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Libor in arrears convexity adjustment simple example

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Libor in arrears

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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