Forward-Rate Correlation in Caplets, Caps, and Swaptions
Summary
The document separates the effect of correlation on an individual caplet from correlation among forward rates along the yield curve. A caplet pays based on one forward rate at its expiry. Since its payoff depends on that rate alone, correlation with other forward rates does not directly change the caplet payoff. A cap is a collection of such individual options, so its constituent caplets likewise do not require modeling correlation between different forwards to define their payoffs.
Correlation still matters for instruments whose payoff depends jointly on multiple rates. A swaption is described as an option on a forward swap, which combines forward rates; its value can therefore depend on their correlation, much like an option on a basket. The answers also note that neighboring forward rates tend to be correlated, with the relationship varying by maturity. These are structural distinctions about payoff dependence; the document offers no quantitative model, calibration method, or empirical estimates for the correlations.
Key ideas
- An individual caplet payoff depends on its own underlying forward rate, not other forwards.
- A cap aggregates caplets, so correlation between distinct forwards does not directly determine each caplet payoff.
- A swaption depends on a forward swap comprising multiple rates, making forward-rate correlation relevant.
- Neighboring forward rates are correlated, and the stated relationship can vary across the curve.
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# Caplets volatility questions
# Caplets volatility questions
Is that correct to assume that all Caps/floors are insensitive to correlation between FRA and why? I find it to be a strong assumption and I don't get very much why some people tell me this. If a 3 months tenor fixed in 6 months have an increased volatility, I would assume that the 3 months tenor in 9 months will tend to have an increase volatility. Please correct me if I am wrong. Kind regards
## Answer by David (score 3, accepted)
https://quant.stackexchange.com/a/79509
Your question actually covers two different aspects:
- The impact of correlation on (say) the 6x9 caplet, i.e. an option on a 3m forward rate expiring in 6m time.
- Correlation between different parts of the yield curve, e.g. 6x9, 9x12 FRAs.
The answer to 1) is zero. The payoff is written in terms of only the 3m forward rate expiring in 6m time. By definition, it doesn't depend on any other forward rate, so correlation between different FRAs, e.g. 6x9, 9x12, cannot impact its the caplet payoff.
The answer to 2) is non-zero. Neighbouring points of the yield curve are correlated, with higher correlation for longer dated maturities. For example, consider a tenor structure $T_1,...,T_N$. Then the correlation between neighbouring forward rates, $(F_i,F_{i+1})$ increases with $i$.
## Answer by user68819 (score 2)
https://quant.stackexchange.com/a/79455
A cap is a collection of options on individual forwards, I.e. a cap is made of caplets so caplets are insensitive to the correlation of forwards. Swaptions are options on fwd swaps which are made up of fwds, kind of like an option on a basket, so sensitive to the correlation of fwds.
Looks for a trade called a Wedge, ye shall find detailed answers there.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.