Convexity Adjustment: Futures Rate vs. Forward Rate
Under stochastic interest rates, the futures rate on a Eurodollar-style contract and the corresponding forward rate (the rate implied by today's discount bonds) are not the same -- even though they reference the same period.
Assume the short rate $r_t$ follows a Gaussian (Hull-White) process:
$dr_t = \theta(t)\,dt + \sigma\,dW_t$
under the risk-neutral measure $\mathbb{Q}$, where $\sigma > 0$ is constant.
The futures rate $f^{\text{fut}}(t, T)$ is the rate locked in today via a futures contract settling at time $T$. The forward rate $f(t, T, T+\delta)$ is the rate implied by the ratio of zero-coupon bond prices $P(t,T)/P(t,T+\delta)$.
- Explain intuitively why the futures rate differs from the forward rate when rates are stochastic, even though they coincide under deterministic rates.
- Show that, under the Gaussian short-rate model, the convexity adjustment takes the form:
$f^{\text{fut}}(t, T) - f(t, T, T+\delta) = \text{(a positive correction that depends on } \sigma^2 \text{)}$
Derive or justify the sign and the key factors this correction depends on.
- Which direction does the adjustment go -- is the futures rate higher or lower than the forward rate? Why does this make economic sense in terms of daily margining?
Hints
- Think about the key mechanical difference between a futures contract and a forward rate agreement -- what happens to cash flows before maturity?
- The futures rate is $E^{\mathbb{Q}}[r_T]$ while the forward rate is $E^{\mathbb{Q}^T}[r_T]$. Consider what the change of measure between $\mathbb{Q}$ and $\mathbb{Q}^T$ does to the drift of $r_t$.
- Under the Hull-White model, the Girsanov shift from the risk-neutral to the $T$-forward measure adds a drift term proportional to $\sigma \cdot B(t,T)$, where $B(t,T) = T - t$. Integrate this shift to get the convexity correction.
Worked Solution
How to Think About It: This is one of the classic "interview gotchas" in rates. A futures contract and a forward rate agreement (FRA) both let you lock in a borrowing rate for a future period, but they have a crucial mechanical difference: futures are marked to market daily, while FRAs settle once at maturity. When rates are stochastic, this daily settlement creates a systematic bias. Before writing any math, you should be able to say: "Futures rates are higher than forward rates because of the correlation between rate moves and the value of daily margin flows." If you can explain that in 30 seconds, you have passed the interview question. The math just makes it precise.
Quick Sanity Checks: - If $\sigma = 0$ (deterministic rates), there is no adjustment. Futures rate = forward rate. Good -- the adjustment must be proportional to $\sigma^2$. - The adjustment should be positive (futures > forward) for standard interest rate contracts. - The adjustment should grow with time to maturity $T$ and with $\sigma^2$. - Under Gaussian models, the adjustment has a clean closed-form because all distributions are normal.
Derivation:
*Part 1 -- The intuition:*
A futures contract is settled daily. When rates rise, the futures price falls and you must post margin -- but rates are now higher, so the money you borrow to post that margin is more expensive. When rates fall, you receive margin -- but rates are lower, so reinvesting that margin earns less. This is an asymmetric pain: you lose more on the bad days than you gain on the good days. To compensate for this disadvantage, the market demands a higher locked-in rate on the futures contract than on the corresponding FRA. Hence futures rate > forward rate.
Under deterministic rates, the daily P&L is known in advance, so there is no reinvestment risk and the two rates coincide.
*Part 2 -- The formal argument:*
The futures rate is the $\mathbb{Q}$-expectation (risk-neutral) of the spot rate at time $T$:
$f^{\text{fut}}(t, T) = E^{\mathbb{Q}}_t[r_T]$
This holds because the futures price is a $\mathbb{Q}$-martingale (no cost of carry due to daily margining).
The forward rate, on the other hand, is the expectation of $r_T$ under the $T$-forward measure $\mathbb{Q}^T$:
$f(t, T) = E^{\mathbb{Q}^T}_t[r_T]$
The two measures differ. The change of measure from $\mathbb{Q}$ to $\mathbb{Q}^T$ involves the Radon-Nikodym derivative:
$\frac{d\mathbb{Q}^T}{d\mathbb{Q}}\bigg|_{\mathcal{F}_t} = \frac{P(t,T)}{P(0,T) \cdot B_t}$
where $B_t = \exp\left(\int_0^t r_s\,ds\right)$ is the money-market account.
Under the Hull-White (Gaussian) model, the Girsanov shift from $\mathbb{Q}$ to $\mathbb{Q}^T$ introduces a drift adjustment. Specifically, the Brownian motion under $\mathbb{Q}^T$ is:
$dW_t^T = dW_t + \sigma B(t,T)\,dt$
where $B(t,T)$ is the bond volatility coefficient. For the simple constant-$\sigma$ model, $B(t,T) = T - t$.
This drift shift means:
$E^{\mathbb{Q}}_t[r_T] - E^{\mathbb{Q}^T}_t[r_T] = \sigma^2 \int_t^T B(s,T)\,ds = \sigma^2 \int_t^T (T - s)\,ds = \frac{1}{2}\sigma^2(T-t)^2$
So the convexity adjustment is:
$f^{\text{fut}}(t,T) - f(t,T) = \frac{1}{2}\sigma^2(T-t)^2$
*Part 3 -- Direction and economic sense:*
The futures rate is higher than the forward rate. The adjustment $\frac{1}{2}\sigma^2(T-t)^2$ is always positive.
Economically: the long futures position suffers from the correlation between rate changes and the cost of financing margin calls. When rates rise (bad for the long), the cost of funding the margin call also rises. When rates fall (good for the long), the reinvestment rate on the margin inflow also falls. This systematic disadvantage must be offset by a higher locked-in futures rate. The effect is purely driven by volatility ($\sigma^2$) and grows quadratically with time to expiry, which is why long-dated Eurodollar futures require substantial convexity adjustments.
Answer: Under a Gaussian short-rate model, the convexity adjustment is:
$f^{\text{fut}}(t,T) - f(t,T) = \frac{1}{2}\sigma^2(T-t)^2$
The futures rate exceeds the forward rate by this amount. The adjustment is positive because daily margining creates a reinvestment risk that is adversely correlated with rate movements. The correction is proportional to $\sigma^2$ and quadratic in time to maturity $(T-t)^2$.
Intuition
The convexity adjustment is fundamentally about the interaction between daily margining and stochastic rates. A futures contract forces you to realize gains and losses every day, and you must finance those flows at the prevailing rate. When rates are correlated with the contract's value -- which they are by definition for interest rate futures -- this creates a systematic bias. You always reinvest gains at worse rates and finance losses at worse rates. The market compensates for this by setting the futures rate above the forward rate.
This is one of the most practically important results in rates trading. For short-dated contracts (under 2 years), the adjustment is small -- a few basis points. But it grows quadratically with maturity, so for a 10-year Eurodollar strip the cumulative adjustment can be 50-100+ basis points. Ignoring it means your curve is systematically wrong. In practice, traders calibrate $\sigma$ from swaption or cap volatilities and apply this adjustment when building forward curves from futures prices. The Gaussian model gives the cleanest formula, but the principle -- futures rate > forward rate due to margining -- holds in any stochastic rate model.