To my understanding the value of an FRA option is identical to that of a caplet of equal maturity, strike and tenor. A volatility surface of cap implied volatilities is generally available, and from that you can strip the caplet volatilities that make up each of the caps that constitute the surface. But the caplets typically have a tenor of 3 or 6 months, whereas I presume an FRA option may be of any tenor like 9 months or 3 years (is that correct?). If so how do we find what a 9 month or 3 year caplet volatility would be when all we have to go on is a cap surface built from caplets with 3 month tenors (and the stripped caplet surface)?
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1$\begingroup$ Caplets on liquid rates (G10) exist up to three years quite commonly. On USD rates, you can get caplets up to 30 years of maturity (with the underlying rates maturity of 1m, 3m or 6m). $\endgroup$– Jan StullerOct 12, 2020 at 14:11
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2$\begingroup$ As you have already stated, somewhat liquid CapFloor (or bootstrapped caplet/floorlet) surfaces are available for cap(let) tenors of 3M (short end) and 6M (long end); and these are quoted for expiries of up to ten years (or more). If you want to price a FRA option with a non-standard tenor, say 2Y, then you will need to make use of more information than what the CF surface offers as you need some information about the (implied) correlation structure of your FRAs, i.e. you need to incorporate information from the Swaption surface as well. $\endgroup$– KermittfrogOct 12, 2020 at 14:15
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$\begingroup$ @Kermittfrog Yes you seem to understand me perfectly. Can you point me in the right direction regarding this? You also refer to FRA options with non-standard tenors, am I right to take that to mean that most FRA options that are traded are in fact on caplet tenors (i.e 3M or 6M)? $\endgroup$– OscarOct 12, 2020 at 14:20
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$\begingroup$ @JanStuller Yes that I have, but if you were to value an FRA option with 10 year maturity but say a 2 year rates maturity, as you call it, how would you go about it? $\endgroup$– OscarOct 12, 2020 at 14:22
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1$\begingroup$ @JanStuller Perhaps I'm mixing it up here. What I mean is what wikipedia calls an Interest Rate Guarantee. So what I mean is an option to enter into a loan at a fixed rate for a predetermined period (say 2 years) in a predetermined time (the maturity, so say 10 years). So basically just a caplet. I wrote FRA option because this is what I thought was the common name for it but perhaps it is not. The underlying is just the interest rate. $\endgroup$– OscarOct 12, 2020 at 15:27
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1 Answer
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I think it is necessary to be more precise on the terminology here, so my answer will be a bit longer.
Firstly, we need to distinguish the FRA, the FRA option and the caplet/floorlet. A FRA basically locks in a future LIBOR fixing for you, e.g., the 1x7 FRA strike allows you to lock in the 6m LIBOR that will fix in 1m from now. Note that it is a linear derivative with symmetric payoff. Anyways, here the first ambiguity can arise in terms of terminology -- what is the maturity of the FRA, and what is the tenor? I follow your notation and say that the FRA has maturity 1m and tenor 6m (as it refers to the 6m LIBOR).
Now, caplets and floorlets are single-period calls and puts on LIBORs, respectively. Hence, buying a 1m caplet and selling a 1m floorlet on the 6m LIBOR, is equivalent to a FRA (to be precise: almost equivalent -- we have some differences in settlement practices). This can be seen as some sort of put call parity for interest rate derivatives.
Lastly, here you are referring to FRA options, i.e., calls/puts not on the LIBOR itself, but rather a linear derivative thereof. As a mnemonic, you can think of it being similar to a swaption, which is an option on a IRS (another linear derivative on LIBORs). In fact, some people call FRAs single-period IRSs (although they are not 100% congruent). Either way, the optionality now brings in another dimension. For instance, you can have an option living for 2 years that allows you to enter a 1x7 FRA in 2 years from now. In this case, you would still be interested in the volatility of 6m LIBOR. But what is maturity now? The 2y maturity of the option? Or the 1m "lifetime" of the FRA? So please make sure that you're 100% clear on the timeline of this transaction and if/how caplet surface can be used for your task.
In any case, your question, however, reveals (at least to me) an interesting challenge for practicioners, namely: how can we extract the vol of a 3m LIBOR if we only have quotes on the vol of 6m LIBOR, or vice versa?
- This thread might be of interest for you in this context.
- I have also seen tries to compare 3m vs. 6m realized vols and extract a spread, which can then be applied to current market quotes to find the vol of the non-standard tenor.
As a last side note, it will be very interesting to see how the market in interest rate vols will develop once LIBORs are discontinued and replaced by overnight risk-free rates (backward vs. forward looking, and volatility "accrual" in this context).
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$\begingroup$ The thread you linked certainly offers a very simple solution to the problem and is in fact what I would be looking for I think. I think that equation won't hold up in the Bachelier/normal model with normal volatility though, which is needed because I am dealing with negative rates. $\endgroup$– OscarOct 12, 2020 at 15:28
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$\begingroup$ I have not checked all the maths to test what impact this would have on the rebasing formula provided, but you can approximate Bachelier vol using Black vol times (FK)^0.5 ("Hagan approximation" - not to be confused with the approximation derived for SABR). $\endgroup$– KevinTOct 13, 2020 at 7:09