Finding the ATM Forward Rate for Caps and Floors
Summary
The document explains how the at-the-money level for an interest-rate cap or floor is determined and why volatility quotes are often organized around it. The relevant level is the forward swap rate for a swap with the same maturity, schedule, and other terms as the cap and floor. Cap-floor parity implies that their price difference equals the value of that swap; the ATM strike is where the swap’s value is zero, making cap and floor prices equal.
Quoting strikes relative to this level keeps the volatility surface meaningful as market rates move, whereas fixed absolute strikes can become less useful. The response also describes the ATM point as roughly corresponding to an even chance of finishing above or below the strike and as the point of maximum option time value. These are broad intuitions, not exact universal claims; matching the swap’s conventions matters, and standard swaps may differ from the instrument underlying the cap or floor.
Key ideas
- The cap or floor ATM strike is the forward swap rate for a swap matching the instrument’s terms.
- Cap-floor parity identifies ATM as the strike where the matching swap has zero value.
- Relative strike quotes remain more useful than fixed absolute strikes as market rates change.
- The matching swap’s schedule and conventions must align with the cap or floor being valued.
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Full text
# Cap/Floor ATM Rate # Cap/Floor ATM Rate This is a question on cap volatility market data. The quotes usually include volatilities for different strike (1%, 2%, ... 5%) and maturities (1Y,2Y,...20Y). One volatility for each combination of strike and maturity. If I want to price a cap with, let's say, a strike of 2% and maturity in 10 years I would use the corresponding volatility from the market data described above. I think I understand it so far. But then I also have volatility quotes for the at-the-money (ATM) rate. How do I know what the current at-the-money rate is? Second, in which situation would I use the ATM quotes? ## Answer by byouness (score 10) https://quant.stackexchange.com/a/42906 The ATM level (ATMF: at the money forward to be more precise) is the one giving you the same price for call and put, or in this case, the same price for cap and floor. So, let us start with writing the cap / floor parity: $$Cap(K) - Floor(K) = Swap(K)$$ where $Swap(K)$ is a swap paying K and that has the exact same characteristics (maturity, schedule, etc.) as the cap and floor. The ATM level is then the one for which $Swap(K) = 0$, and that is, by definition, the swap rate of this swap. Now, why do people define strikes with respect to this ATM level. Roughly speaking, it's the level such that you have a 50/50 chance of ending above or bellow of it at expiry. So, it makes more sense to define a set of strikes relative to this ATM level rather than having absolute strikes that can be useless after a move of market parameters. Another interesting property is that the ATM level is where the option's time value is maximal. ## Answer by BerndSchmitz (score 2) https://quant.stackexchange.com/a/75334 The "exact same characteristics" part is important. The swap will never be exactly the standard swap. E.g. for EURIBOR 6M index it will be a swap with a 6M and ACT360 fix leg while the standard swap has a 12M D30/360 fix leg. Furthermore IBOR cap/floors always skip the first caplet/floorlet as the corresponding rate is already fixed.
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