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The theorem says if $U$ is a numeraire and let $\mathbb{Q}^U$ be the corresponding measure. Then for every tradable asset $S$, the relative price $S_t/U_t$ is a martingale under $\mathbb{Q}^U$. But I don't know the meaning of "tradable asset". For example: in Brigo's "interest rate models" p39:

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In this proof, $\mathbb{E}^T$ means the expectation under the forward measure( the numeraire is $T$-bond $P(t,T)$). $\mathbb{E}$ means the expectation under the risk-neutral measure (numeraire $B(t)=e^{\int_0^tr_s\,ds}$) I think he uses the risk-neutral pricing formula in the first equation: $$\mathbb{E}^T[r_T|\mathcal{F}_t]=r_t/P(t,T)$$ $$\mathbb{E}[e^{-\int_0^Tr_s\,ds}r_T|\mathcal{F}_t]=r_te^{-\int_0^tr_s\,ds}$$

my question is: 1. why the short rate $r_t$ is a tradable asset?

Let $H$ be the payoff function. In practice, I often apply the pricing formula as long as $H\in L^2$ regardless of whether it is a tradable asset or not. I know there is a complete market hypothesis: every derivative in the market is replicable.

my question is: 2 Does complete market implies $\forall H\in L^2$ is a tradable asset?

3 if the market is not complete, can we apply the pricing formula for $r_t$? i.e. how to judge $H$ is replicable or not?

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Who says the short rate is tradable? I think all the formula says is that the expected future short rate, which is the forward rate, which is basically an infinitely tight calendar spread, is (kind of) tradable.

A general rule is if the underlying asset is not tradable then the market is incomplete. The short rate is not tradable --> interest rate market is incomplete which is why the market price of risk appears in the term structure equation. I think Bjork in his book "Arbitrage Theory in Continuous Time" gives an excellent treatment of these topics.

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