Posted by
Toyin Akin-4 on
Oct 22, 2002; 4:04am
URL: http://quantlib.414.s1.nabble.com/RE-Delta-on-Capfloor-tp2259p2262.html
Ah,
Maybe the answer (to the 2.02*notional / 1BP) is that in the following...
> > Now you already know that the put-call parity between cap and floor is
Cap
> > = Floor + Swap.
The rate that the swap is struck at in the formula above is not the strike
of the market instrument used to hedge with.
(Assuming you are using ATM swaps for hedging here.)
Thus you are hedging a cap with an underlying not used to price the cap.
What do you think Xavier?
Regards,
Toyin Akin.
----- Original Message -----
From: "Toyin Akin" <
[hidden email]>
To: <
[hidden email]>
Cc: "'Andre Louw'" <
[hidden email]>; "Perissin Francesco"
<
[hidden email]>; "QuantlibUsers (E-mail)"
<
[hidden email]>;
<
[hidden email]>
Sent: Tuesday, October 22, 2002 9:48 AM
Subject: Re: [Quantlib-users] RE:Delta on Capfloor
> Hi Again,
>
> > A swaption is an option to enter into a swap at a given date.
> > for example if you have the option to enter in a 5Y swap in 1Y then if
you
> > exercice this option once you enter in the swap there is no option
> anymore.
> > In 1Y you enter in a 5Y swap and whatever are the futures rates there is
> no
> > option anymore.
> > This is not exaclty the payoff of a cap: the cap is sum( max(forward -
> > strike,0)) but the payoff of a swaption is max(Swap NPV,0).
>
> Agreed.
>
> > Now you already know that the put-call parity between cap and floor is
Cap
> > = Floor + Swap, this is why the Cap delta is expressed in term of Swap
> BPV.
>
> Again agreed, however when we actually price each part of it, we price a
> portfolio of caplets, floorlets and
> fra's. Where the underlying of each caplet/floorlet is a FRA. It just so
> happens that the sum of all these parts maps
> onto the definition you have provided.
>
> > As you said the swap 1Y is a swap with 4 FRA's but taking a swap at the
> > break even (NPV=0) the swap is also:
> > A Notional exchange + fix flows.
> > From that you can show that the sensitivity of a swap is very close to
> > (swap maturity)*discount factor.
> >
> > Take a 5 Years swap the sensitivity is roughtly 5*exp(-5*0.03)*notional
/
> > BPV = around 43 000 for a 100M 5Y swap.
>
> Okay, my feeling was that it was something like
> Sum(DayCountFraction*DiscountFactor).
>
> Forgive me, I'm probably missing the point somewhere here, but looking at
> your example again.
>
> > 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?
>
> If the caplets were deep ITM (delta of 1) you would now have 2.02*notional
/
> 1BP.
>
> Am I correct to read this as I would need close to 2 units of my swap to
> hedge the cap where the swap is the underlying
> (or the swap expressed in the call-put parity above).
> If so, this does not make sense if the swap is the underlying. If the swap
> is the underlying, I would need 1 unit.
>
> Again I'm probably missing the point somewhere and I apologise for
rambling
> on!!
>
> Best 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: Tuesday, October 22, 2002 9:20 AM
> Subject: Re: [Quantlib-users] RE:Delta on Capfloor
>
>
> >
> > Hi Toyin,
> > A swaption is an option to enter into a swap at a given date.
> > for example if you have the option to enter in a 5Y swap in 1Y then if
you
> > exercice this option once you enter in the swap there is no option
> anymore.
> > In 1Y you enter in a 5Y swap and whatever are the futures rates there is
> no
> > option anymore.
> > This is not exaclty the payoff of a cap: the cap is sum( max(forward -
> > strike,0)) but the payoff of a swaption is max(Swap NPV,0).
> >
> > Now you already know that the put-call parity between cap and floor is
Cap
> > = Floor + Swap, this is why the Cap delta is expressed in term of Swap
> BPV.
> >
> > I absolutely agree that the cap is the sum of caplet and each caplet has
> to
> > be valued separately but if you take the B&S valuation for each caplet,
> the
> > B&S RAW delta is between 0 and 1 but is expressed in term of Swap BPV.
(at
> > least this is what banks are doing in France and Germany!)
> >
> > For example a delta of 0.5 for one caplet is not 0.5 EUR but 0.5 EUR per
> > Swap BPV. It means that from a delta of 0.5 you find the delta in
currency
> > multiplying the B&S delta by the swap BVP then you can sum the delta of
> > each caplet and compare this value to the delta of a swap or a future or
> > even a swaption if you wish. I think this is what Andre was trying to
do.
> >
> > How I found the swap BPV: simple.
> >
> > As you said the swap 1Y is a swap with 4 FRA's but taking a swap at the
> > break even (NPV=0) the swap is also:
> > A Notional exchange + fix flows.
> > From that you can show that the sensitivity of a swap is very close to
> > (swap maturity)*discount factor.
> >
> > Take a 5 Years swap the sensitivity is roughtly 5*exp(-5*0.03)*notional
/
> > BPV = around 43 000 for a 100M 5Y swap.
> >
> > bye
> > Xavier
> >
> >
> >
> >
> >
> > "Toyin Akin"
> > <
[hidden email]> To:
> <
[hidden email]>
> > Sent by: cc:
"'Andre
> Louw'" <
[hidden email]>, "Perissin Francesco"
> >
[hidden email]
> <
[hidden email]>, "QuantlibUsers \(E-mail\)"
> > eforge.net
> <
[hidden email]>,
> >
> <
[hidden email]>
> > Subject:
> Re: [Quantlib-users] RE:Delta on Capfloor
> > 21/10/2002 16:01
> >
> >
> >
> >
> >
> >
> > 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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