# Difference Between FRA and IR Future pricing [closed]

[original question was to know the difference between IR fut and FRA] Turns out that FRA and IR Futures are just the OTC and Futures counterpart of the same underlying, that is, the interest rate. However, the 2 differ in terms of pricing, as will any 2 contracts with exactly same terms except their settlement(daily MTM vs payment at maturity). So, my question is, is there a way to quantify this difference?

The key differences are:

• Futures are traded on exchanges and settled via mark-to-market (MTM) margin accounts with the exchange. FRAs are over-the-counter (OTC) products with collateral exchanges based on the MTM of the trade and subject to the credit support annex (CSA) agreement between the bilateral counterparties
• FRAs can be written as any tenor out of any date, e.g. 3x9 14th is an FRA that starts on the 14th day of the 3rd month from today and runs for a tenor of 6months (9 minus 3). IR futures have specific settlement dates that corresponding to IMM dates, which are defined as the third wednesday in a month, and usually always only run for a 3M tenor.
• The prices of an IR future is quoted in price terms as 100 - rate, whereas an FRA is quoted in rate terms.
• The implied rate for FRAs and IR futures (assuming you align settlement dates) are broadly the same and never differ by a few basis points. The difference is due to the products having different gammas. IR futures always have the same amount of risk per contract regardless of the rate, so have no gamma. FRAs have gamma so for the same notional the risk is larger for lower rates. Since gamma is valuable FRAs are usually over-received and IR futures are usually oversold, hence the small difference in their implied rates - FRA rates are generally lower than implied futures rates.
• @ attack68 your last paragraph is wrong. The cash that is posted on the collateralized FRA receives interest (usually Fed Funds) whereas the futures cash flows receive no interest. The convexity adjustment is a real economic effect. – dm63 Jul 2 '18 at 11:22
• @dm63 happy to remove the para until I can better formulate my intentions. In your scenario when you receive interest on the FRA collateral, the MTM value of the FRA will worsen overnight as the settlement is now one day closer in discounting terms, so I don't understand the notion of economic gain here. Note the delta of the FRA changes o/n due to increased discounting and therefore to maintain delta neutrality overnight you have to reduce the FRA position, so if the price returns to the executed level the your earned interest is negated by the small loss on the rehedge. This ignores gamma. – Attack68 Jul 2 '18 at 16:10
• Also note that Eurex and LCH net margin accounts across products, so IR futures, FRAs and IRSs are all grouped together and interest paid on those margin account balances is OIS. I suspect we both understand the same concept and are just discussing semantics. I have always valued convexity in terms of gamma. – Attack68 Jul 2 '18 at 16:13

An IR futures is a futures contract. Therefore it is exchange traded and PnL is reflected every day on the margin account.

With futures you usually have the expectation under the risk neutral measure for pricing: $$f_{t,T} = E[r_T|F_t].$$ You can calculate this using a properly calibrated interest rate model and Monte Carlo e.g.

The FRA is as far as I know OTC and PnL is exchanged in the end of the period. The mark-to-market price is the corresponding foward rate.

Both prices are usually close.

In the case of IR futures you can reinvest gains (as you get paid during before maturity), furthermore you get paid during the trading time of the futures and not just in the end. Therefore, you have an adjustment to the forward rate - often called the convexitiy adjustment (which you can calculate explicitely in some IR-models) - search for "Convexity Adjustments to Eurodollar Futures".

In general, the convexity adjustment represents the difference between the forward and the futures price.

• I don't understand how you relate convexity to reinvested gains and also dm63s suggestion of OIS interest. In a negative interest rate environment such as EUR, are you suggesting that convexity value would be opposite to that in a positive rate environment because the reinvestment yields negative interest? This certainly isn't true in practice and is why I expressed convexity valuation in terms of gamma and delta changes rather than any form of cash. But I am very keen on understanding other intuitive reasoning (if its correct). – Attack68 Jul 3 '18 at 9:36
• as a quick answer: the conxexity adjustment certainly is more. All in all you have to calculte the expectation to get it right and then you can look at the difference to the FRA. of cours you don't gain always ...I should clarify this – Ric Jul 3 '18 at 9:41