I am trying to understand the connection between market completeness and risk neutral measures.

A market is complete if and only if the equivalent martingale measure is unique. But if I change to the $T$-forward measure ($\mathbb{Q}^T$), I have an equivalent measure to the money market account numéraire measure ($\mathbb{Q}$), so two different but equivalent measures. So the market is not complete?


The market is complete iff there is a unique risk-neutral measure: when every contingent claim is attainable, its unique no arbitrage price is the cost of the replicating portfolio.

In the case of an incomplete market, you no longer have a unique price for unattainable contingent claim, but rather a range of prices : $\left(-p\left(-G\right), p\left(G\right)\right)$ (see Bouchard & Chassagneux for notation) which correspond to the sub- and super-replication prices (lowest and highest prices beyond which the buyer or seller can make an arbitrage) for the European payoff $G$. If $\mathcal{Q}$ is the set of risk-neutral measures, $-p \left(-G\right) = \inf\limits_{\mathbb{Q} \in \mathcal{Q}}{\mathbb{E}^\mathbb{Q} \left(e^{- \int_0^T{r_t \mathrm{d}t}}G\right)}$ and $p \left(G\right) = \sup\limits_{\mathbb{Q} \in \mathcal{Q}}{\mathbb{E}^\mathbb{Q} \left(e^{- \int_0^T{r_t \mathrm{d}t}}G\right)}$ : there, you can see that there are infinitely many risk-neutral measures, and to each one corresponds a price within the price range.

The $T$-forward measure is not another risk-neutral measure: the asset used as numeraire is different (the $T$-zero coupon bond vs. the money market account). For any risk-neutral measure $\mathbb{Q} \in \mathcal{Q}$, you can get an equivalent $T$-forward measure $\mathbb{Q}^T \in \mathcal{Q}^T$, and you have similarly that $-p \left(-G\right) = B\left(0, T\right)\inf\limits_{\mathbb{Q}^T \in \mathcal{Q}^T}{\mathbb{E}^{\mathbb{Q}^T} \left(G\right)}$ and $p \left(G\right) = B\left(0, T\right)\sup\limits_{\mathbb{Q}^T \in \mathcal{Q}^T}{\mathbb{E}^{\mathbb{Q}^T} \left(G\right)}$.

If the market is complete, the set of risk-neutral measures is $\mathcal{Q} = \left\{\mathbb{Q}\right\}$, the set of $T$-forward measures is $\mathcal{Q}^T = \left\{\mathbb{Q}^T\right\}$ and the price of a European contingent claim with payoff $G$ is given by $$ V_0 = \mathbb{E}^\mathbb{Q} \left(e^{- \int_0^T{r_t \mathrm{d}t}}G\right) = B \left(0, T\right)\mathbb{E}^{\mathbb{Q}^T} \left(G\right) $$


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