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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)

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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) enter image description here

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)

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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 $R_n\,.$

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  • $\begingroup$ Hi, Yes I saw that as well, but I found it very strange that he didn't mention it again in this about calculating the drifts, since the very point of that is to price more complicated products. Unless I've misunderstood, you don't even need the drifts to find swaption prices. If I were to price a Bermudan Swaption for instance, would the information in Björk be enough? I find the par rates corresponding to the Bermudan expiration maturities, calculate the drifts under the measure where the final par rate's drift is 0, MC simulate, and price the payoff? Or would I need correlations for this? $\endgroup$
    – Oscar
    Commented Feb 11, 2022 at 16:21
  • $\begingroup$ At this moment I think (without having looked at Björk's swap rate model in detail) that even with correlations it is not necessarily the best approach to price Bermudan swaptions. Keep it simple and think about this: A Bermudan swaption is clearly worth more than a European swaption but clearly less than the corresponding cap/floor with strike being the fixed rate of the swap that underlies the option. This tell us that the Bermudan swaption has characteristics of a cap/floor as well as of a European swaption. $\endgroup$
    – Kurt G.
    Commented Feb 11, 2022 at 19:13
  • $\begingroup$ Björk's model is probably perfect for European swaptions but we might as well start with a Libor market or even a short rate model (Hull White). The matter is how well those models calibrate to caps/floors and swaptions. If HW calibrates well it is probably the simplest model of them all. $\endgroup$
    – Kurt G.
    Commented Feb 11, 2022 at 19:13
  • $\begingroup$ But a European swaption can be priced (in this framework anyway, according to Björk) with Black's model, the starting point of this section was to replicate the Black-schooles formula with the swap market model, and the point of it all (or so I thought) was that we can then use this model to price more exotic derivatives (that black's model can't handle). I'm maybe failing to see what the point of it all is, if we wouldn't use this model (or the one in Tuckman?) to price a Bermudan Swaption then what is it even good for? $\endgroup$
    – Oscar
    Commented Feb 11, 2022 at 19:21
  • $\begingroup$ In a nutshell : the advantage of the LMM is that it does not need to be calibrated to caps/floors . The advantage of Björk's swap rate model is that it does not need to be calibrated to European swaptions. A short rate model needs to be calibrated to both. In my opinion the disatvantage of those market models is that as soon as you need a more universal pricing measure (because your product is exotic) you have a lot of book keeping to do with all those drifts. In a short rate model you can work with the risk neutral measure and a single drift. $\endgroup$
    – Kurt G.
    Commented Feb 11, 2022 at 20:22

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