# From one period to multi period risk neutral pricing

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 $$Q1= Q2$$?