This is (I think) a really basic problem, but I'd be grateful if someone could explain it to me Ladybird book fashion, or point me at a more appropriate (simpler) forum .I've always thought of the convexity bias in Eurodollar futures as being related to the fact that EDs are effectively linear; no matter whether the underlying rate is 1% or 10%, a one basis point change generates a profit or loss of USD 25. FRAs by contrast are discounted instruments so the final payoff is linked to the underlying rate. So if you sell an FRA (long the market) and sell an ED (short the market) you should be able to benefit from the convexity. This seems to me to have nothing to do with margin - and I'm obviously wrong but I don't know why. I put together a spreadsheet where I sold a three-month FRA with a PV01 of USD 25, and then sold an ED. Messed around with possible final settlement prices and (obviously?) the strategy is always profitable with profits increasing with rate changes. So far so simple (?). What has this got to do with margin? I vaguely recall (I'm getting old) Hull etc. going on about funding losses at high rates and investing profits at low rates, but this doesn't seem to affect the underlying issue.Lots of articles from time to time pointing out that as FRAs are margined, the convexity bias is disappearing - but why? The profits from the FRA/ED arb would appear to have nothing to do with margin.I am fairly sure I'm being incredibly stupid but I be grateful if somebody could (gently) point out my error.Fatman

The problem is partially one of language. You are correct in observing that there should be a (usually very small) convexity adjustment for the linear settlement of ED futures at maturity. The bulk of the textbook "convexity adjustment" is really properly described as a covariance adjustment. If you go back to Fischer Black's '76 original argument for why a futures price equals the corresponding forward price, you will see that he requires the assumption of deterministic interest rates. When trying to relax that assumption, you find that the (risk neutral) covariance of the daily P/L on the futures contract and the overnight interest rate enters into the equation determining the futures price. The effect is a bit subtle, since the futures price is recursively defined (today's futures price is set so that the PV of tomorrow's P/L is zero).

I'm incredibly grateful for your reply and I apologise if I have misunderstood. Are there in fact two adjustments? One for the linear/non-linear mismatch, and then one (the 'textbook adjustment'(?)) for the correlation between interest rate changes and margining? Or is it that I don't really understand covariance? (I am unfamiliar with Fischer Black's 76 argument although I think there is a similar one in Hull in an appendix that I ignored). Most of the literature that I have read looks at either one or the other, and seems to conflate them. One (admittedly badly written) blog has the linear/non-linear difference at the root of the problem but then says that it is "monetarized" (sic) by the margin. Of course I should not read blogs but I am happy to be confused by anybody (although I generally find this forum to be happily accurate). Surely the convexity exists even if there is no margin at all? Why (if at all) does adding margin eliminate it? I can see that arbitrage between the two might be impracticable due to needing to post margin in two systems, but it would still make sense for someone looking to buy the market to sell an FRA rather than buy a future.Or would it?

There are really two separate effects. The one you have focused on is real, but just too tiny to worry much about (unlike the similar phenomenon in CMS swaps, where it can be highly material). If you wanted to arb the covariance effect in ED futures, the proper place to look would be to trade them against swaps. This usually takes the form of a "self-arb", where a system that generates yield curves from unadjusted (or improperly adjusted) ED futures prices generates day zero profits from executing swaps at market levels. There are probably people hanging around at this forum who saw this first hand at the New York office of a certain Swiss bank back in the 90's. It doesn't work so well with centrally cleared swaps, though...

Thank you again for the input.My conclusions (please correct if wrong)."Convexity bias" refers to two seperate, unconnected, issues; the linear/non-linear mismatch and the correlation between the forward rate and the margin funding rate (which you call the "covariance effect").The linear/non-linear mismatch is significant in CMS but less so in ordinary swaps, so that an arbitrage is impractical.The margin funding bias is the one normally referred to in the literature. The statements that connect the two elements of bias (and there are lots of them) are incorrect.The emergence of margining for OTC products (such as FRAs) has eliminated the margin bias between forwards and futures. (I would love someone to give me a numerical example of the original bias; despite statements that make superficial sense (borrowing at lower levels, lending at higher levels), I've yet to see a worked example).There is a lot of confusion (not least mine) around what is going on here.The HTML input for this forum is garbage. Is WYSIWIG too proletarian for exotic quants? Fatman

These effects are not entirely unconnected, IMHO.For an in-depth discussion of the subject, as well as an estimate of the effect of these adjustments when they used to exist, I'd refer you to Galen Burghardt's "Eurodollar Futures and Options Handbook". There's a lot of detail in there (although some of it is somewhat dated). Furthermore, there have been somewhat curious, and significant, recent developments in the world of futures convexity.

Thanks for this. I don't have the book and am trying to put something together in the next 24 hrs (I'm an ex-trader turned financial writer). Can you point me to more accessible stuff (or web excerpts)?

If you send me a PM with your email address, I can send you the Burghardt & Hoskins paper from 1994, which is, basically, where the whole thing started.