Sign of European Swaption Delta

MrKlugh · Post by **MrKlugh** » Tue Nov 06, 2012 6:33 pm

Hi,

Shouldn't the delta of all ATM payer swaptions (different expiries) have the same sign? I am using a pricing toolkit that seems to give negative deltas for payer swaptions with long expiries. Any idea of what's the problem?

Cheers

silverside · Post by **silverside** » Tue Nov 06, 2012 9:51 pm

are you talking about parallel delta forward delta or bucket delta?

MrKlugh · Post by **MrKlugh** » Wed Nov 07, 2012 10:55 am

Each bucket is bumped and the delta I refer to is the sum of all buckets

Nonius · Post by **Nonius** » Sat Nov 10, 2012 5:46 am

from which perspective? if I buy a payer I am short. I make money if rates go up. doesn't matter if I'm at the money, out of the money, or in the money. So, I'd expect that to be like long a put on a bond. so, yeah, I'd say that should be negative delta. but it does depend on shitloads of semantics.

MrKlugh · Post by **MrKlugh** » Mon Nov 12, 2012 12:14 pm

I agree that being long a payer is being short the market. It's also equivalent to a put on a bond. If rates go up, bond prices go down and the put's PnL should be positive. Looks like a positive delta to me (or negative DV01). The delta of a put is negative wrt prices not rates. Or am I losing the plot?

MrKlugh · Post by **MrKlugh** » Mon Nov 12, 2012 12:42 pm

I've just checked with Bloomberg's SWPM and indeed I get a positive delta for payers. There seems to be a change of sign in the long end but I'd say it's due to extrapolation errors (which is probably what I need to investigate in the system I was talking about in the first place...)

Bloomberg's default yield curve goes out to 50Y. Delta of a 45Yx5Y payer > 0 but Delta of 46Yx5Y or 50Yx5Y payer < 0. It all happens at the 50Y point which is probably not a coincidence.

Any comments much appreciated.

Cheers

silverside · Post by **silverside** » Mon Nov 12, 2012 9:55 pm

doh "read the question carefully"

I suggest you build a toy model in excel/VBA with the following 3 methods

- keep 55y swap rate same as 50y swap

- keep 55y zero same as 50y zero

- keep 50y5y same as 45y5y

and... well... see what happens. plot some nice graphs of the term structure to aid your thought process.

a variant of this question is actually going to affect the long end of the EUR curve to some extent (the Solvency II "risk-free" curve has a weird extrapolation formula defined beyond 20y, luckily no-one is trying to arb it until they know the implementation date for SII).

MrKlugh · Post by **MrKlugh** » Tue Nov 13, 2012 10:58 am

I don't understand why it's so difficult to answer a question with... an answer. If there's obviously something I dont get, a couple of clear sentences would just do the job... And if I came to the forum it's because I am not 100% sure of my thought process.

Instead of trying to sound smart, just explaining would help more. Just my 2 cents.

pj · Post by pj » Tue Nov 13, 2012 12:27 pm

Hi,

My 2 cents.

It seems that the delta should

be staying of the same sign,

but it decreases with a longer maturity.

I get

3.43168E-06 for 20 years

1.66757E-06 for 50 years

5.30512E-07 for 100 years.

(Which makes the sense to me

when thinking what delta is. It shouldn't flip)

But I am using the third party software.

Maybe some weird market data?

Does it happen with the constant

vol and constant yield as well?

In that case you should pointedly

ask the vendor.

Or change it.

HTH

Jim · Post by **Jim** » Tue Nov 13, 2012 3:38 pm

I can think of two causes for the observations you describe. The first is calculating risk measures by bumping curve inputs. When you bump one input to a curve generator and leave others constant, you end up measuring sensitivity to your curve generator and not necessarily your sensitivity to the market. For example, suppose you build your curve with some short dated instruments and, say, the 4-, 5-, and 7-year swap rates. If you bump the 5-year swap rate up 1 b.p. and freeze all other inputs at their original values, in forward rate terms the 4 to 5 year forward rate will have to go up by five basis points (to make the new 5-year swap price correctly) and the 5 to 7 year forward will have to go down by 2.5 b.p. to ensure the 7 year swap prices at the frozen 7-year rate. Depending upon the dates spanned by the swap underlying the swaption, the underlying swap rate could go up or down giving you apparently anomalous delta measurements.

A second possible cause is an inability to disentangle delta (exposure from a change in underlying) from rho (exposure from a change in discounting). Long-dated options often have large exposures to discounting. The rho exposures in short dated buckets could easily have a different sign from the exposure to change in underlying seen in the longer maturity buckets.