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RE:Delta on Capfloor

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Andre Louw-2

RE:Delta on Capfloor

Hi all,

I'm still trying to get behind the total delta on a capfloor. To recap, the
delta on a capfloor should relate back to the RATIO between it's sensitivity
and that of a swap with an equivalent structure. Using the
example Xavier gave:

> The sensi of a 5Y 100 M swap at the break even is
> rougthly 45 000, the sensi of a 5Y 100M cap at the same strike is 2 times
less: 22 500 because
> here the delta is 0.5.

On the same note, a deep in the money cap's sensitivity tends closer and
closer to that of the swap's (1.0).

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). Thinking
about it, this does not really make sense.

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 1.5!
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!

What am I missing?

Any help appreciated.

Andre

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Perissin Francesco

RE: 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 obtained 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 1.5!

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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Toyin Akin-4

Re: 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 obtained 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 1.5!
>
>
> 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
> --
> ############################### DISCLAIMER

#################################
>
> This message (including any attachments) is confidential and may
be
> privileged. If you have received it by mistake please notify the sender
by
> return e-mail and delete this message from your system. Any
unauthorised
> use or dissemination of this message in whole or in part is
strictly
> prohibited. Please note that e-mails are susceptible to change. Banca
del
> Gottardo (including its group companies) shall not be liable for the
> improper or incomplete transmission of the information contained in
this
> communication nor for any delay in its receipt or damage to your
system.
> Banca del Gottardo (or its group companies) does not guarantee that the
> integrity of this communication has been maintained nor that
this
> communication is free of viruses, interceptions or interference.
>
>
############################################################################

>
>
> -------------------------------------------------------
> This sf.net email is sponsored by:ThinkGeek
> Welcome to geek heaven.
> http://thinkgeek.com/sf
> _______________________________________________
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Toyin Akin-4

Re: RE:Delta on Capfloor

In reply to this post by Andre Louw-2

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

> > >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 obtained

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 1.5!
> >
> >
> > 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
> > --
> > ############################### DISCLAIMER
> #################################
> >
> > This message (including any attachments) is confidential and may
> be
> > privileged. If you have received it by mistake please notify the sender
> by
> > return e-mail and delete this message from your system. Any
> unauthorised
> > use or dissemination of this message in whole or in part is
> strictly
> > prohibited. Please note that e-mails are susceptible to change. Banca
> del
> > Gottardo (including its group companies) shall not be liable for the
> > improper or incomplete transmission of the information contained in
> this
> > communication nor for any delay in its receipt or damage to your
> system.
> > Banca del Gottardo (or its group companies) does not guarantee that the
> > integrity of this communication has been maintained nor that
> this
> > communication is free of viruses, interceptions or interference.
> >
> >
>

############################################################################

> >
> >
> > -------------------------------------------------------
> > This sf.net email is sponsored by:ThinkGeek
> > Welcome to geek heaven.
> > http://thinkgeek.com/sf
> > _______________________________________________
> > Quantlib-users mailing list
> > [hidden email]
> > https://lists.sourceforge.net/lists/listinfo/quantlib-users
>

Xavier.Abulker

Re: RE:Delta on Capfloor

In reply to this post by Andre Louw-2

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 obtained
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
> > --
> > ############################### DISCLAIMER
> #################################
> >
> > This message (including any attachments) is confidential and

may
> be
> > privileged. If you have received it by mistake please notify the
sender
> by
> > return e-mail and delete this message from your system. Any
> unauthorised
> > use or dissemination of this message in whole or in part is
> strictly
> > prohibited. Please note that e-mails are susceptible to change.
Banca
> del
> > Gottardo (including its group companies) shall not be liable for
the
> > improper or incomplete transmission of the information contained in
> this
> > communication nor for any delay in its receipt or damage to your
> system.
> > Banca del Gottardo (or its group companies) does not guarantee that
the
> > integrity of this communication has been maintained nor that
> this
> > communication is free of viruses, interceptions or interference.
> >
> >
>
############################################################################

-------------------------------------------------------
This sf.net email is sponsored by:ThinkGeek
Welcome to geek heaven.
http://thinkgeek.com/sf
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Toyin Akin-4

Re: 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.

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

> 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
> > > --
> > > ############################### DISCLAIMER
> > #################################
> > >
> > > This message (including any attachments) is confidential and
> may
> > be
> > > privileged. If you have received it by mistake please notify the
> sender
> > by
> > > return e-mail and delete this message from your system. Any
> > unauthorised
> > > use or dissemination of this message in whole or in part is
> > strictly
> > > prohibited. Please note that e-mails are susceptible to change.
> Banca
> > del
> > > Gottardo (including its group companies) shall not be liable for
> the
> > > improper or incomplete transmission of the information contained in
> > this
> > > communication nor for any delay in its receipt or damage to your
> > system.
> > > Banca del Gottardo (or its group companies) does not guarantee that
> the
> > > integrity of this communication has been maintained nor that
> > this
> > > communication is free of viruses, interceptions or interference.
> > >
> > >
> >
>

############################################################################

>
> > >
> > >
> > > -------------------------------------------------------
> > > This sf.net email is sponsored by:ThinkGeek
> > > Welcome to geek heaven.
> > > http://thinkgeek.com/sf
> > > _______________________________________________
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> > > [hidden email]
> > > https://lists.sourceforge.net/lists/listinfo/quantlib-users
> >
>
>
>
> -------------------------------------------------------
> This sf.net email is sponsored by:ThinkGeek
> Welcome to geek heaven.
> http://thinkgeek.com/sf
> _______________________________________________
> Quantlib-users mailing list
> [hidden email]
> https://lists.sourceforge.net/lists/listinfo/quantlib-users
>
>
>
>
>
> *************************************************************************
> Ce message et toutes les pieces jointes (ci-apres le "message") sont
> confidentiels et etablis a l'intention exclusive de ses destinataires.
> Toute utilisation ou diffusion non autorisee est interdite.
> Tout message electronique est susceptible d'alteration.
> La Fimat et ses filiales declinent toute responsabilite au
> titre de ce message s'il a ete altere, deforme ou falsifie.
> ********
> This message and any attachments (the "message") are confidential and
> intended solely for the addressees.
> Any unauthorised use or dissemination is prohibited.
> E-mails are susceptible to alteration.
> Neither Fimat nor any of its subsidiaries or affiliates shall
> be liable for the message if altered, changed or falsified.
> *************************************************************************
>

Toyin Akin-4

Re: RE:Delta on Capfloor

In reply to this post by Andre Louw-2

Whoops

regarding this paragraph

> 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.

It's not 3*notional / 1BP, but 2.02*notional / 1BP.
Basically greater than 1.

Regards,
Toy.

----- 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: Monday, October 21, 2002 3:01 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.
>
> 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.

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

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
> > > > --
> > > > ############################### DISCLAIMER
> > > #################################
> > > >
> > > > This message (including any attachments) is confidential and
> > may
> > > be
> > > > privileged. If you have received it by mistake please notify the
> > sender
> > > by
> > > > return e-mail and delete this message from your system. Any
> > > unauthorised
> > > > use or dissemination of this message in whole or in part is
> > > strictly
> > > > prohibited. Please note that e-mails are susceptible to change.
> > Banca
> > > del
> > > > Gottardo (including its group companies) shall not be liable for
> > the
> > > > improper or incomplete transmission of the information contained

in
> > > this
> > > > communication nor for any delay in its receipt or damage to your
> > > system.
> > > > Banca del Gottardo (or its group companies) does not guarantee that
> > the
> > > > integrity of this communication has been maintained nor
that
> > > this
> > > > communication is free of viruses, interceptions or interference.
> > > >
> > > >
> > >
> >
>
############################################################################

> >
> > > >
> > > >
> > > > -------------------------------------------------------
> > > > This sf.net email is sponsored by:ThinkGeek
> > > > Welcome to geek heaven.
> > > > http://thinkgeek.com/sf
> > > > _______________________________________________
> > > > Quantlib-users mailing list
> > > > [hidden email]
> > > > https://lists.sourceforge.net/lists/listinfo/quantlib-users
> > >
> >
> >
> >
> > -------------------------------------------------------
> > This sf.net email is sponsored by:ThinkGeek
> > Welcome to geek heaven.
> > http://thinkgeek.com/sf
> > _______________________________________________
> > Quantlib-users mailing list
> > [hidden email]
> > https://lists.sourceforge.net/lists/listinfo/quantlib-users
> >
> >
> >
> >
> >
> >

*************************************************************************

> > Ce message et toutes les pieces jointes (ci-apres le "message") sont
> > confidentiels et etablis a l'intention exclusive de ses destinataires.
> > Toute utilisation ou diffusion non autorisee est interdite.
> > Tout message electronique est susceptible d'alteration.
> > La Fimat et ses filiales declinent toute responsabilite au
> > titre de ce message s'il a ete altere, deforme ou falsifie.
> > ********
> > This message and any attachments (the "message") are confidential and
> > intended solely for the addressees.
> > Any unauthorised use or dissemination is prohibited.
> > E-mails are susceptible to alteration.
> > Neither Fimat nor any of its subsidiaries or affiliates shall
> > be liable for the message if altered, changed or falsified.
> >

*************************************************************************
> >
>

Xavier.Abulker

Re: RE:Delta on Capfloor

In reply to this post by Andre Louw-2

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.

Toyin Akin-4

Re: 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

> > 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

that
> > > this
> > > > communication is free of viruses, interceptions or interference.
> > > >
> > > >
> > >
> >
>
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>
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> > >
> >
> >
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> >

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>

Toyin Akin-4

Re: RE:Delta on Capfloor

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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