Risk-Neutral Pricing Across Periods with Changing Measures

Summary

The document examines how to combine one-period risk-neutral pricing relations into a multi-period price when only short-maturity risk-free bonds are available. It writes successive pricing steps using expectations under measures labeled for each period, then questions whether the nested expectations can be collapsed into a single expectation with accumulated discounting. The central issue is the relationship between those measures and the conditioning information at each date.

The text presents the pricing equations but does not resolve the question or establish that the two measures are equal. In general, iterated expectations require a consistent numeraire and an appropriate change-of-measure relationship; discount factors and stochastic interest rates also need to remain inside the correct conditional expectation. The example is therefore a useful prompt about measure consistency, but it is not a complete derivation or evidence that the proposed collapse is valid as written.

Key ideas

• Multi-period pricing can be built by applying one-period pricing relations sequentially.
• The example uses a different risk-neutral measure label for each period.
• Nested expectations cannot be removed solely by assuming the measure labels are interchangeable.
• A valid collapse depends on conditioning, discounting, and consistent measure construction.

Tags

# From one period to multi period risk neutral pricing


# 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 $Q1= Q2$?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.