Tradability, Numeraires, and Risk-Neutral Pricing in Interest Rate Models
Summary
The document explores the meaning of a tradable asset in the numeraire pricing theorem, using short rates and interest rate models as examples. Under a measure associated with a chosen numeraire, the price of a tradable asset divided by that numeraire is a martingale. The question is whether the short rate itself qualifies, and whether market completeness makes every square-integrable payoff tradable or replicable.
The answer cautions that the short rate is not generally a directly tradable asset. Instead, expected future short rates relate to forward rates, which can be viewed approximately through traded rate instruments or tight calendar spreads. It also links a nontradable underlying to market incompleteness and the appearance of a market price of risk in term-structure models. This is a brief conceptual response, not a formal proof or a full account of completeness assumptions; the claim about forward rates is qualified and should not be read as exact equivalence.
Key ideas
- Numeraire pricing applies the martingale property to the price of a tradable asset relative to the numeraire.
- A short rate is not ordinarily itself a traded asset.
- Forward rates are related to expected future short rates and can be approximated through traded interest rate instruments.
- The response associates nontradable underlyings with market incompleteness and a market price of risk.
- The discussion does not prove that completeness makes every square-integrable payoff replicable.
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Full text
# condition of risk neutral pricing
# condition of risk neutral pricing
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:
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?
## Answer by user34971 (score 1)
https://quant.stackexchange.com/a/42296
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.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.