# Libor in arrears convexity adjustment simple example

Libor in arrears

simplified Libor in arrears payoff: pay at time 1 1-year Libor reset at time 1 F(1)

$\frac{NPV(0)}{P_1(0)}=\mathbb{E}^1 (\frac{F(1)}{P_1(1)})$

where $\mathbb{E}^1$ is measure with numeraire $P_1(t)$
change measure from time 0 to time 1 (time while F(t) is changing)
with girsanov formula :

$\frac{P_1(0)}{P_1(1)}dP_1=\frac{P_2(0)}{P_2(1)}dP_2$

so we get

$NPV(0) = P_2(0) E^2 ( \frac{F(1)}{P_2(1)})$

but $\frac{1}{P_2(1)}=1+F(1)$

so $NPV(0)=P_2(0)\mathbb{E}^2(F(1))+P_2(0)\mathbb{E}^2(F^2(1))$

under $\mathbb{E}^2$ F(1) is martingale i.e. $\mathbb{E}^2 F(1)= F(0)$

to calculate $\mathbb{E}^2(F^2(1))$ we must introduce dynamics for F(1)

for example black-scholes where under $\mathbb{E}^2$ : $dF(t)=\sigma F(t) dW_t$

so $F(1)=F(0) e^{\sigma W_1 - 0.5 \sigma^2 }$ and therefore

$\mathbb{E}^2 F^2(1)=F^2(0) e^{-\sigma^2 } \mathbb{E}^2 e^{2 \sigma W_1 } = F^2(0) e^{\sigma^2 }$

for CMS convexity adjustment use linear model i.e. Facebook

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