Why Identical Payoffs Must Have the Same Price
Summary
The document examines a proof of the law of one price using a state price vector. If two assets deliver identical cash flows in every state, applying the same state prices to each asset gives the same value. The questioner worries that multiple possible state price vectors could assign different values to identical assets.
The response clarifies that a state price represents the price of a specific contract paying one unit of numeraire in one state and nothing in other states. Under that definition, the vector lists prices for these state-contingent claims; it is not an arbitrary choice of coefficients for each asset. The key implication is that any one valid pricing rule values identical payoff vectors equally. The exchange offers a conceptual explanation, not a full proof of when state prices exist or are unique. In incomplete markets, distinct state price vectors may still price some nonidentical payoffs differently.
Key ideas
- A state price is the price of a claim paying in one specified state.
- Identical state-by-state payoffs have equal prices under any common pricing vector.
- The law of one price does not require a unique state price vector.
- Different valid state prices can matter for payoffs that are not identical.
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Full text
# Show if Arrow price vector $\pi$ exists, then the law of one price hold
# Show if Arrow price vector $\pi$ exists, then the law of one price hold
Now, the proof I have read goes like this:
Take assets 1 and 2, entirely identical. By assumption there is a pricing vector, i.e. $\sum_s\pi_sd^1_s=q^1$ and $\sum_s\pi_sd^2_s=q^2$ where $d^i_j$ is the payoff of asset $i$ in state $j$ and $\pi_i$ is the $i$-th element of the pricing vector $\vec{\pi}$. Since $d^1_s=d^2_s$ for all $s$ (identical assets), we must have $q^1=q^2$.
My question about this proof is that the assumption does not say only one $\vec{\pi}$ exists. It seems possible to me that there can be another price vector and we would have got a different price for the same asset. For example, price vector $(1,2,3)$ and $(2,3,4)$.
## Answer by funnycrab (score 1)
https://quant.stackexchange.com/a/27794
By pricing vector I assume you refer to the state price vector, whose definition can be found here.
According to the above definition, the price of a particular state specifically refers to the the price of a contract paying exactly one unit of numeraire if that state does occur and zero unit otherwise. And state price vector is thus the vector of state prices for all states.
So it seems to me that this state price vector is kind of normalized, only one specific state price vector exists for a given measure. Thus if two assets are identical (providing exactly same cash flows under any state), their price should be equal to each other no matter what. I think this is the point of the law of one price.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.