State Price Uniqueness and Market Completeness
Summary
The document discusses two fundamental asset pricing claims: the law of one price is equivalent to the existence of a state price vector, and, when the law holds, uniqueness of that vector is equivalent to market completeness. The response focuses on the latter relationship and sketches why completeness implies uniqueness. If two risk-neutral measures assign different probabilities to an event, the event’s indicator payoff would receive different prices under the two measures, conflicting with its replicability in a complete market.
The reverse direction is illustrated by suggesting a trinomial model: demonstrate incompleteness by failing to hedge a call, then calculate the set of risk-neutral measures. This is a conceptual proof outline rather than a full mathematical derivation. It also uses state price vector and risk-neutral measure somewhat interchangeably and assumes pricing and discounting conventions that are not fully specified, so the argument needs careful formalization in a general setting.
Key ideas
- In a complete market, every measurable payoff, including an event indicator, can be replicated.
- If two risk-neutral measures price an event indicator differently, the payoff would not have a unique price.
- A trinomial model can illustrate how incompleteness is associated with multiple risk-neutral measures.
- The response outlines the uniqueness-completeness relationship but does not fully prove the general theorem.
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Full text
# How to prove the "Law of one price" theorem?
# How to prove the "Law of one price" theorem?
There are two subparts to Fundamental Asset Pricing theorem.
- The Law Of One Price (LOOP thereafter) holds if and only if there exists a state price vector.
- In a market in which the LOOP holds, the state price vector is unique if and only if the market is complete.
How can we prove (1) and (2) mathematically?
## Answer by Phun (score 1)
https://quant.stackexchange.com/a/16554
I just comment your second point, because in the definition i now of the LOP the state price vector (martingale measure) is involved.
- assuming the LOP holds then: state price vector is unique <=> the market is complete.
to proof "=>" look at the trinomial model, show that the model is not complete by trying to find a hedge for a call, afterwards calculate the set of risk-neutral measures.
"<=" market complete => for each derivate $D$ (including indicator variables for each measurable subset) a hedge $H$ exists such that $$D_t = \mathbb{E}_t^Q[D_T e^{r_{T-t}}] ~~~~~~~~~~~~~~(1)$$ for any risk-neutral measure $Q$ and an $t\geq0$ where $r$ is the discount-rate. Now observe $D_0$ is deterministic (not random) and the same whatever $Q$ we use for pricing. Now assume there a two different risk neutral measure $Q_1,Q_2$ such that $Q_1(A)\neq Q_2(A)$ for some measurable subset. If we "hedge" the indicator variable $1_A$ we get $$\mathbb{E}_0^{Q_1}[1_A e^{r_{T-t}}]=Q_1(A)$$ and $$\mathbb{E}_0^{Q_2}[1_A e^{r_{T-t}}]=Q_2(A).$$ But, this conflicts $(1)$ since the above would induce $\mathbb{E}_0^{Q_1}[1_A e^{r_{T-t}}]\neq\mathbb{E}_0^{Q_2}[1_A e^{r_{T-t}}]$ , hence there exists only one risk neutral measure $Q$ and therefore only one state price vector.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.