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A 3 month LIBOR that fixing at $T$, paying in 3 months does not have a convexity adjustment.
However, 3 month LIBOR fixing at $T$, paying in 6 months needs a convexity adjustment.
How is this shown mathematically and how would one compute the value of this adjustment?
Let us denote $\delta$, the Libor's tenor (e.g. 3M), $P(t, T)$ the price of a zero coupon bond price paying 1 unit of currency at $T$, and $L_t(T, T + \delta)$ the forward 3M Libor starting at $T$ and ending at $T+\delta$, seen from $t$:
$$L_0(T, T + \delta) = \frac{1}{\delta} \left(\frac{P(0, T)}{P(0, T + \delta)} - 1 \right)$$
The numéraire corresponding to the $T+\delta$ forward measure is $P(t, T + \delta)$, and the Libor expressed above is a martingale under this measure. So that, if we consider a payment of this libor at its end date $T+\delta$, it has the price:
$$\begin{aligned} \mathbb{E}^{\mathbb{Q}}\left[ e^{\int_0^{T+\delta}r(u)du} L(T, T+\delta) \right] &= P(0, T+\delta) \times \mathbb{E}^{T+\delta}[L(T, T+\delta)] \\ &= P(0, T+\delta) \times L_0(T, T+\delta) \end{aligned}$$
Both terms are known at $t = 0$.
This is how vanilla interest rates swaps for example are valued, you read the Libor forward from today's curve, assume that it is precisely this forward that will be paid or received, and discount with the zero coupon.
Now, if the payment is made at a different date: $T' \neq T+\delta$ (e.g. $T + 2\delta$) then:
$$\begin{aligned} \mathbb{E}^{\mathbb{Q}}\left[ e^{\int_0^{T'}r(u)du} L(T, T+\delta) \right] &= P(0, T') \times \mathbb{E}^{T'}[L(T, T+\delta)] \\ &\neq P(0, T') \times L_0(T, T+\delta) \end{aligned}$$
This is because the Libor rate is not a martingale under the $T'$ forward measure but under the $T+\delta$ forward measure as we have seen above. To make the forward Libor $L_t(T, T+\delta)$ appear, we need to change measures:
$$\begin{aligned} \mathbb{E}^{T'}[L(T, T+\delta)] & = \mathbb{E}^{T+\delta} \left[L(T, T+\delta) \times \frac{d\mathbb{Q}^{T'}}{d\mathbb{Q}^{T+\delta}} \right]\\ &= \mathbb{E}^{T+\delta} \left[L(T, T+\delta) \right] + \mathbb{E}^{T+\delta} \left[L(T, T+\delta) \left(\frac{d\mathbb{Q}^{T'}}{d\mathbb{Q}^{T+\delta}} -1 \right)\right] \\ &= L_0(T, T+\delta) + \underbrace{\mathbb{E}^{T+\delta} \left[L(T, T+\delta) \left(\frac{d\mathbb{Q}^{T'}}{d\mathbb{Q}^{T+\delta}} -1 \right)\right]}_{\text{convexity adjustment term: } Conv(T+\delta, T')} \end{aligned} $$
We have expressed the value of this Libor payment as the product of the zero coupon and Libor forward as in the vanilla case, but this time with an adjustment term:
$$ \mathbb{E}^{\mathbb{Q}}\left[ e^{\int_0^{T+\delta}r(u)du} L(T, T+\delta) \right] = P(0, T+\delta) \times \left( L_0(T, T+\delta) + Conv(T+\delta, T') \right) $$
The numéraires of the two probability measures involved are the prices of the zero coupon bond prices with maturities $T'$ and $T+\delta$, so that:
$$\begin{aligned} \frac{d\mathbb{Q}^{T'}}{d\mathbb{Q}^{T+\delta}} &= \frac{P(T, T')}{P(0,T')} \times \frac{P(0, T+\delta) }{P(T, T+\delta) }\\ &= \frac{P_T(T+\delta, T')}{P_0(T+\delta, T')} \end{aligned} $$
Leading to the final expression:
$$ Conv(T+\delta, T') = \mathbb{E}^{T+\delta} \left[L(T, T+\delta) \left(\frac{P_T(T+\delta, T')}{P_0(T+\delta, T')} - 1 \right) \right] $$
To explicit the value of this term, you need a model for your zero coupon bond prices, or equivalently for the Libor rates.
Because they involve the expectation of a product, convexity adjustment terms will involve the covariance between the numéraire and the quantity being evaluated.
In this specific case, a correlation of 1 is typically assumed, and the convexity adjustment term will depend on the Libor's volatility.
Here is an example assuming a lognormal Libor.
Consider a date sequence \begin{align*} 0 \leq t_0 \leq T_s < T_e < T_p, \end{align*} where $t_0$ is the valuation date, $T_s$ is the Libor start date, $T_e$ is the Libor end date, and $T_p$ is the payment date. Let $\Delta_s^e = T_e-T_s$. For $0\le t \le T_s$, define \begin{align*} L^e(t, T_s, T_e) = \frac{1}{\Delta_s^e}\bigg(\frac{P(t, T_s)}{P(t, T_e)}-1 \bigg), \end{align*} where $P(t, \mu)$ is the price at time $t$ of a zero-coupon bond with maturity $\mu$ and unit face value. Moreover, let $\Delta_e^p = T_p-T_e$. For $0\le t \le T_e$, define \begin{align*} L^p(t, T_e, T_p) = \frac{1}{\Delta_e^p}\bigg(\frac{P(t, T_e)}{P(t, T_p)}-1 \bigg). \end{align*} Furthermore, let $Q^{T_e}$ denote the $T_e$-forward measure, and $Q^{T_p}$ denote the $T_p$-forward measure. We assume that, under $Q^{T_p}$, for $0\le t \le T_s$, \begin{align*} dL^e &= \mu(t) dt + \sigma_e L^e dW_t^e, \tag{1}\\ dL^p &= \sigma_p L^P\left(\rho dW_t^e + \sqrt{1-\rho^2} dW_t^p \right), \end{align*} where, $\sigma_e$, $\sigma_p$, and $\rho$ are constant, while $\{W_t^e, t \ge 0\}$ and $\{W_t^p, t \ge 0\}$ are two independent standard Brownian motions.
Let $E^{T_p}$ be the expectation operator under the $T_p$-forward measure $Q^{T_p}$. We seek the value defined by \begin{align*} P(t_0, T_p)E^{T_p}\big(L^e(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big). \end{align*}
Note that, for $0 \le t \le T_e$, \begin{align*} \eta_t \equiv \frac{dQ^{T_e}}{dQ^{T_p}}\big|_{t} &= \frac{P(t, T_e)P(t_0, T_p)}{P(t_0, T_e)P(t, T_p)}\\ &= \frac{1+\Delta_e^pL^p(t, T_e, T_p) }{1+\Delta_e^pL^p(t_0, T_e, T_p) }. \end{align*} Then \begin{align*} d\eta_t &= \frac{\Delta_e^pL^p(t, T_e, T_p) }{1+\Delta_e^p L^p(t_0, T_e, T_p) }\sigma_p \left(\rho dW_t^e + \sqrt{1-\rho^2} dW_t^p \right)\\ &=\frac{\sigma_p\Delta_e^pL^p(t, T_e, T_p) }{1+\Delta_e^p L^p(t, T_e, T_p) }\eta_t\left(\rho dW_t^e + \sqrt{1-\rho^2} dW_t^p \right). \end{align*} Consequently, \begin{align*} \hat{W}_t^e = W_t^e - \int_0^t \frac{\rho\sigma_p\Delta_e^pL^p(s, T_e, T_p) }{1+\Delta_e^p L^p(s, T_e, T_p) }ds \end{align*} is a standard Brownian motion under the $T_e$-forward measure $Q^{T_e}$. Therefore, \begin{align*} \mu(t) &= -\frac{\rho\sigma_e\sigma_p\Delta_e^pL^p(t, T_e, T_p)}{1+\Delta_e^p L^p(t, T_e, T_p)}L^e(t, T_s, T_e)\\ &\approx -\frac{\rho\sigma_e\sigma_p\Delta_e^pL^p(t_0, T_e, T_p)}{1+\Delta_e^p L^p(t_0, T_e, T_p)}L^e(t, T_s, T_e) \end{align*} in $(1)$ above, given that $L^e$ is a martingale under $T_e$-forward measure $Q^{T_e}$. That is, under the $T_p$-forward measure, \begin{align*} dL^e(t, T_s, T_e) &\approx L^e(t, T_s, T_e)\left(-\frac{\rho\sigma_e\sigma_p\Delta_e^pL^p(t_0, T_e, T_p)}{1+\Delta_e^p L^p(t_0, T_e, T_p)} dt + \sigma_e dW_t^e\right), \end{align*} and \begin{align*} L^e(T_s, T_s, T_e) &\approx L^e(t_0, T_s, T_e)\exp\bigg(-\frac{\rho\sigma_e\sigma_p\Delta_e^pL^p(t_0, T_e, T_p)}{1+\Delta_e^p L^p(t_0, T_e, T_p)}(T_s-t_0) \\ &\qquad\qquad\qquad\qquad\qquad -\frac{\sigma^2}{2}(T_s-t_0) + \sigma \big(W_{T_s}^e -W_{t_0}^e\big) \bigg) \end{align*}
Moreover, \begin{align*} &\ P(t_0, T_p)E^{T_p}\big(L^e(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big) \\ \approx&\ P(t_0, T_p)L^e(t_0, T_s, T_e) \exp\left(-\frac{\rho\sigma_e\sigma_p\Delta_e^pL^p(t_0, T_e, T_p)}{1+\Delta_e^p L^p(t_0, T_e, T_p)}(T_s-t_0) \right)\\ \approx&\ P(t_0, T_p)L^e(t_0, T_s, T_e)\left(1- \frac{\rho\sigma_e\sigma_p\Delta_e^pL^p(t_0, T_e, T_p)}{1+\Delta_e^p L^p(t_0, T_e, T_p)}(T_s-t_0)\right). \end{align*} Here, the term \begin{align*} - \frac{\rho\sigma_e\sigma_p\Delta_e^pL^p(t_0, T_e, T_p)L^e(t_0, T_s, T_e)}{1+\Delta_e^p L^p(t_0, T_e, T_p)}(T_s-t_0) \end{align*} can be treated as the convexity adjustment.
The other two answers do a good job of explaining, within the context of mathematical financial models, why a convexity adjustment is necessary, but I think a more tangible perspective can also be useful.
Consider two forward rate agreements (FRA) to receive fixed and pay floating, with the same fixing date $T_s$ and end date $T_e$. The first pays on the end date, and the second pays some time later at $T_p = T_e + \Delta$.
The second contract will be struck at a lower rate (the convexity adjustment). Why?
When realised interest rates are higher than the market currently prices, you must make a payment at $T_e$ on the first contract, and must finance this loss until $T_p$ at a high interest rate. When realised interest rates are lower than market prices, the first contract receives cash at $T_e$ but can only invest it at a low interest rate. The second contract therefore has an advantage, so the forward rate must be adjusted downward to compensate.
More concretely, assume that both contracts are struck at the same forward rate $K$. At some time $t < T_s$ the expected value of the first contract at $T_e$ is
$$ V_1 = K - F $$
where $F$ is the forward rate to be fixed at $T_s$ (which is an unknown quantity at this point). The expected value of the second contract is
$$ V_2 = (K - F) e^{-R\Delta} $$
where $R$ is the discount rate for the period from the end date to the payment date, which will be known at $T_e$. The variables $F$ and $R$ are unknown, but related -- if the forward rate is high, the discount rate is likely to be high as well.
Consider a simple two-scenario model where
$$ F = \begin{cases} K+\delta F & \textrm{with probability 1/2} \\ K-\delta F & \textrm{with probability 1/2} \end{cases} $$
$$ R = \begin{cases} R_0+\delta R & \textrm{with probability 1/2} \\ R_0-\delta R & \textrm{with probability 1/2} \end{cases} $$
Then you can show that
$$ {\rm E}(V_1) = 0 $$
$$ {\rm E}(V_2) \approx \Delta e^{-R_0 \Delta} {\rm Cov}(\delta F, \delta R) > 0 $$
If there is no convexity adjustment this represents an arbitrage (you would receive fixed in the second contract and pay fixed in the first contract, with the plan to take payment in the first contract at $T_e$ and close out the second contract at market rates at the same time).
The second contract must be struck at a lower forward rate to remove this arbitrage opportunity -- this is the convexity adjustment, and it is approximately the negative of ${\rm E}(V_2)$ in the above equation. This shows three things about the size of the convexity adjustment --
