Derive a mathematical equation for Eurodollar future rate

Question

If we suppose that r(t) follows a Vasicek model, which is: $$dr(t) = (\mu - \kappa r(t))dt + \sqrt\sigma dW(t)$$ How to derive an expression for Eurodollar future rate?

Answer 1

there are many ways to solve Vasicek system, for me personally I markov short rate approach. Without going into the details of proofs:

Note that eurodollar future is calculated under risk neutral Q measure of libor rate at each settlement $t_{fix}$ (on three months interval each)

libor rate $l(t_{fix}) = \frac{1}{tenor} e^{A_{diff} - B_{diff} * r(t_{fix})} $ where $A_{diff} = A(t_0 - t_{fix}) - A(t_1 - t_{fix}) $ and similarly to $B_{diff}$. $t_0$ and $t_fix$ is starting time of current eurodollar settlement and $t_1$ is its time of maturity (eg, $t_0 = 0.25, t_{fix} = 0.25, t_1 = 0.5$)

A and B are calculated using the following system:

$\frac{dB}{dt} = kB - 1$

$\frac{dA}{dt} = \mu B - \frac{1}{2} \sigma B^2 $

denote $ A_{diff} - B_{diff} * r(t_{fix})$ = $\psi$ (just for typing purpose)

Thus it became clear that to calculate eurodollar rate = $E^Q(l(t_{fix}) = \frac{1}{tenor} e^{E(\psi) + \frac{1}{2}\sigma(\psi)^2}$, you already have A and B and just need mean and variance of short rate process, which are very straightforward under defined vasicek model.

mean: $r_0e^{-kt_{fix} + \mu (\frac{1-e^-t_{fix}}{k})} $

variance: $\frac{\sigma}{2k} (1-e^{-2k t_{fix}})$

also don't forget to -1 inside the expectation after rate