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This would be my explanation for the reason that convexity adjustments must exist:

Futures are margined daily, such that if a trader is paid a future and rates goes up then money is paid into their margin account, and if rates goes down then money is taken from their margin account, daily, so that we have two outcomes from a position:

  1. Paid position: Rates go up, so money is paid into the margin account, and said money can be reinvested at the newer, higher rates. Rates go down, so money is taken from the margin account, but can be borrowed back at the newer, lower rates.

  2. Received position: Rates go up, so money is taken from the margin account, and said money can be borrowed back at lower rates. Rates go down, so money is paid into the margin account, and can be reinvested at higher rates.

Is this where the idea of convexity arises from? The fact that the daily margining creates a clear advantage to utilising futures instead of the corresponding FRA?

Thus, in order to offset this advantage from investing in futures over FRAs, a convexity adjustment is implemented such that (in a naive sense):

$$FRA=Futures-Convexity.$$

If this is not correct or I haven't fully understood, then please correct me.

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This has been posted a few times now, so I will invest the time on a full response.

FRA / Futures convexity has nothing to do with profits/losses being immediately recognised on the future through margin settlement, and potential reinvestment, whilst deferred on the FRA.

Although the opposite seems to be a very common belief amongst many practitioners (possibly due to bad wording in Hull:Options, Futures, Derivatives) it is not correct.

Let me characterise this with some different intuitive arguments:

  1. FRAs and Futures have similar reinvestment patterns. If you own a future and it makes \$10000 on day 1 you will receive \$10000 margin, which you invest overnight at OIS and receive some interest, say \$1. On day 2 the future is worth \$10000 and you have \$1 in your pocket.

    If you own an FRA and it makes \$10000 you will receive \$10000 collateral. You invest this overnight and receive \$1 OIS interest, but you owe \$1 in interest to the collateral poster. On day 2 the discount factors for the FRA cashflows are 1 day closer to expiry and discounted by 1-days OIS interest less. The FRA is valued at \$10001 and you have \$0 in your pocket.

  2. Imagine that your future was instead a fixed CFD and settled on the same date as the FRA. But it was a collateralised CFD. This would result in exactly the same convexity value, but it is independent of having an immediate margin exchange of profits/gains and any reinvestment.

  3. Assume that everyday the future closed at the same price so that no profit exchange was ever made (but this was coincidental). Then the reinvestment argument would not be able to make any gains since there is never any gain to reinvest. However there is an important difference if the intraday volatility is either zero, low, or high, and a portfolio continuously delta hedges, even if at the end of the day it remains at where it started.

FRA / Futures convexity is risk based and its value can be intuitively understood in a way similar to the replicating portfolio of an option, to determine its price.

What does this mean? First consider the formula for the settlement of a paid (bought) FRA:

$$ P = v N d \frac{r - R}{1+d r} $$

Where $v$ is the (ois) discount factor for the settlement date of the FRA, $r$ is the floating rate, $R$ the fixed rate and $d$ the day count fraction. The key here is the term $(1+dr)^{-1}$; when rates go up you make money on the position but this term discounts your contract settlement more heavily. In fact the settlement is paid upfront but is implicitly assumed to be discounted from the end of the FRA at the settled FRA rate, and this is the term that gives the product asymmetry.

So how is this risk related? Consider a portfolio where you have paid (bought) a FRA and hedged it by buying a future. The value of your portfolio is as follows:

$$ P = v N d\frac{r-R}{1+rd} - \tilde{N}d(F-r) $$

where $F$ is the equivalent futures rate at which you have traded a nominal amount, $\tilde{N}$ of futures to delta hedge. Your portfolio will satisfy two properties initially:

$$ P_{t=0} = 0, \quad \frac{\partial P}{\partial r}_{t=0} = 0 $$

So this means that you trade a specific amount of futures initially to be delta hedged:

$$ \tilde{N}|_0 = f(N, d, r, v)|_0 $$

But the issue now is what happens to the risk of your portfolio as rates change?

The risk of your futures always remains constant:

$$\frac{\partial P_{futures}}{\partial r} = \tilde{N} d $$

but the risk on your FRA changes to be dependent upon the prevailing rate, $r$:

$$\frac{\partial P_{fra}}{\partial r} = \frac{vNd}{1+dr} \left (1-\frac{d(r-R)}{(1+dr)} \right ) $$

In fact, more than that it actually depends on previous rates (and the OIS rate) that impacts the discount factor $v$, whilst the risk on the future always remains constant.

You end up with the scenario that when rates increase for this portfolio you need to buy more FRA to remain delta hedged, but you are buying at higher prices. If the price falls again then you sell it back to remain delta hedged costing yourself money. So this is a process that is entirely dependent upon the volatility.

Conversely if you sell a FRA then when rates fall you have too large a position and can pay the FRA back to lower your delta, since you are doing this at more favourable rates your continuous delta hedging is generating an arbitrage profit against the future.

Hence futures are always oversold relative to FRAs that are oversold. The rates on FRAs are naturally lower than those for futures and the difference is called the convexity bias.

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  • $\begingroup$ Attack68: would it be fair to say though that in practice, larger chunk of the FRA convexity comes from the (OIS) discounting, rather than the Libor-discounting term? People commonly say "The longer the FRA maturity, the larger the convexity adjustment relative to Futures". Assume the underlying is 3-m Libor. If the entire convexity came from the Libor-discounting over the 3-m period between setting of the FRA and the underlying Libor maturity, then the FRA maturity would play no part in the convexity. Clearly, this is not true (again, longer-dated FRA = higher convexity). @dm63 $\endgroup$
    – Jan Stuller
    Mar 15, 2021 at 12:17
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    $\begingroup$ Yes, simplistically if rates move in parallel and OIS/LIBOR has correlation 1, then $\frac{\partial v}{\partial r_i}$, where $r_i$ is any OIS rate along the curve upto the FRA settlement, will be the dominant contributor. The longer the maturity the more risk change there is going to be on the FRA for a 1bp parallel shift, and hence more potential gains/losses from the structure. $\endgroup$
    – Attack68 ♦
    Mar 15, 2021 at 19:56
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It’s an ok explanation until the last 2 paragraphs. You should say :” in order to offset this advantage from SHORTING futures versus the FRA....”. And then the equation is $$FRA rate = Futures\space Implied\space Rate - Convexity\space Adjustment$$

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    $\begingroup$ So my confusion comes from the fact that there is an advantage from shorting the futures, but not an advantage when buying the futures. Why is there an advantage in shorting but not buying? $\endgroup$
    – quanty
    May 25, 2019 at 15:04

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