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
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2 Answers
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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 $i$.
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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.
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1$\begingroup$ Ok seems clearer, but do we agree that you can’t assume that even if you have a reset on the same forward rate, this is not correct to assume the caplet independence? (Ie. The variance of the caplets is the sum of those two caplet’s variance) $\endgroup$– JohnGaltCommented May 21 at 16:08
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1$\begingroup$ The vols may indeed be correlated, so will changes in the fwds, but that has nothing to do with the caplets payoff. The caplet is a single option on a forward rate. $\endgroup$ Commented May 21 at 16:24
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1$\begingroup$ That was my true question. I am validating a model were a guy make this assumption. I am hesitating to do a challenger model with caplet vol being correlated. You helped a lot! Many thanks. $\endgroup$– JohnGaltCommented May 21 at 16:28
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$\begingroup$ But if you got a parallel shift on your interest rate curve, everything will move the same way including the payoff isn’t it? $\endgroup$– JohnGaltCommented May 21 at 20:28
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$\begingroup$ Yep..but the fact is a single caplet depends on a single fwd rate. Has absolutely nothing to do with other fwds. So parallel, zig zag, whatever the movement is, the caplets are independent. The cap vol however Is a function of its price which is a function of all the caplet prices (vols) $\endgroup$ Commented May 21 at 21:07