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### CMS replication

Posted: **December 20th, 2010, 8:51 pm**

by **DocToc**

Hi all,Was just wandering if some one could out line cms Swap replication. I noticed that online most people go on to discuss CMS Cap/Floor replication - no one really stops to explain the CMS swap replication. It might be blaringly obvious but I seem to be missing it.So say I have a 2y swap paying 10y CMS and receiving 6m EURIBOR, how do I replicate this using forward swaps (I understand the convexity adjustment needs to be handled with swaptions), more interested in using the forward swaps (so hedging the CMS payment without the convexity correction).Its probably a dim question but I seem to be missing something in my understanding.Thanks,Doc

### CMS replication

Posted: **December 20th, 2010, 9:09 pm**

by **rot13**

Can't you use a strip of forward 10Y swaps directly?

### CMS replication

Posted: **December 20th, 2010, 11:07 pm**

by **katastrofa**

Isn't the replication argument derailed by counterparty risk? I mean: in theory I can replicate an exotic option with a strip of vanilla options, but in practice I will not get all vanillas from the same counterparty, so my CVA is different and the PVs of the exotic and vanilla portfolio are no longer equal, so the replication is not 100%. How big error do I make? for long maturity trades, the probability of default can be even 10%. Does it mean my error could be of the same order as well?

### CMS replication

Posted: **December 22nd, 2010, 10:25 am**

by **prodiptag**

If you already know cms cap and floor replication, swap should be relatively straight forward - simply use the put call parity. the swap thus manufactured using a caps and floors will give you the instruments you need to hedge (mainly fwd swaps plus series of swaptions strangles)

### CMS replication

Posted: **December 22nd, 2010, 5:02 pm**

by **piterbarg**

QuoteOriginally posted by: katastrofaIsn't the replication argument derailed by counterparty risk? I mean: in theory I can replicate an exotic option with a strip of vanilla options, but in practice I will not get all vanillas from the same counterparty, so my CVA is different and the PVs of the exotic and vanilla portfolio are no longer equal, so the replication is not 100%. How big error do I make? for long maturity trades, the probability of default can be even 10%. Does it mean my error could be of the same order as well?a dealer would hedge cms using options bought from other dealers and as such, collateralized. so no CVA on the hedge

### CMS replication

Posted: **December 22nd, 2010, 6:33 pm**

by **DocToc**

Thanks for your replies everyone,@Prodiptag - I dont understand the forward swap part. Could you outline the replication strategy in a bit of detail as I seem to be missing something there. I understand that you need the swaptions as there is a volatility component from the convexity adjustment that you need to make. How do the forward swaps fit in exactly..Assume we have an n year cms swap on 10y CMS (paying CMS annually) for the i th payment:I would hedge this by using an i-1 year 10y forward swap (along with some swaptions to take care of the convexity)..BUT, what about all the future payments of this hedge..Pls let me know if I am being unclear..I am not really using this to dynamically replicate anything I want to understand how the risk is bucketed on the curve//DocToc

### CMS replication

Posted: **December 22nd, 2010, 9:06 pm**

by **katastrofa**

@piterbargThanks, I forgot about collateralization

### CMS replication

Posted: **December 23rd, 2010, 6:36 am**

by **prodiptag**

take an example of a n year swap of CMS10y against a fixed rate, say K. now pick out a single swaplet, say the 5th year payout -> this can be decomposed in to a caplet and a floorlet on CMS10y, both struck at K. Assuming you are familiar with cms caps and floors replications, you can figure out that to hedge the caplet you will need a series of 5y10y payer swaptions, starting and K and higher strikes. similarly to hedge the floorlet, you need 5y10y receivers starting at K and down below. the resultsant basket is a 5y10y forward swap at K and a series of 5y10y swaptions collars at other strikes (plus a K payout on the other leg). sum it over all the swaplets and you get the entire hedge you need.

### CMS replication

Posted: **December 23rd, 2010, 7:23 pm**

by **DocToc**

prodiptag - that was great! So CMS swaps must have some reasonably large skew exposure right?

### CMS replication

Posted: **December 24th, 2010, 5:02 am**

by **prodiptag**

sorry, my typo misled you, it is NOT "swaptions collars" but "swaptions strangles" (see my 1st post). so not much skew exposure, but reasonably high vol of vol exposure ("convexity" exposure in other words)

### CMS replication

Posted: **December 24th, 2010, 6:20 pm**

by **DocToc**

if you want to make the cms swap then isn't cms swap = cms cap - cms floor so you are long the cap short floor. so you would buy payer and sell receivers thereby giving you a collar..? I might be missing something but this is really helpful on the whole..

### CMS replication

Posted: **December 25th, 2010, 6:32 am**

by **prodiptag**

yes, you are long the cap, short the floor -> that means long payers at higher strikes and also long receivers at lower strikes. You can revisit the replications papers you have seen, to convince yourself. else intuitively, on the cap side when swap rates goes up, dv01 goes down, so you BUY swaptions for a LONG cap position. on the floor side, when swap rates goes down dv01 goes up, so you SELL swaptions for a LONG floor position

### CMS replication

Posted: **December 27th, 2010, 8:15 pm**

by **DocToc**

prodiptag, to replicate the long cms floor you'd still buy atm recievers and sell further otm recievers right? this is what i gather from cms floor replication. let me know if this is incorrect.

### CMS replication

Posted: **December 28th, 2010, 4:50 am**

by **prodiptag**

yes, long floor at strike K -> long rcvr at K + short rcvrs at strikes below.

### CMS replication

Posted: **December 28th, 2010, 8:40 pm**

by **DocToc**

Thanks Propditag - I think I get it now. So just a few questions for confirmation:(1). Say I am long a CMS cap at K, the caplets all have different atm's obviously - at the moment (although I haven't checked) say we are considering 10Y CMS on a 5year cap. This is likely to have quite a small gamma relative to where it was in the past. I assume that the close to expiry caplets have K(i) << K, these caplets would be the ones which would have the most gamma if we the 10Y forward swap was hovering near their strike right? And even if the 10Y forwards were quite high for longer periods these wouldn't necessarily have much gamma?Just want to see if I understand this to some level..Let me know what you think.THanks,DocToc