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