All martingale measures price the attainable claim equally

Question

Background Information:

This question is from Lectures on Financial Mathematics: Discrete Asset Pricing.

Theorem 3.2 First Fundamental Theorem of Asset Pricing - Suppose $\nu$ is any measure such that $S/S^{0}$ is a $\nu$-martingale. For an attainable claim $X$ with replicating strategy $\phi$ and $0\leq t\leq T$, we have $$V_t(\phi) = E_{\nu}\left(X\frac{S_t^{0}}{S_T^{0}}|\mathcal{F}_t\right)$$

Question:

Prove that:

1. All martingale measures price the attainable claim equally, and

2. if there is a martingale measure, then all replicating strategies for a given claim have the same value at all times.

I am sort of confused even where to begin, some guidance or suggestions may help.

Answer 1

For Question 1, let $\phi$ be a replicating strategy, that is, $V_T(\phi) = X$. Then for any two martingale measures $u$ and $v$, from the First Fundamental Theorem of Asset Pricing, \begin{align*} E_u\left(X\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right) = V_t(\phi), \end{align*} and \begin{align*} E_v\left(X\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right) = V_t(\phi). \end{align*} That is, all martingale measures price the attainable claim equally.

For Question 2, let $\mu$ be the martingale measure. Moreover, let $\phi$ and $\psi$ be two replicating strategies, that is, $V_T(\phi)= V_T(\psi)=X$. Then, for any time $t$, \begin{align*} V_t(\phi) &= E_{\mu}\left(V_T(\phi)\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right)\\ &= E_{\mu}\left(X\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right)\\ &= E_{\mu}\left(V_T(\psi)\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right)\\ &= V_t(\psi). \end{align*} That is, all replicating strategies for a given claim have the same value at all times.