Why Interest Rate Futures and FRAs Have Different Implied Rates

# Difference Between FRA and IR Future pricing


# Difference Between FRA and IR Future pricing












[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?

## Answer by Attack68 (score 2)

https://quant.stackexchange.com/a/40582

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.

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/40580

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.