitions.

Key ideas

  • In the cited notation, V(a,b) indexes option expiry at T(a) and the underlying swap end at T(b).
  • T(0) denotes the first Libor tenor date rather than calendar time zero.
  • V(0,1) therefore describes the nearest available swaption point under that convention.
  • Reading the indices this way clarifies the maturity and tenor layout of the calibration table.

Tags

Fixed incomeOptionsVolatilityDerivatives pricing
Full text
# LMM. Calibration to swaptions by Brigo and Morini. Volatility of swaption that matures at T=0


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












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 by lkjldfkjhljk (score 2)

https://quant.stackexchange.com/a/15485

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.