# How is calculated the futures/forward convexity adjustment for FX?

I could find lots of stuff online for IR derivatives but it seems there isn't too much on FX for this specific adjustment.

The futures/forward convexity adjustment comes from the covariance between rates and the index. For a future/forward that settles on an index $$I_T$$ on expiry $$T$$ the future price is $$F_{\text{fut}} = \mathbb{E}^{\mathbb{P}}\left[I_T \right]$$, where $$\mathbb{P}$$ is the risk neutral measure, and the forward price is $$F_{\text{fwd}} = \mathbb{E}^{\mathbb{Q}^T}\left[I_T \right]$$, where $$\mathbb{Q}^T$$ is the $$T$$-forward measure. Applying the change of measure $$d\mathbb{P}/d\mathbb{Q}^T = e^{\int_0^T r_t dt} D(T)$$ you get $$F_{\text{fut}} = \mathbb{E}^{\mathbb{Q}^T}\left[e^{\int_0^T r_t dt} D(T) I_T \right] = \mathbb{E}^{\mathbb{Q}^T}\left[ I_T \right] + \mathbb{E}^{\mathbb{Q}^T}\left[(e^{\int_0^T r_t dt} D(T)-1) I_T \right] \\= F_{\text{fwd}} + D(T)\mathbb{COV}^{\mathbb{Q}^T}\left[e^{\int_0^T r_t dt}, I_T \right]$$ When $$I_T$$ is an FX index or an equity index market practice seems to disregard the covariance term.

You can however get an estimate of its magnitude using a simple Hull & White model with volatility $$\sigma_r$$ and no mean reversion for $$r_t$$, and an exponential brownian motion for $$I_T$$ with volatility $$\sigma_I$$ and correlation $$\rho$$ between the two brownians, the formula above becomes $$F_{\text{fut}} = F_{\text{fwd}} e^{\sigma_r \sigma_I \rho T^2/2}$$ With say $$\sigma_r = 50$$ bps, $$\sigma_I = 10\%$$, $$\rho = 25\%$$ and $$T=5$$ you would get $$F_{\text{fut}} = F_{\text{fwd}} \times 1.0006$$, so it is not entirely negligible.

• thanks! Makes sense from a mathematical prospective, I was wondering what is the economic/financial reason why futures rates are usually higher than forwards? – Student Mar 26 at 17:20
• if positive correlation between index and rates, gains on upward move of index are more likely to be reinvested at a higher rate, losses at a lower rate. – Antoine Conze Mar 26 at 17:40
• Usually interest rate futures have price given by 100-R, so if the underlying rate decreases I get excess margin and I can invest it at lower rates in this case. Can you clarify which IR futures contracts you have positive price-rates correlation so that when price goes up I can invest the extra margin at a higher rate? thanks – Student Mar 27 at 16:38
• your OP was on FX futures. Obviously for an IR future that settles on 100-R the index and IR are negatively correlated. – Antoine Conze Mar 29 at 7:02

The futures/forward convexity adjustment for non-interest rate futures only tends to matter for futures with maturities greater than a year (which tend to be part of bespoke structures and not traded in size on screen). You can get a closed form solution in a GBM-Ho-Lee hybrid model if you don't mind grinding though some partial differential equation work you will find that the convexity adjustment is proportional to $$\frac{1}{2}\rho\sigma_r\sigma_xT^2 + \frac{1}{3}\sigma_r^2T^3$$

So we need a convexity adjustment even if the two Brownian motions are uncorrelated, since the short rates still drives the drift of the underlying process which means there will be a terminal correlation between it and the cash account.

• Hi. Just a comment, here $\sigma_x$ is the volatility of the spot price in the hybrid model, and needs to be calibrated from market implieds $\sigma_I$ which are Black & Scholes volatilities of forward prices. Hence the additional term in the convexity adjustment formula. – Antoine Conze Mar 25 at 9:53
• We have the closed form solution for that calibration though (dbl check me as I did this on scrap paper now)? $\sigma_I^2 = \sigma_x^2+2\rho\sigma_r\sigma_x(T-t)+\sigma_r^2(T-t)^2$. I don't think that removes the extra adjustment term? – river_rat Mar 25 at 15:17
• the correlation between $r$ and the spot also has to be calibrated from the correlation between $r$ and the forward, or vice-versa – Antoine Conze Mar 25 at 15:25