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My question is why we need instantaneous forward rate $f_t$? What is the usage?

I know that some stochastic rate models model this one, and we easily integrate to get spot rate $r_t$, since $f_t=(tr_t)’$. Is it true that the model for $f$ is more convenient? Any other reasons?

Also, why use ATMf vol for caps/floors… ATM vol is not enough? I suppose there are no caps on instantaneous forward rate…

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  • $\begingroup$ 30 years after studying HJM I see $f_t=(tr_t)'$ for the first time. Is $r_t$ even differentiable? $\endgroup$ Commented Jun 11, 2024 at 9:26
  • $\begingroup$ Here, $r_t$ is the discount curve as a function of $t$, which is usually differentiable $\endgroup$ Commented Jun 25, 2024 at 10:46
  • $\begingroup$ So I should not call it as spot rate proces $\endgroup$ Commented Jun 25, 2024 at 10:55
  • $\begingroup$ I still don't follow. Before we baptize $r_t$ you should come around with a precise definition. "Discount curve" could be anything, from discount factors to their logs and what not. $\endgroup$ Commented Jun 25, 2024 at 11:28

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Instantaneous forward rates and for that matter continuously compounded zero coupon rates are usually used for modelling simplicity.

ATM rates don't really exist for rates options. So they are usually quote as ATMF, where ATMF is not At the money forward continously compounded, more At the money (par) forward.

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  • $\begingroup$ Do you mean that ATMF has nothing to do with forward rate? $\endgroup$ Commented Jun 10, 2024 at 21:09
  • $\begingroup$ Btw, why atm doesn’t exist for rate option? The swap rate is atm in the sense of the cap/floor parity $\endgroup$ Commented Jun 10, 2024 at 21:12
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    $\begingroup$ Re your first comment - no, I wrote its ATM (par) forward not ATM (instantaneous) forward. $\endgroup$ Commented Jun 10, 2024 at 21:41
  • $\begingroup$ A 1y par swap (just going back to libors) has a first fixing today effective t+2. If you trade a 1y cap with an immediate fixing today it would be a bit silly. So a 1y cap on say 3m underlying, is made up of 3 caplets not 4. $\endgroup$ Commented Jun 10, 2024 at 21:52

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