Future Convexity Adjustment
A convexity and delay adjustment formula was published in Options, Futures and Other Derivatives by John C Hull in the mid 90s.
In Hull's book, the convexity adjustment is said to arise because futures are marked-to-market daily, so their expected value under the risk-neutral measure is different from the forward price, which is settled "at maturity", but this appears to be a mischaracterization of the pricng logic: for a future, terminal PnL at maturity $T_1$ is $(L(T_1,T_2)-K)$ whereas for a FRA contract, the terminal payoff has the correct discounting $\frac{L(T_1,T_2)-K}{1+\delta_i L(T_1,T_2)}$.
Both the FRA and the future may be subject to daily margin calls or collateral calls, but the comparison with the FRA shows that the terminal rate of the contract has not been discounted for $T_2$ to $T_1$, and this is the reason that the future contract is not to be priced with forwards, but needs a convexity adjustment.
We derive our formula here in the simplest risk neutral HJM framework with appropriate numéraire changes.
With usual notations, let $P(t,T)=\exp\left(-\int_t^T f(t,u)du\right)$ be the $T$ maturity zero coupon bond price at time $t$, $A_t$ a tradeable asset. We are looking at the future and forward rate for the period $[T_1,T_2]$ seen from a time $t < T_1<T_2$.
Under the measure $Q_1$ associated with $P(t,T_1)$ as a numeraire, we have:
$$ \frac{A_t}{P(t,T_1)} = E^{Q_1}\left(\frac{A_{T_1}}{P(T_1,T_1)} \Big| \mathcal{F}_t \right) $$
$$= E^{Q_1}\left(A_{T_1} | \mathcal{F}_t \right)$$
Under the measure $Q_2$ associated with $P(t,T_2)$ as a numeraire, we have:
$$ \frac{A_t}{P(t,T_2)} = E^{Q_2}\left(\frac{A_{T_1}}{P(T_1,T_2)} \Big| \mathcal{F}_t \right) $$
The change of measure from measure ${Q_2}$ to measure ${Q_1}$ can be written:
$$ \frac{dQ_1}{dQ_2} = \frac{P(t,T_2)}{P(t,T_1) P(T_1,T_2)} = \exp\left(-\int_{T_1}^{T_2} f(t,u)-f(T_1,u) du\right) $$
The libor rate $L(T_1,T_2)$ verifies this relation at $T_1$: $$1+\delta L(T_1,T_2) = \frac{1}{P(T_1,T_2)}= \exp\left(\int_{T_1}^{T_2} f(T_1,u) du\right)$$
The libor is a $Q_2$ martingale for $t<T_1$ its expectation is independant of volatility:
$$ E^{Q_2}\left(1+\delta L(T_1,T_2) | \mathcal{F}_t \right) = E^{Q_2}\left(\frac{1}{P(T_1,T_2)} \Big| \mathcal{F}_t \right)$$
$$=\frac{P(t,T_1)}{P(t,T_2)} = \exp\left(\int_{T_1}^{T_2} f(t,u) du\right)$$
The future rate is a $Q_1$ expectation of $L(T_1,T_2)$. A numeraire change enables us to link it to the forward, which is the $Q_2$ martingale above:
$$ 1+\delta F= E^{Q_1}\left(1+\delta L(T_1,T_2) | \mathcal{F}_t \right) = E^{Q_2}\left(\frac{1}{P(T_1,T_2)} \frac{dQ_1}{dQ_2} \Big| \mathcal{F}_t \right) = E^{Q_2}\left(\frac{P(t,T_2)}{P(t,T_1) P(T_1,T_2)^2} \Big| \mathcal{F}_t \right)$$
We note that $\frac{P(t,T_1)}{P(t,T_2)} = E^{Q_2}(\frac{1}{P(T_1,T_2)})$, if we write $X=\int_{T_1}^{T_2} f(T_1,u)du$, and assuming rates normal (i.e. the volatility $\sigma(t,T)$ of all forward $f(t,T)$ is assumed to be deterministic, thus, bond prices are lognormal), $X \sim N(\mu_X,\sigma_X)$ we get:
$$ 1+\delta F = E\left( \frac{e^{2X}}{E(e^{X})} \right) = e^{2\mu_X + 2\sigma_X^2 - (\mu_X+\frac{1}{2} \sigma_X^2)} = e^{\mu_X + \frac{1}{2}\sigma_X^2 + \sigma_X^2} = E(e^{X})e^{\sigma^2} = (1+\delta L)e^{\sigma_X^2} $$
where $\sigma_X^2 = Var\left(\int_{T_1}^{T_2} f(T_1,u)du \Big| \mathcal{F}_t \right)$. Under the further most simplifying assumption that all forward $f(t,T)$ have a constant volatility $\sigma(t,T) = \sigma_r$, we get:
$$ \sigma_X^2 = \sigma_r^2 (T_2-T_1)^2 (T_1-t) $$
and therefore the expected future $F_t(T_1,T_2)$ which will be paid on $T_1$ needs to be adjusted for convexity: $$ 1+\delta F_t(T_1,T_2) = (1+\delta L)e^{\sigma_r^2 (T_2-T_1)^2 (T_1-t)} $$
Generalisation to Forward Timing Adjustment for Normal Rates
We can generalize this formula to get the rate paid at any time $T \in [T_1,T_2]$.
$$ E^{Q_T}\left(1+\delta L(T_1,T_2) | \mathcal{F}_t \right) = E^{Q_2}\left(\frac{1}{P(T_1,T_2)} \frac{dQ_T}{dQ_2} \Big| \mathcal{F}_t \right) = E^{Q_2}\left(\frac{1}{P(T_1,T_2)}\frac{P(t,T_2)}{P(t,T) P(T,T_2)} \Big| \mathcal{F}_t \right) $$
- The first term is $e^X$ with $X = \int_{T_1}^{T_2} f(T_1,u)du$
- The second term is the martingale $\frac{e^Y}{E(e^Y)}$ with $Y = \int_{T}^{T_2} f(t,u)-f(T,u)du$
We have $$ E^{Q_T}\left(1+\delta L(T_1,T_2) | \mathcal{F}_t \right) = E\left(\frac{e^{X+Y}}{E(e^Y)}\right) = E(e^X) e^{cov(X,Y)} $$ with $E(e^X) = 1+\delta L(T_1,T_2)$
$X,Y$ are gaussian, their variance and covariance is obtained by writing $X$ and $Y$ as ito integrals assuming a normal non mean-reverting forward diffusion:
- $X = \mu_X + \int_t^{T_1} \sigma_r (T_2-T_1) dW_s$
- $Y = \mu_Y + \int_t^{T} \sigma_r (T_2-T) dW_s$
- $\sigma_X^2 = \sigma_r^2 (T_2-T_1)^2 (T_1-t)$
- $\sigma_Y^2 = \sigma_r^2 (T_2-T)^2 (T-t)$
- $cov(X,Y) = \sigma_r^2 (T_2-T)(T_2-T_1)(T_1-t)$
Injecting the covariance term, we obtain a more general formula, which generalizes for payment at $T \in [T_1,T_2]$ and matches our formula for the boundary cases: $$E^{Q_T}\left(1+\delta L(T_1,T_2) | \mathcal{F}_t \right) = (1+\delta L_t(T_1,T_2)) e^{\sigma_r^2 (T_2-T_1)(T_2-T) (T_1-t)}$$
Generalisation to Forward Timing Adjustment for HJM Model
These formulae can be extended for the general case of HJM models: $$cov(X,Y) = \int_t^{T_1} v(u,T_1,T_2) v(u,T,T_2) du$$ with $v(s,t,u) = v(s,u)-v(s,t)$ and $v(t,T) = \int_t^T \sigma(t,u) du$ is the bond volatility. It is also linked to the zero coupon bond risk neutral SDE $$\frac{dP(t,T)}{P(t,T)} = rdt - v(t,T) dW_t$$ and to forward SDE by:$ $$ df(t,T) = v(t,T)\sigma(t,T) dt + \sigma(t,T) dW_t $$