SDF Uniqueness and Complete Markets in Asset Pricing
Summary
The document examines a proposed derivation that would make the stochastic discount factor equal to the inverse of an asset’s expected return. The apparent contradiction is that the SDF can vary across future states, while the proposed expression is known at the current time. The resolution given is that the SDF is not generally unique when markets are incomplete, so the derivation cannot identify it from the pricing relation for one security.
A two-state example illustrates the point: one security pays different amounts across the states, and two distinct state-contingent discount factors both reproduce its price. The example shows why pricing a limited set of securities does not determine a unique SDF. Uniqueness requires complete markets, with enough linearly independent securities to span possible outcomes. The document does not fully develop the distinction between physical and risk-neutral measures or address the question’s discounted cash flow interpretation, so those topics require further explanation.
Key ideas
- The SDF is generally not uniquely determined in incomplete markets.
- A single security’s price can be consistent with multiple state-contingent discount factors.
- Complete markets require enough linearly independent securities to span the possible outcomes.
- An asset’s expected return alone does not generally identify the SDF.
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# Confused About Ex-Ante vs. Ex-Post Pricing Representation
# Confused About Ex-Ante vs. Ex-Post Pricing Representation
This is going to be a really simple question, but I am confused by it. The basic pricing formula is $p_t=E^p_t(m_{t+1}X_{t+1})$, where $p$ is the physical measure. We can also say that $R_{t+1}=\frac{X_{t+1}}{p_{t}}$ ex-post. This is just the definition of returns. This means that $E^l_t(R_{t+1})=\frac{E^l_t(X_{t+1})}{p_t}$, and thus $p_t=\frac{E^l_t(X_{t+1})}{E^l_t(R_{t+1})}$. This is with respect to any measure $l$, including the physical measure(it is just a weighted average). If we choose $l=p$(the arbitrary measure equal to the physical measure) then, by uniqueness of the SDF we can say that $m_{t+1}=\frac{1}{E_t^p(R_{t+1})}$. Obviously there must be something wrong with this reasoning, since it concludes that $m_{t+1}$ is not time-t random. I think the problem must be an ex-post vs. ex-ante kind of thing, basically taking the arbitrary measure expectation on the third line is not as simple as it sounds, but I was wondering if anyone had a better explanation.\
I found this especially confusing because my understanding of the basic discounted cash flow model in corporate finance comes from the $p_t=\frac{E^l_t(X_{t+1})}{E^l_t(R_{t+1})}$ expression. But there must be some reason why this expression only makes economic sense if the arbitrary measure is the risk neutral measure.
## Answer by Matthew Gunn (score 1)
https://quant.stackexchange.com/a/40043
To get uniqueness of the SDF, you need complete markets. In a world with $n$ possible outcomes, complete markets would require $n$ linearly independent securities. Without complete markets, the SDF is not unique.
#### Simple Example: 2 possible outcomes, 1 security
Imagine there are two states of the world and I represent probability measures and random variables as two-dimensional vectors. Let:
$$ \mathbf{p} = \begin{bmatrix}\frac{1}{2} \\ \frac{1}{2} \end{bmatrix} \quad \mathbf{x} = \begin{bmatrix}3 \\ 1 \end{bmatrix} $$
$$ \mathbf{m}^{(a)} = \begin{bmatrix} \frac{1}{3} \\ 1 \end{bmatrix} \quad \quad \mathbf{m}^{(b)} = \begin{bmatrix} \frac{1}{2} \\ \frac{1}{2}\end{bmatrix}$$
Let's imagine the price of security $x$ is 1. Both $\mathbf{m}^{(a)}$ and $\mathbf{m}^{(b)}$ correctly price the security since $\sum_i p_i m^{(a)}_i x_i = \sum_i p_i m^{(b)}_i x_i = 1$.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.