## quantlib-users

 Re: [Quantlib-users] RE:Delta on Capfloor From: - 2002-10-16 07:28:11 ```Hi Andre, I think you're mixed up saying that the swap sensitivty is equivalent to a cap sensitivity. the plain vanilla swap is equivalent to a bond, a notional exchange + fix flows. there's almost no convexity and the payoff is linear. Now I presume you know the shape of an option payoff: not linear at all! e.g 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 and not 1. now take a deep in the money cap, don't you find that the cap sensitivity tends to the swap sensitivity? Regards Xavier Andre Louw To: "QuantlibUsers (E-mail)" Sent by: cc: quantlib-users-admin@... Subject: [Quantlib-users] RE:Delta on Capfloor eforge.net 16/10/2002 07:50 Hi, Francesco wrote: >> I could not understand why you need to calculate Delta NUMERICALLY. Francesco, a bit of confusion, I actually meant analytically! >> In order to get the the cap delta starting from caplet >> deltas, you simply need to sum them up, since the cap is a sum of caplets... >> (and given the additive propriety of the derivative operator) Xavier wrote: >> from what I saw in the major european trading rooms on IR derivatives it's >> even simpler: >> the total delta is just the sum the delta for each caplet. You need to >> weight these deltas if the notional changes during the life of the cap ( >> amortization e.g) OK, I hear what you guys are saying, BUT: Firstly, I have the 'delta' as showing the ratio between the option's sensitivity to a 1 basis point move in the strike versus the underlying instrument's sensitivity to a 1 basis point move in the yield curve. Now back to capfloors - according to me the delta should show the ratio between the capfloor's total 01 sensitivity vs that of an equivalent swap? Alas, the sum of the caplet's delta is not event close to this! Have I got myself totally mixed up or what? Andre ------------------------------------------------------------------------- This e-mail is intended only for the use of the individual or entity named above and may contain information that is confidential and privileged, proprietary to the company and protected by law. If you are not the intended recipient, you are hereby notified that any dissemination, distribution or copying of this e-mail is strictly prohibited. Opinions, conclusions and other information in this message that do not relate to the official business of our company shall be understood as neither given nor endorsed by it. ------------------------------------------------------- This sf.net email is sponsored by: viaVerio will pay you up to \$1,000 for every account that you consolidate with us. http://ad.doubleclick.net/clk;4749864;7604308;v? http://www.viaverio.com/consolidator/osdn.cfm _______________________________________________ Quantlib-users mailing list Quantlib-users@... 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. ************************************************************************* ```
 RE: [Quantlib-users] RE:Delta on Capfloor From: Perissin Francesco - 2002-10-21 10:13:54 ```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. ############################################################################ ```
 Re: [Quantlib-users] RE:Delta on Capfloor From: Toyin Akin - 2002-10-21 10:54:29 ```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" To: "'Andre Louw'" ; "QuantlibUsers (E-mail)" 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 > _______________________________________________ > Quantlib-users mailing list > Quantlib-users@... > https://lists.sourceforge.net/lists/listinfo/quantlib-users ```
 Re: [Quantlib-users] RE:Delta on Capfloor From: Toyin Akin - 2002-10-21 11:13:46 ```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" To: "Perissin Francesco" ; "'Andre Louw'" ; "QuantlibUsers (E-mail)" 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" > To: "'Andre Louw'" ; "QuantlibUsers (E-mail)" > > 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 > > _______________________________________________ > > Quantlib-users mailing list > > Quantlib-users@... > > https://lists.sourceforge.net/lists/listinfo/quantlib-users > ```
 Re: [Quantlib-users] RE:Delta on Capfloor From: - 2002-10-21 12:27:32 ```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" To: "Perissin Francesco" , "'Andre Sent by: Louw'" , "QuantlibUsers \(E-mail\)" quantlib-users-admin@... 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" To: "Perissin Francesco" ; "'Andre Louw'" ; "QuantlibUsers (E-mail)" 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" > To: "'Andre Louw'" ; "QuantlibUsers (E-mail)" > > 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 > > _______________________________________________ > > Quantlib-users mailing list > > Quantlib-users@... > > 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 Quantlib-users@... 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. ************************************************************************* ```
 Re: [Quantlib-users] RE:Delta on Capfloor From: Toyin Akin - 2002-10-21 14:01:13 ```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: To: "Toyin Akin" Cc: "'Andre Louw'" ; "Perissin Francesco" ; "QuantlibUsers (E-mail)" ; 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" > To: "Perissin Francesco" , "'Andre > Sent by: Louw'" , "QuantlibUsers \(E-mail\)" > quantlib-users-admin@... > 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" > To: "Perissin Francesco" ; "'Andre Louw'" > ; "QuantlibUsers (E-mail)" > > 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" > > To: "'Andre Louw'" ; "QuantlibUsers (E-mail)" > > > > 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 > > > Quantlib-users@... > > > 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 > Quantlib-users@... > 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. > ************************************************************************* > ```
 Re: [Quantlib-users] RE:Delta on Capfloor From: Toyin Akin - 2002-10-21 14:10:50 ```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" To: Cc: "'Andre Louw'" ; "Perissin Francesco" ; "QuantlibUsers (E-mail)" ; 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. > The swap 1Y is a swap with 4 FRA's. > > Regards, > Toyin Akin. > > ----- Original Message ----- > From: > To: "Toyin Akin" > Cc: "'Andre Louw'" ; "Perissin Francesco" > ; "QuantlibUsers (E-mail)" > ; > > 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" > > To: > "Perissin Francesco" , "'Andre > > Sent by: Louw'" > , "QuantlibUsers \(E-mail\)" > > quantlib-users-admin@... > > > 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" > > To: "Perissin Francesco" ; "'Andre Louw'" > > ; "QuantlibUsers (E-mail)" > > > > 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" > > > To: "'Andre Louw'" ; "QuantlibUsers (E-mail)" > > > > > > 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 > > > > Quantlib-users@... > > > > 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 > > Quantlib-users@... > > 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. > > ************************************************************************* > > > ```
 Re: [Quantlib-users] RE:Delta on Capfloor From: - 2002-10-22 08:21:27 ```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" To: Sent by: cc: "'Andre Louw'" , "Perissin Francesco" quantlib-users-admin@... , "QuantlibUsers \(E-mail\)" eforge.net , 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: To: "Toyin Akin" Cc: "'Andre Louw'" ; "Perissin Francesco" ; "QuantlibUsers (E-mail)" ; 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" > To: "Perissin Francesco" , "'Andre > Sent by: Louw'" , "QuantlibUsers \(E-mail\)" > quantlib-users-admin@... > 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" > To: "Perissin Francesco" ; "'Andre Louw'" > ; "QuantlibUsers (E-mail)" > > 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" > > To: "'Andre Louw'" ; "QuantlibUsers (E-mail)" > > > > 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 > > > Quantlib-users@... > > > 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 > Quantlib-users@... > 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. > ************************************************************************* > ------------------------------------------------------- This sf.net email is sponsored by:ThinkGeek Welcome to geek heaven. http://thinkgeek.com/sf _______________________________________________ Quantlib-users mailing list Quantlib-users@... 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. ************************************************************************* ```
 Re: [Quantlib-users] RE:Delta on Capfloor From: Toyin Akin - 2002-10-22 08:48:23 ```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: To: "Toyin Akin" Cc: "'Andre Louw'" ; "Perissin Francesco" ; "QuantlibUsers (E-mail)" ; 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" > To: > Sent by: cc: "'Andre Louw'" , "Perissin Francesco" > quantlib-users-admin@... , "QuantlibUsers \(E-mail\)" > eforge.net , > > 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: > To: "Toyin Akin" > Cc: "'Andre Louw'" ; "Perissin Francesco" > ; "QuantlibUsers (E-mail)" > ; > > 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" > > To: > "Perissin Francesco" , "'Andre > > Sent by: Louw'" > , "QuantlibUsers \(E-mail\)" > > quantlib-users-admin@... > > > 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" > > To: "Perissin Francesco" ; "'Andre > Louw'" > > ; "QuantlibUsers (E-mail)" > > > > 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" > > > To: "'Andre Louw'" ; "QuantlibUsers (E-mail)" > > > > > > 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 > > > > Quantlib-users@... > > > > 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 > > Quantlib-users@... > > 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. > > ************************************************************************* > > > > > > ------------------------------------------------------- > This sf.net email is sponsored by:ThinkGeek > Welcome to geek heaven. > http://thinkgeek.com/sf > _______________________________________________ > Quantlib-users mailing list > Quantlib-users@... > 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. > ************************************************************************* > ```
 Re: [Quantlib-users] RE:Delta on Capfloor From: Toyin Akin - 2002-10-22 10:03:58 ```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" To: Cc: "'Andre Louw'" ; "Perissin Francesco" ; "QuantlibUsers (E-mail)" ; 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: > To: "Toyin Akin" > Cc: "'Andre Louw'" ; "Perissin Francesco" > ; "QuantlibUsers (E-mail)" > ; > > 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" > > To: > > > Sent by: cc: "'Andre > Louw'" , "Perissin Francesco" > > quantlib-users-admin@... > , "QuantlibUsers \(E-mail\)" > > eforge.net > , > > > > > 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: > > To: "Toyin Akin" > > Cc: "'Andre Louw'" ; "Perissin Francesco" > > ; "QuantlibUsers (E-mail)" > > ; > > > > 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" > > > To: > > "Perissin Francesco" , "'Andre > > > Sent by: Louw'" > > , "QuantlibUsers \(E-mail\)" > > > quantlib-users-admin@... > > > > > 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" > > > To: "Perissin Francesco" ; "'Andre > > Louw'" > > > ; "QuantlibUsers (E-mail)" > > > > > > 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" > > > > To: "'Andre Louw'" ; "QuantlibUsers (E-mail)" > > > > > > > > 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. 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