Duration of a receiver swaption

[joeguirg]

Hi,
I recently interviewed for a mbs analytics role and was asked whether the key rate duration on a 2X5 receiver swaption was either negative, zero or positive with respect to the 2-year key rate and the 7-year key rate.

I answered that the duration was negative with respect to the 2-year key rate and positive with respect to the 7-year key rate. My reasoning was as follows:

1. In general, a receiver swaption implies you are long duration since receiving fixed and paying float is equivalent to being long a bond.

2.2x5 swaption can be thought of as an option on a 5 year swap starting 2 years forward (i.e. forward starting swap).

3. This swap can be replicated by going long a 7-year bond and short a 2 year bond.

4. Therefore, this forward starting swap described in 3. has negative duration with respect to the 2-year key rate and positive duration with respect to the 7-year key rate.

5. Therefore, the swaption (since it can be thought of as a long call on the forward starting swap) also has negative duration with respect to the 2-year key rate and positive duration with respect to the 7-year key rate.

I'm not sure whether this is correct but this is my intuition. Is this correct? Thanks.

 

[QTMGMT]

The replicating portfolio you describe in 3. for the swap would leave you simply receiving fixed after 2 years from today. The swap you are replicating would have you buying a 5yr bond two years from now (receiving fixed) and financing it with short term securities, which you continuously roll over until the maturity of the bond (paying floating).
So it doesn't seem like 3. is the replicating portfolio.

 

[joeguirg]

The replicating portfolio you describe in 3. for the swap would leave you simply receiving fixed after 2 years from today. The swap you are replicating would have you buying a 5yr bond two years from now (receiving fixed) and financing it with short term securities, which you continuously roll over until the maturity of the bond (paying floating).
So it doesn't seem like 3. is the replicating portfolio.

I'm using a heuristic. That is, a long position in a receiver swap is nearly equivalent to a long position in a fixed rate bond. Ceteris paribus, the sign of the duration on both instruments is certainly the same.

 

[financeguy]

Yeah to be clear here you're just rationalizing the sign of the delta hedge, so that should be correct. I'm not a swaptions expert but it seems right to me.

 

[QTMGMT]

Yes got it but it seems you are drawing a conclusion based on a replicating portfolio which as I understand it, is misspecified. I thought about it for a few minutes and couldn't come up with the replication portfolio (not a swaptions expert clearly but it is an interesting thought experiment). So instead I used the following logic.

If you draw an analogy to a european equity call option, the underlying is the swap which starts two years from now (you will enter the swap if it is more valuable than the swap given the prevailing rates at that time and otherwise you won't like you will exercise the equity call if it is better than paying prevailing market price for the stock or you won't). So all else equal, if the 7 year rate increases suddenly the yield curve will be steeper between 2 and 7 years, implying higher yields for 5 year bonds two years from now compared to what you have negotiated. that would be bad for your underlying swap and so bad for the swaption because you want higher yields. ( just speaking in terms of rate sensitivity and you can convert that to your interpretation of what "long druation" or "sign of duaration" actually mean because it depends on who you're talking to).

i.e. all else equal, 7 year yield up swaption down

If you extend the same logic to the 2 year rate then an increase would mean a flatter yield curve between 2 and 7 years, implying a lower 5 yr bond yield 2 years from now compared to what you have negotiated which would be good for the underlying swap and so good for your swaption.

i.e. all else equal, 2 yield up swaption up.

But I couldn't reconcile this: why should the 2 year matter at all if the swap won't start until after 2 years.

anyway interesting question let us know if you determine the answer and the logic also.

PS my favorite go-to in interviews when I don't know the answer to something like this is just distract by saying well there is basis risk even if you come up with the mathematically correct hedge ratio once markets go crazy that will all fall out the window. a lot of markets guys still don't consider dynamic correlations and if you're good at talking you can take the attention away from the fact that you don't know the answer.

Good Luck

 

[financeguy]

The 2y swap matters because you care about the forward starting rate: the 7y rate tells you about how much interest you get for the cumulative period 0 day - 7y. You only care about how much is apportioned to the period 2y-7y. So depending on what the 2y rate is, more or less cumulative interest is backloaded to the 2y-7y period instead of the 0 day - 2y period. To hedge the forward starting swap dynamic, you'd receive fixed (and pay floating) on a 7y swap and pay fixed on a 2y swap in even notional. The idea here is the 2y swap undoes what the 7y swap does for you in the first two years, so for the first two years basically nothing happens, then after the 2y swap rolls off you're left with a 5y swap - which is what you want. (Note that since these are even notionals, you would still have net received DV01 exposure since 7r has more interest rate risk per bp than 2y - which is also a pretty intuitive result if you are going into a forward starting received position. So if both 2y and 7y rates went up by equal amounts, you'd still lose money on a 2x5 receiver.) It should be clear then that the 2y matters - and that you can directly hedge the 2y exposure. To delta hedge the 2x5 receiver, you'd do the opposite of what we described above, you'd receive 2y and pay 7y in even notionals.

 

[JonahLiu]

From my understanding, \[ (1+2yr)^2 * (1+F(5yr))^5 = (1+7yr)^7 \], so you have an approximate formula like this:\[ F(5yr) = 7yr - 2 yr \], so when 2yr goes up, F(5yr) drops, which makes your receiver swaption more valuable, suggesting a positive relationship with 2-yr rate and thus a negative duration. And vice versa, when 7yr rate goes up, F(5yr) increases, which makes receiver swaption less valuable, suggesting an inverse relationship with 7-yr rate and thus a positive duration. I think it's correct. Please point it out if any part goes wrong. Thanks!