Why Forward-Rate and Swap-Rate LMM Drifts Differ
Summary
The document asks how drift specifications in the Libor Market Model (LMM) relate when one treatment describes forward rates and another describes par swap rates. It also asks whether a volatility term marked with an asterisk represents a Black volatility. The answer stresses that forward rates and par swap rates are different modeled quantities, so their drift expressions should not be expected to match directly. It also notes that one presentation sets aside correlations among Wiener processes for simplicity.
The answer cites a statement that such correlations do not affect swaption prices in the simplified discussion, while they can matter for pricing more complex products. It suggests that the starred volatility notation may be a typo and that nearby notation likely refers to a forward-rate Black volatility. These are brief interpretive comments rather than a derivation: the source does not reconcile the equations algebraically, establish the notation conclusively, or specify the limits of the pricing claim. Readers should treat the notation explanation as tentative.
Key ideas
- Forward rates and par swap rates are distinct modeled quantities, so their drift terms need not coincide.
- The cited treatment omits correlations among Brownian drivers to simplify the presentation.
- The answer says such correlations may matter for more complex products even if excluded in a swaption discussion.
- The interpretation of the starred volatility as a typo is tentative and is not derived in the document.
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# Reconciling different specifications of drifts in the LMM
# Reconciling different specifications of drifts in the LMM
I've been going through the book "Fixed Income Securities" by Bruce Tuckman which gives the following definitions of the drift terms (after showing it for a specific example with 3 forward rate)
This expression includes the correlation between the forward rates.
To find some different (and more mathematical) presentations of LMM I turned to "Arbitrage Theory in Continuous time" by Björk which gives the below drift terms (that don't include correlations)
It seems to me that they are modelling different things however, with Bruce modelling the forward rates while Björk is modelling the par swap rates $R_n^N$.
How do we reconcile these two specifications with each other? Particularly when it comes to correlations. How come we get away with not needing it for one specification and not the other, and since we do, how can we reconcile them with each other?
As a side question, if anyone happens to know, I'm unsure what exactly Björk is specifying with the $\sigma^*$ term in 27.69. Is it a black vol like $\sigma_{n,{N+1}}$ or what is it referring to? (the proof didn't make it clear to me)
## Answer by Kurt G. (score 1)
https://quant.stackexchange.com/a/69844
Björk writes a few pages earlier: "We can also allow for correlation between the various Wiener processes but this will not affect the swaption prices. Such a correlation will however affect the pricing of more complicated products." Clearly, Björk decided to make it not too complicated. Putting correlations aside : forward rates and par swaprates are indeed different things - as you wrote. I don't expect their drifts to agree.
The star at $\sigma_{j+1}$ looks like a typo to me. Looking at the relations just before his proof $\sigma_n$ looks like the Black volatility of the forward rate $R_n\,.$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.