s: contact me by email and I can get you a paper with a pretty good method (i think its the best, but what do I know?)

QuoteOriginally posted by: gcThe article by Adams made me think that a better solution would have been instead of generating a instantaneous forward curve that is smooth and that at the same time matches the observed values for the swaps. The only problem, is that since the instantaneous forward rates are not observable, and since Adams suggests using quartic splines (that depend globally on all the points), I was left without clear idea on how to replace the bootstrapping with an alternative algorithm...Adams himself shows how to retrieve the zero rates from a splied instantaneous forward curve, and claims that it returns good swap rates, but in the article I have doesn't give any hint on how to compute the instantaneous forward rates in first place ....gcOk, I will try to be clearer this time: if the forward is (locally) a quartic, its integral is a quintic i.e. is locally a quintic. Hence is of the form Requiring continuity of f and its first three deivatives we get 6n-10 equations in 6n-6 unknowns. But these are equations in the coefficients and the rates. The forwards have 'disappeared'. We then fill up the last 4 degrees of freedom with your choice of 4 from (and I am sure one can come up with some other possibilities).The first three are Adams' choices. Thus, he leaves one degree unresolved. This you can see towards the end of his paper: if you know one of the rates (r_1 is his example) you get the rest.If there is a dense set of known or bootstrapped rates, this is probably quite a good method, as it makes the forward very smooth, and amongst other things this is great for interest rate models. However, if the set of known nodes is even vaguely sparse (such as in an emerging market such as my own) then it just crashes badly. There is nothing in this method (or many other 'pure mathematical' methods) that ensure financial fundamentals are preserved: for example, that the function r(t) t is increasing (which it must be, as the exponent of this is the capitalisation function).

Does anyone have the corrected version of"Fitting yield curves and forward rate curves with maximum smoothness" by Adams, Kenneth J. and Donald R. Van Deventerand "Estimation of the Yield Curve and the Forward Rate Curve Starting from a Finite Number of Observations" by Delbaen, F and Sabine Lorimier.Trying to implement the ideas in the paper by KainGuan Lim titled " Computing Maximum Smoothness Forward Rate Curves"Thanks

a lot of different aspects of the topic have been explored here, but i assure you that a 4th order ("quartic spline") is the way to go as far as interpolation methodology.... in practice, if you only use cubic (i.e. smooth spot rates) then hedge funds will relentlessly pick you off by exploiting the non-smoothness of your forward rates....

Hi All,Great thread! I've looked on related threads and am still having problems with short end interp. If someone could provide input, it would be greatly appreciated. here it goes.It involves interpolation of cash rates with the stripped out forwards in eurodollar futures. This is my process... please identify any flaws:1. Obtain cash rates (ON, 1M, 2M, 3M) and futures (out two 2 years or red/mid-greens).2. convert to continuous rates using (#period*ln(1+r/#period))3. adjust for convexity using HULL's approximation (inputs: Bloomberg's mean rate and vol) and subtracting this value from each continous futures rate.4. Linearly interpolating the rate between cash rates for the the start date of the first futures contract and apply the forward rate formula to determine the rate 3months hence and so on for the remaing futures.My problem is that i fail to meet the no-arbitrage relationship between the 2 year rate derived from stripping out futures and the 2 year swap rate. I am off by several bps. Any help is greatly appreciated.

Problem solved... the tool... simple reflection.

Has anyone had any success implementing a constrained optimization approach whereby we minimize the sum of squared jumps in the forward curve while constraining the solution to satisfy the bootstrapping constraints? It seems that this nonlinear optimization problem should work well and provide a good solution, but my initial attempts to impliment it have not converged. I think it's just a matter of having a robust optimization package to use, and I imagine that anyone with the Matlab optimization toolbox could make it work fairly easily using a quasi-Newton method or something of the sort. Any thoughts?

bigslick - dunno how you ended up resolving the convexity-adjusted ed strip vs the mkt swap rates but i can tell you that from a trading perspective this issue is a constant thorn in the side.... no matter how generally consistent your ed convexity adjustment algorithm, you will find that when the market is moving (esp when it is moving really fast!) that your "strip" generated rate will not match the market.... this is generally due to the fact that 2y swaps tend to trade off of 2y trsys + the 2y swap spread, and the liquidit in ed futures has deteriorated horribly, so often in fast markets the swap rate is trading "ahead" of the strip... also from a liquidity perspective it is nearly impossible to move a large bundle of ed futures, whereas 2y swap will be trading in massive size in 1/4 increments....the only reason that i think that dealer even use ed futures beyond fronts in their curve is because the 3m frequency of the contract provides a lot more accurate granularity w/r/t rate fluctuations in the future.... in other words, fed movements in the short rate are aggresively predicted for the first two years of the ed curve, and this information is necessary - otherwise you would be splining or quartic-ing between 1y and 2y swap rates, and you would not be caputuring the the proper rate expectation shape in-between.... when i was dealing swaps i literally had my curve reverse-solving a ed pack bundle adjustment to match to the market swap rates.... this way i would reprice the par market rates while preserving the information in the ed curve... a bit clugy but it seemed to work well...."

gc,If you're still looking for a simple way to replace your log-linear interpolation method... try the following.Interpolate the par-rates (i.e. the swap rates) to the required cashflow dates. Then bootstrap the zero-rates from these synthetic rates.Linear interpolation gives a "sawtooth" forward rate, so use a spline instead.

QuoteOriginally posted by: johnself11a lot of different aspects of the topic have been explored here, but i assure you that a 4th order ("quartic spline") is the way to go as far as interpolation methodology.... in practice, if you only use cubic (i.e. smooth spot rates) then hedge funds will relentlessly pick you off by exploiting the non-smoothness of your forward rates....From my experience, I have seen a lot documented on cubic splines but less on quartics. The advantage of the former is that the problem reduces to a tridiagonal system. I have not worked it out but I expect QS to lead to a larger banded matrix, hence trickier in all respects. At some stage round-off error may play a role.The question of the boundary conditions is a delicate issue.What about NON-POLYNOMIAL/nonlinear approx? exponential, rational functions. These are sometimes better in approximtaing certain kinds of functions. What I have missed is: What are the smoothness properties of the original curve that is to be interpolated upon: C0, C1, C2 etc.

> Adams suggests using quartic splines (that depend globally on all the points), gc,This phenomenon is called polynomial snaking: As the degree increases the polynomial spreadsover more sub-intervals. Polynomials can be unpredictable creatures at times.Some functions are NOT well approximated by polynomals about are well approximated by rational functions, i.e. quotients of two polys (see NR page 114).I do not know what the type of function you use ...

to johnself11,i found the same result... during periods of increased volatility the 2yr ed strip tends to lag movement in the 2yr swap rate (most likely for the reasons that you mentioned with ed liquidity being the major reason for the lag). the solution i came up with is remarkably similar to yours... reprice the ed contracts so that the difference between the ed strip and the 2yr swap rate is arb'd out. during less volatile periods the relationship holds true, but of course the day i first tried modelling it, the caveat was in play. i've inquired with other swappers and the use of quartic has never come up... it probably should, but my take is the bid/ask is pretty wide when you get that far out on the swap curve and it seems to compensate the dealer for the additional uncertainty (supply/demand seem to dominate that far out the curve) associated with forward rate.. but what do i know i'm just a newbie swapper.question, do you know of any papers or examples that display an application of the splining methods i.e. quartic spline method.thx,pd

bigslick - cool that we both came up with the same ED "bumping" process.... i know a lot of traders who still trade blindly off the strip and get clipped for 1/4 bps left and right.... as to the quartic spline issue, i agree that the bid/offer in the long-dated interpolated rates SHOULD compensate for interpolation uncertainty but i am afraid this is not the case... just take a look at Barclay's Barx swap trading screen (i think it is BARX <GO> on BBG).... they are making, say, 17y rates less than 1bps wide for 50mm-75mm up! moreover, the hedge funds will magnify this ridiculously small bid/offer by asking you to deal the 1y rate 16y forward if they think your interp method is off... the leverage of this forward rate to the 17y spot rate is astronomical, and this is why you need to not only make sure that your spot rates are smooth (cubic) but that your forward rates are as well (quartic).... i agree not all dealers are yet doing this but the money bleed from the hedgies is causing them to wise-up quickly.......oh and sorry i don't know of any such papers to recommend.....

Hi,I am also actually looking around qualitative issues to decide which, between different interpolation methods I hold, is the better. I basically get several methods, as linear interpolation on ZC rates, to polynomial forms of instantaneous forward rates. The only criterium I used to check out till now, is the smoothness criterium for (both spot & forward) market curves. As some method that I implement are relying on pretty elaborated numerical methods, I should now be interested in comparing the set of methods with regard to a stability issue i.e. stability of pricing and hedging wrt different market configurations, i.e. randomly-generated market curves’ shape 1: should this be a a suitable analysis/methodology?2: What kind of IRD product should emphasize, if any, impacts of the stripping mode on the (delta, gamma) Greeks?Note to experts: help consisting in pointing on a quite simple to implement payoff / interest rate model for MC simulations, should be much appreciated!Regards;