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Girsanov theorem finance

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to change the probability measure one uses Girsanov theorem (formula):

\frac{N_a(0)}{N_a(T)} d \mathbb{P}_a=\frac{N_b(0)}{N_b(T)} d \mathbb{P}_b

N_a is numeraire (price of any non dividend paying asset, usually bond or bank account) and it’s correspoding probability measure is P_a

N_b is another numeraire with it’s probability measure P_b

Applications

expectation of short rate is forward rate under T-forward measure

– prove that E^T is a expectation under T-forward measure (where bond E^{RN}(e^{-\int_t^Tr_sds}|F_t)=e^{-\int_t^Tf_s(t)ds}
1) differentiate both parts by T
2) change measure from Risk-neutral to T-forward measure from time t to time T (as usual rule apply girsanov formula between times when the process is stochastic ,in this case not between 0 and t for example)

Black Scholes formula

to calculate the term S_t is numeraire

Libor In Arrears Convexity adjustment

Simple example libor in arrears convexity adjustment to change from T1 measure to T2 measure

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