How to prove the "Law of one price" theorem?

Question

There are two subparts to Fundamental Asset Pricing theorem.

1. The Law Of One Price (LOOP thereafter) holds if and only if there exists a state price vector.
2. 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 1

I just comment your second point, because in the definition i now of the LOP the state price vector (martingale measure) is involved.

2. 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.