For a one period economy, we have the price of an asset as:
$ p_0 = E^Q [p_1 * \frac {B0}{B1}] $
where $B0 = e^{-r_0}$ = time 0 price of risk free bond maturing at time =1 and $r_0$ is known at t0. And B1=1
Now lets say the economy is 2 period, but the only risk free instruments are 1 period bonds. Then you can price the asset as follows:
$ P_0 = E^{Q1} [P_1 * \frac {B0}{B1}] $
$ P_1 = E^{Q2} [P_2 * \frac {B1}{B2}] $
Notice $E^{Q1}$ vs $E^{Q2}$ in the two equations. These can be combined:
$ P_0 = E^{Q1} [ \frac {B0}{B1} * E^{Q2} [P_2 * \frac {B1}{B2}]] $
$= E^{Q1} [ e^{-r_0} * E^{Q2} [P_2 *e^{-r_1}]] $
Now I see that the final step is just collapsed into this:
$ P_0 = E^{Q1} [ P_2 *e^{-(r_0 + r1)}]] $
My question is how can you remove $E^{Q2}$ in the end? Do we need to assume the $Q1= Q2$?
