Girsanov theorem finance

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