| Математика
Key words: FRA, EuroDollars, convexity, interest rates, margining, interest rate volatility
Futures and forward convexity adjustment

Are futures and forwards the same derivative instruments? The answer can be “Yes” or “No”.

1. We want to participate in interest rate change over the time. If so, we can use futures(for example, Eurodollars traded in the International money Market pit of Chicago Mercantile Exchange, CME or simply Merc) or forwards taking into account that futures are exchanged traded, standardized instruments and forwards are OTC (over-the-counter), custom contracts. If we knew interest rates would raise we would buy FRA or sell Eurodollar (ED) futures. Therefore they have the same function, i.e. give possibility to participate in interest rates fluctuations and to lock an interest rate payment. Intuitively, they are the same.

2. As long as FRAs are not cleared or aren’t under ISDA/CSA agreement with daily margining, forwards are not linear instruments. In contrast, futures are not exactly forward rate contracts, as they are linear derivatives and daily settled. Therefore the answer could be “No”.

To explain why they are different, we need to start from convexity. This is a desirable property in a bond, meaning that if yields fall, prices rise at an accelerating rate and when yields rise, prices fall at a decelerating rate. This is known as positive convexity.

Based on the same forward rate, Eurodollars have daily margining and Profit and Loss (PnL) is assumed to be calculated on a daily basis while PnL for FRAs contracts is paid at the final settlement day. We can conclude that the PnL of the future is calculated without discounting and PnL of the forward is discounted value. In other words, FRAs and EDs payoff respond differently to interest rate volatility.

Taking into account the fact that one basis point of EDs is defined to be worth 25 USD at all times, the value of EDs contract changes linearly while by contrast, FRAs have a convex shape. FRAs holders enjoy the benefit of being owners of positive convexity, ED holders don’t. Therefore the implied rate in EDs is higher than the ”true” forward rate and this is known as future/forward convexity adjustment:

ED rate = “true” FRA forward rate + convexity adjustment

Convexity adjustment depends on the volatility of the forward rates, time to maturity t and T (equal t+3 months):

Convexity adjustment = (volatility*volatility)/2 * t *T

It is also possible to explain convexity adjustment in another way. We know ED contract is a margin contract. So we deposit only a small amount of money in order to open ED contract. On a daily basis, the account will be deposited or charged by PnL amount. If rates go down, the price goes up thus positive PnL can be withdrawn from the margin account and invested at lower rates. If rates go up, the price goes down and instead of investing money at higher rates, in order to keep the contact opened, we need to deposit some amount of money that equals our daily negative PnL. Therefore we need some compensation for these adverse characteristics of ED compared to FRAs. The compensation is a lower price of ED which means ED contracts must have higher interest rate.

The convexity adjustment gets larger as maturity increases and this makes long dated contracts to be less attractive due to “unknown” volatility of the long dated interest rates.

The settlement structure of the Eurodollar contract is another reason for convexity bias as it is written in the article “Convexity adjustment, part 2”.