Hi all!Just to put stuff into perspective - I am a swap trader not a quant so my mathematical abilities are reasonably limited (no offence to any PhD swap traders out there)..I have a par swap curve. Today my friendly LDI guy calls me and asks me to price him a Zero coupon IRS - as they do.I price and send, and he chuckles - tells me Im way off (this has happened 3x now)..I am starting to think that there is something missing from my price. Is there some kind of convexity that I am not adjusting for? I do know that prices for zero's have gotten increasingly wide due to the LIBOR/OIS issues and the fact that the LDI market is one way etc.. but that is not the point of this question. He was asking me for where I see mid, so the fact that people are skewing their prices is irrelevant..So anything you guys can think of?[Assume that my price is directly off a bootstrapped curve with SONIA, on the correct basis, assume GBP two way collateral etc].

Why don't you ask your friendly LDI guy to get an idea of what other people are doing to arrive at their pricing?If I had to guess, it's either convexity or something to do with the use of the appropriate discount curve.

He's friendly - not that friendly and I have a sneaking suspicion that he has no idea/cares as to how people are arriving at their prices, but it seems that I am an outlier (so assumedly wrong).It could potentially be discount curves, but AFAIK these blokes have a 2way GBP Cash CSA (0 threshold) with everyone on the street.So I am guessing it is convexity that I am missing. The only thing is I am unsure how this convexity arises..i.e. is it something like CMS convexity which I need to add on (and never have been)?Appreciate the question is pretty vague but any pointers would be appreciated.

You mention GBP cash collateral and I'm assuming that the swap itself is also in GBP. You use a multi-curve / collateral discounting framework with one discounting and one "forwarding" curve and a spread between them. The zero-coupon you are pricing has a fixed leg and, I guess, a floating leg which is Libor compounded over the swap life, for example 10 GBP-LIBOR-6M compounded over 5 years. To price the floating payment, what most people do is to compute the LIBOR-6M forwards, compose them - product (1 + \delta_i L_i) - and discount them. This way to proceed is correct only under quite strong hypothesis on your curves, for example that the spread between the OIS and LIBOR curves is constant through time (i.e pre 2007 behavior). This hypothesis is not very important if there is only one composition, for example on the LIBOR-3M leg of a USD basis swap LIBOR-3M v LIBOR-6M where two 3M are compounded into 6M (with FLAT compounding type for the spread, but this is another story). When you go to zero-coupons where there is more than a composition of 2 rates but maybe 20 over 10 years, then the impact maybe bigger.If I can refer to my book (I don't know any other reference about this question, but would be happy to add another reference if someone can provide one) "Interest Rate Modelling in the Multi-Curve Framework", this is discussed in Section 6.3. The hypothesis that simplifies the computation as described above is the hypothesis called SO^CPN in the book (Section 2.5, page 23). Note that the same type of correction should be applied to FRA with settlement method "FRA discounting" (ISDA convention name). For the FRA, I remember only one computation on the impact in a paper by F. Mercurio.To model the actual convexity adjustment, you need a term structure model with the dynamic of the spread between the two curves. I have never seen (or heard about) any results on the computation of that adjustment, but would be happy to read about it someone has done the analysis.

Thanks for the reply Marc.Yes these are GBP swaps with GBP cash collateral and no threshold in the CSA, so pretty standard from a margining perspective. I would normally do as you say and obviously my forward curve would be the 6m GBP LIBOR curve for the floating leg in the case that he wanted a swap vs 6s and SONIA would be my discounting curve.From what you said (I haven't got your book unfortunately) is that the convexity is driven by OIS-LIBOR spread and that needs a model to be computed? I agree that this is a possible source of convexity BUT then I would also say a plain vanilla IRS has a similar issue.If I pay and mkt sells off 100bp, I will be up a lot of money, my margin account receives a lot of cash but then I am paying a higher rate of interest on this cash too which in itself is a -ve convexity in a way, but I don't naturally adjust (at least knowingly) for this convexity that I have effectively sold anywhere, and I hope my 10y swap prices are not off too!maybe I am barking up the wrong tree here but appreciate the input and any comments..!

going back to this..Lets keep things simple and assume that my forwarding curve is the same as my discounting curve (so pre-OIS-iBOR basis setup).If i set up a portfolio of a zero coupon swap ,assume i pay fixed 100m 30y @ x bps and hedge this statically with a 30y PAR swap @ y bps, DV01 neutral. My portfolio is net short convexity, correct?Therefore, wouldn't i have to make a convexity correction to the zero's price at the outset i.e. I would need to be compensated for selling this convexity and therefore would not pay y bps and instead something lower? I understand that the standard IRS is positively convex and that convexity is already in the rate 'x' which I receive, but my issue is that I am now hedging a different product with my standard swap with a different convexity. Therefore, my payoff from the portfolio will depend on volatility, isnt this issue quite similar to hedging CMS with a forward swap?I might be talking absolute @E$$ as I have never seen anything written about this and therefore assume I am missing something pretty basic in my reasoning. EDIT: my definition of a ZC swap is where the fixed leg is compounded up and paid at maturity and likewise with the ffunding/floating leg

I have no solution, but what I always saw with long dated zerocoupon bonds, was that they were impossible to be priced with simple discounting using the issuer's curve. It was due to long dated exposure on a single rate. I think it is also observable on US Treasuries and STRIPS, where value of principal only part does not sum with interest part to value of original treasury. I am also very much interested in someone posting solution to those problems.

Right, think I made some progress (but again would ideally need some confirmation)..In the simple 1-curve case, the portfolio I described in my last post carries+rolls positively therefore compensating for the short vol position we are getting thrown into. Makes sense as the GBP curve at least is pretty flat at the long end, so from the PAR swap which I receive I carry positively (roll doesn't contribute much) and the Zero which has no cash flows till expiry has no carry and very little roll down. @Pimpel could this be your issue too?

QuoteOriginally posted by: DocTocgoing back to this..Lets keep things simple and assume that my forwarding curve is the same as my discounting curve (so pre-OIS-iBOR basis setup).EDIT: my definition of a ZC swap is where the fixed leg is compounded up and paid at maturity and likewise with the ffunding/floating legSo is it compounded as in OIS, or averaged as in case of FF swap?Quote@Pimpel could this be your issue too? My issues are rather tiny as I don't have such 30y exposures, therefore quite difficult to observe/ bifurcate from other problems.