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 $Q1= Q2$?

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