Why is the numeraire in the LGM model tradeable?

Question

I'm trying to understand the LGM model, which Hagan defines as follows. The state variable $X$ evolves according to $$dX(t) = \alpha(t) dW^N(t)$$ wrt the numeraire $$N(t) = \frac{1}{P(0,t)} e^{H(t)X(t)+H^2(t)\int_0^t\alpha^2(s)ds}.$$ The functions $H$ and $\alpha$ are deterministic and can be chosen (almost) arbitrarily.

I would like to understand why $N$ is even eligible as a numeraire in the first place. It is positive, but without any other assumption I don’t see how it must be a tradeable asset.

Also the SDE for $X$ depends on $W^N$, which depends on $N$, which in turn depends on $X$. I understand that assuming everything is well defined and $N$ is a valid numeraire we can derive the explicit form under $Q$ by Girsanov, but isn’t the definition a bit circular?

Answer 1

The confusion is that you think that we define the numeraire as this exponential function... It is not the case. We give the numeraire properties to $N$, then we model it. Similar to any other model.

All we know is that $N$ is positive, and we have $$\frac{V_t}{N_t}=E^{N}\left[\frac{V_T}{N_T}|\mathbb{F_t}\right]$$ where $V_t$ is a tradable asset.

$N$ can be the money market account, a zero coupon bond paying at time $T$... or it can stay "no-name".it is actually irrelevant to define what it is exactly, I know it can be misleading not to be able to picture the numeraire.

Whether it is defined or not, we always need to model the numeraire, that is the second step. When we model a zero-coupon bond, we make sure it has properties that do not violate what we defines in the first paragraph. Same thing with this numeraire $N$.

As far as the measure is concerned, if it is circular, then it is the same problem for all models. When you use the simplest short-rate model under the risk-neutral measure, the numeraire is the money market account , and it is a function of the short-rate model...

It is the same story here, we define first a numeraire $N$ ( just naming it and give it all the numeraire properties), and we say that the numeraire is driven by an SDE $X$ under the measure that makes all the ratio $\frac{V_t}{N_t}$ martingale where $V$ is an asset.