to change the probability measure one uses Girsanov theorem (formula):
is numeraire (price of any non dividend paying asset, usually bond or bank account) and it’s correspoding probability measure is
is another numeraire with it’s probability measure
Applications
expectation of short rate is forward rate under T-forward measure
– prove that is a expectation under T-forward measure (where bond
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 is numeraire
Libor In Arrears Convexity adjustment
Simple example libor in arrears convexity adjustment to change from T1 measure to T2 measure