Deriving Eurodollar Futures Rates from a Vasicek Short-Rate Model

Summary

The note outlines a way to express a Eurodollar futures rate when the short rate follows a Vasicek process. It treats the futures settlement rate as the risk-neutral expectation of the three-month LIBOR rate at its fixing date. LIBOR is written as an exponential affine function of the short rate, using coefficient functions A and B evaluated across the accrual interval. Those coefficients follow differential equations associated with the Vasicek model.

The proposed calculation reduces the expectation to the mean and variance of the relevant Gaussian quantity, then applies the lognormal exponential moment formula. The response gives formulas for the short-rate mean and variance and notes that the futures quote convention requires subtracting one after the expectation. It is a sketch rather than a complete derivation: notation is compressed, and it does not discuss measure changes, contract-specific convexity adjustments, or conventions that may distinguish a futures rate from a forward rate. These details matter in practical valuation.

Key ideas

• The approach models the settlement-date LIBOR rate as an exponential affine function of the short rate.
• The Vasicek model supplies coefficient functions through differential equations.
• The expectation of the exponential rate can be computed from the Gaussian variable’s mean and variance.
• The response identifies a quote-convention adjustment after taking the expectation.
• The derivation is abbreviated and omits practical contract and measure details.

Tags

# Derive a mathematical equation for Eurodollar future rate


# 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?

## Answer by numerairX (score 2, accepted)

https://quant.stackexchange.com/a/43079

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.