Future Convexity Adj
A convexity and delay adjustment formula was published in Options, Futures and Other Derivatives by John C Hull in the mid 90s. 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 $\frac{P(t,T_2)}{P(t,T_1)} = E^{Q_2}(P(T_1,T_2))$, if we write $X=\int_{T_1}^{T_2} f(T_1,u)du$, and assuming rates normal (which makes bond prices 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 $\sigma_X^2 = Var\left(\int_{T_1}^{T_2} f(T_1,u)du \Big| \mathcal{F}_t \right)$, further assuming that forwards have a constant volatility $\sigma_r$, 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: $$ 1+\delta F_t(T_1,T_2) = (1+\delta L)e^{\sigma_r^2 (T_2-T_1)^2 (T_1-t)} $$
Forward Timing Adjustment
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: $$E^{Q_T}\left(1+\delta L(T_1,T_2) | \mathcal{F}_t \right) = (1+\delta L_t(T_1,T_2)) e^{\sigma_r^2 (T_2-T_1)(T_2-T) (T_1-t)}$$