Why Rate Models Use Instantaneous Forwards and ATMF Quotes

Summary

The note gives a brief motivation for using instantaneous forward rates in interest-rate models: they are often chosen for modeling simplicity. It relates the instantaneous forward rate to the derivative of maturity times the continuously compounded spot rate, so integrating the forward curve can recover spot-rate information. The source does not develop a particular stochastic model or show a worked calculation, so the practical modeling advantage is stated generally rather than demonstrated in detail.

It also clarifies market terminology for caps and floors. Their rates options are typically quoted at the money forward, using the par forward rate as the relevant reference, rather than an undefined or generic at-the-money spot rate. The answer distinguishes this convention from an “at the money forward” interpretation based simply on a continuously compounded forward rate. The discussion is concise and does not explain cap or floor volatility surfaces, calibration, or why a specific quote convention is used across markets.

Key ideas

• Instantaneous forward rates and continuously compounded zero-coupon rates are commonly used for modeling simplicity.
• The instantaneous forward rate can be expressed as the derivative of maturity multiplied by the spot rate.
• Integrating the forward-rate representation gives spot-rate information.
• Caps and floors are generally quoted relative to the par forward rate rather than a generic spot at-the-money rate.

Tags

# Why use instantaneous forward rates?


# Why use instantaneous forward rates?












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…

## Answer by user68819 (score 1)

https://quant.stackexchange.com/a/79681

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