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I am trying to work out a convexity adjustment function for my favorite trader. He is the kind of guy who likes to see things on a practical point of view.I have read the Pat Hagan "Convexity conundrums" article which seems fine to me. But I can not link the formulas to what the trader has on his mind.What the trader says is something like:"Let's say that you receive a CMS leg in a year. If you want to hedge that, you do a forward starting swap. But as time goes by, the duration (PV01, annuity or whatever you want to call it) of your swap changes. So you need swaptions! The idea is to say that if you can value your hedge, you can value the convexity correction.Let's say that the rates go up 10 bps... You need a payer swaption to hedge (the right notional being (1/newPV01 - 1/previousPV01) )You would need such an option if it goes up 20bps, 30bps , and so on...If the rates go down, you would have to do the same thing with a receiver swaption.The sum of all these swaptions is the convexity correction"I am not even sure that I have understood everything, but maybe that makes sense for one of you guys.When I take a closer look at the previous explanation it looks definitely like a discrete integral of swaptions ( just like Pat Hagan suggests)! But something seems to be wrong with the notional, or I have missed something.The mathematical answer of Pat is : I understand the math in Pat's article but can't put words on the f and the G functions.If somebody could help...Thanks.

Hi there,I am going to preach for myself here but then it makes you feel good sometimes: I have posted an intuitive explanation for the CMS convexity adjustement in a previous thread. You will probably find it by searching on my posts about CMS.Hope this will help.

...easiest way to explain this to him is to put a hypothetical trade on between you two, where you receive the fixed swap and the fixed-dv01 CMS....wait a couple of months and compare NPV's..... he'lll get the point real quick.....

QuoteOriginally posted by: ClopinetteHi there,I am going to preach for myself here but then it makes you feel good sometimes: I have posted an intuitive explanation for the CMS convexity adjustement in a previous thread. You will probably find it by searching on my posts about CMS.Hope this will help.I fail to find ur post u mentioned , can u give me the link? thx in advance

QuoteI am not even sure that I have understood everything, but maybe that makes sense for one of you guys.it is really easy. First choose (somehow discretized) rate curve scenarios w.r.t. which you build your hedge / replication portfolio (parallel movements, hull white scenarios ... as in pat hagans paper). Start with a zero payoff scenario corresponding to a swap rate k (e.g. for a cms caplet start with the scenario generating a swap rate equal to the strike of the caplet, a swaplet can be written as the difference between cap and floor and so on...). Move to the next scenario up corresponding to a swap rate k + d. Use a swaption with strike k to replicate the payoff in this scenario. Move up again one scenario and use a swaption with strike k + d to replicate the payoff (taking into account the first swaption, note that the second swaption does not destroy anything for the first scenario). And so on to some reasonable limit. Then do down scenarios if applicable. Then value the swaption basket on your modelled swaption smile.Pat Hagans formulas do the same in a limiting and more condensed form.

a question. is it correct, that in case of fixing = payment the exact scenario set applied does not matter, but only the set of corresponding swap rates? Which would mean that in this case the pricing is independent of the particular scenarios chosen and only bounds and discretization of the swap rates affect the pricing. Only if fixing < payment, P( fixing, payment) (for the timing adjustment) and the swap rate have an effect on pricing, so it matters, if e.g. only parallel movements are applied or hull white movements or whatever. On the other hand one could expect the pricing still stable w.r.t. the chosen scenarios, at least if payment and fixing are not too far away from each other. This would then mean, that this pricing model is (in these cases) very close to pure arbitrage pricing, isn't it, at least if one had swaption prices very far otm?another question concerning the last point. cms coupon prices are made by traders, so one could assume that "real" hedge portfolios are used, so that in particular there is a market standard which strikes (in which boundaries, with what distance) are used for hedging, and also perhaps standards for vol extrapolation and minimum prices for swaptions, if far otm ... making the market prices consistent with the replication method. ?

I think the scenarios and the 'model' used to generate the scenarios do influence the price / adjustment even when there is no timing adjustment as in your situation where fixing date = payment date. Scenarios - for a given 'model' and assuming the upper bound scenario is a sufficiently high swap rate that the corresponding replicating swaption premium is negligible, the price will change as the scenario discretisation is increased from coarse to fine, but will converge after say 50 to 100 scenarios. Model - I think the choice of model impacts the G function (to use Hagan's notation). This function in turn influences the required notional for each scenario swaption in the replication portfolio and hence the final price. Putting it another way, if you choose a flat yield ('bond math') type model versus a non-parallel shift Hull-White type model, for a given swap rate the corresponding implied annuity term, driven by bond prices implied form the chosen model, will differ. G is a function of the annuity term (and the timing adjustment bond price if there was one) and impacts the swaption replication portfolio notionals. That said, I think the difference is pretty small though.

 Hi Guys, I would like to know how did you calibrate the correlation in the Hull-White convexity adjustment form basically in the timing adjustment?
From my understanding one need to retrieve the forward swap rate ie ex (10Y1Y, 10Y2Y etc) and correlate its evolution with the forward curve like EURIBOR 6 month? Does that make sense? Is an historical method a good proxy like between 99 and 2003? Any recommandation would be highly appreciate! Thanks,