Posted by
Toyin Akin-4 on
Oct 21, 2002; 8:02am
URL: http://quantlib.414.s1.nabble.com/RE-Delta-on-Capfloor-tp2259p2265.html
Hi all,
> Andre, you're right to compare the swap sensitivity to the cap's one.
> This is the same structure: a Swap is serie of FRA and a caplet is an
> option on FRA. Exactly like comparing the option on Equity and the Equity.
> this is what traders are doing: hedging CAP/FLOOR with Futures and Swap or
> the opposite.
I agree that in the equity case comparing the option on Equity and the
Equity makes sense, the underlying of the
equity option *IS* the equity.
Maybe I'm missing the point somewhere but...
The underlying for a caplet is a single FRA, not a series of FRAs.
An option on a series of FRA's is a swaption.
Several options where the underlyings are FRA's is a cap.
The underlying of a cap is not a swap.
Now I haven't looked at the history of this exchange, but... if the question
you are all asking is that you want to hedge the cap's delta with a swap and
not why does the cap's delta's sum, sum to more than 1, then yes, you could
hedge with any instrument, the most liquid being FRA's, futures, swaps or,
if one really wanted to, any option where you want to introduce gamma/vega.
You just need to take into account the hedge instruments sensitivity.
The reason why we pick Deposit,s FRA's, futures, swaps to hedge is because
these are the most liquid instruments and
our yield curve is composed of these instruments. Thus all we need to do is
to find the right ratio to hedge.
But in answer to the original question... as to why the total delta of a cap
sum could be more than 1... you really have a portfolio of options.
> Taking your example:
> Caplet 1 (6 months) has delta of 0.6 => delta* swap 6M bpv = 0.6* 0.45 *
> notional/1 BP (around that)
> Caplet 2 (9 month) has a delta of 0.5 => delta* swap 9M bpv = 0.5* 0.67 *
> notional/1BP
> Caplet 3 has a delta of 0.4 => delta* swap 1Y bpv = 0.4*0.9 * notional/
1BP
> Summing these gives 0.965*notional / 1BP that is to say almost 100% of the
> 1Y swap sensitivity isn't it?
I don't really have all the calculations for the above and not really sure
how the swap bpv is calculated, but lets say we have a very in the money
option (strikes at 0.5, not assuming JPY), assuming that the swap bpv
remains the same above, and the deltas are all 1.0 we'll now have 3*notional
/ 1BP as the final value. Again (protecting myself here!!) I'm not sure how
you have computed the 0.45 in the 6M swaps's bpv calc above, but I assume
it's independant of the option's strike.
Also is the swap 1Y bpv an ATM swap from spot up to one year? If so you can
see immediatly that we are dealing
with different structures. Caplet 3 is an option on a 3M FRA... one fixing.
The swap 1Y is a swap with 4 FRA's.
Regards,
Toyin Akin.
----- Original Message -----
From: <
[hidden email]>
To: "Toyin Akin" <
[hidden email]>
Cc: "'Andre Louw'" <
[hidden email]>; "Perissin Francesco"
<
[hidden email]>; "QuantlibUsers (E-mail)"
<
[hidden email]>;
<
[hidden email]>
Sent: Monday, October 21, 2002 1:26 PM
Subject: Re: [Quantlib-users] RE:Delta on Capfloor
>
> Hi All,
> Andre, you're right to compare the swap sensitivity to the cap's one.
> This is the same structure: a Swap is serie of FRA and a caplet is an
> option on FRA. Exactly like comparing the option on Equity and the Equity.
> this is what traders are doing: hedging CAP/FLOOR with Futures and Swap or
> the opposite.
>
> Now what you forgot is to multiply each caplet sensitivity by the swap BPV
> (the sensi for 1BP change in the IR curve) because the delta of a caplet
is
> not in % but calculated for a move of the underlying.
>
> Taking your example:
> Caplet 1 (6 months) has delta of 0.6 => delta* swap 6M bpv = 0.6* 0.45 *
> notional/1 BP (around that)
> Caplet 2 (9 month) has a delta of 0.5 => delta* swap 9M bpv = 0.5* 0.67 *
> notional/1BP
> Caplet 3 has a delta of 0.4 => delta* swap 1Y bpv = 0.4*0.9 * notional/
1BP
> Summing these gives 0.965*notional / 1BP that is to say almost 100% of the
> 1Y swap sensitivity isn't it?
>
> Bye
> Xavier
>
>
>
>
> "Toyin Akin"
> <
[hidden email]> To:
"Perissin Francesco" <
[hidden email]>, "'Andre
> Sent by: Louw'"
<
[hidden email]>, "QuantlibUsers \(E-mail\)"
>
[hidden email]
<
[hidden email]>
> eforge.net cc:
> Subject:
Re: [Quantlib-users] RE:Delta on Capfloor
>
> 21/10/2002 13:13
>
>
>
>
>
>
> Hi again,
>
> In addition, a cap structure being a portfolio of caplets does not include
> information regarding the correlation being
> the individual FRA's. You'll need to look at swaptions for this.
>
> I may be wrong here but I wouldn't try to compare a swap's sensitivity to
> that of the cap's.
> The 2 numbers really account for different measures of risk.
>
> compare swap sensitivity with that of swaptions
> compare a FRA's sensitivity to that of caplet's
> a series of FRA's sensitivity (which really is a swap) should thus be
> compared to swaptions.
>
> Regards,
> Toy.
>
> ----- Original Message -----
> From: "Toyin Akin" <
[hidden email]>
> To: "Perissin Francesco" <
[hidden email]>; "'Andre Louw'"
> <
[hidden email]>; "QuantlibUsers (E-mail)"
> <
[hidden email]>
> Sent: Monday, October 21, 2002 11:54 AM
> Subject: Re: [Quantlib-users] RE:Delta on Capfloor
>
>
> > Hi all,
> >
> > Remember, a cap is a series of caplets. thus you are buying a portfolio
> of
> > options.
> > Like any portfolio of options, the sum of the deltas, which will be your
> > total sensitivity if rates move can
> > be well over 1.0. However the delta of any *ONE* caplet will be <= +-
> 1.0.
> >
> > If each caplet is well in the money, then each caplet will have a delta
> of
> > 1. Thus the sum will be above one.
> >
> > Also, a caplet is an option on an individual FRA. Thus you cannot really
> > bring swaps into this.
> > You may be confusing swaptions with caps here.
> >
> > A swaption is an option on a swap.
> >
> > Regards,
> > Toyin Akin.
> >
> >
> > ----- Original Message -----
> > From: "Perissin Francesco" <
[hidden email]>
> > To: "'Andre Louw'" <
[hidden email]>; "QuantlibUsers (E-mail)"
> > <
[hidden email]>
> > Sent: Monday, October 21, 2002 11:13 AM
> > Subject: RE: [Quantlib-users] RE:Delta on Capfloor
> >
> >
> > >
> > > Hi,
> > >
> > > >The answer given originally to my question on calculating the delta
on
> a
> > > >capfloor structure was that it is simply the sum of the delta's on
all
> > the
> > > >caplets (assuming flat nominals, there is no weighting needed).
> > >
> > >
> > > I gave you this answer assuming another definition of delta for an
> > interest
> > > rate instrument, i.e. the change in present value given a parallel
> shift
> > of
> > > 1bp in the used curve. Also, this is quite close to the change obtaine
d
> by
> > > shifting of 1bp the prices of the benchmarks used to bootstrap the
> yield
> > > curve.
> > > This seems to be equal to what you name "sensititvity", is it right?
> > >
> > > Regarding the definition that you are referring to, (i.e. the cap
> > > sensitivity divided by the IRS sensitivity) I have to think a while
> about
> > > it. The example could help.
> > >
> > >
> > >
> > > >e.g I have a 3Mx12M, 3 monthly cap, which results in 3 caplets,
> results
> > as
> > > >follows:
> > > >the 1st strikes at a delta of 0.6, the 2nd strikes at a delta of 0.5,
> the
> > > >3rd at a delta of 0.4. Summing these gives 1.5E
> > >
> > >
> > > Just for understanding, do you mean that the first fixing is in 3
> months
> > ans
> > > last fixing in 9 months? Which curr? Which strike?
> > > In any case, it's not correct to sum the deltas (according to your
> > > definition). I would be tempted to say that you should sum the caplet
> > > sensitivities in order to get the cap sensitivity. Now you could
divide
> > the
> > > result by the IRS sensitivity, isn't it?
> > > Let me know it this can help...
> > >
> > >
> > > >Ok, maybe it's on the Delta with respect to spot: the above caplets
> gives
> > > >0.13, 0.11, 0.08, giving a total of 0.32, (doesn't 'feel' right!),
> > > depending
> > > >on the termstructure and the length of the cap, the sum of the
> > spot-delta's
> > > >quite often goes over 1.0!
> > >
> > >
> > > Andre, I could not understand this... could you try to explain again
> the
> > > above numbers?
> > >
> > >
> > > Ciao
> > > Francesco
> > > --
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