# Derive a mathematical equation for Eurodollar future rate

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?

• If $\sigma$ is a constant, then the use of $\sqrt{\sigma}$ does not appear sensible. – Gordon Dec 14 '18 at 16:47

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

• Hi, when I do the parameter fit for the dataset on a cross sectional basis, which is a particular day, I find that the fit and the actual data is very close. Is this collinearity and how to improve that in this case? – Qing Dec 14 '18 at 20:57
• Hi Qing, I assumed your actual data is different eurodollar settlements, so there you go the fitted curve is supposed to be close to the actual data. – numerairX Dec 14 '18 at 21:15
• May I ask how to do the fit process for Affine model to fit the Eurodollar future rate curve? – Qing Dec 16 '18 at 2:10