numeraire paradox

Consider a one-period market with exactly 2 possible time-T outcomes w1 and w2.

Among the tradable assets is G. At termination, G_T(w1) = $6 or G_T(w2) = $12. Under G-measure, we are given Pr(w1) = Pr(w2) = 50%. It seems at time-0 (right now) G_0 should be $9, but it turns out to be $7! Key – this Pr is inferred from (and must be consistent with) the current market price of another asset [1]. Without another asset, we can’t work out the G-distro. In fact I believe every asset’s current price must be consistent with this G-measure Pr … or arbitrage!

Since every asset’s current price should be consistent with the G-Pr, I feel the most useful asset is the bond. Bond current price works out to Z_0 = $0.875. This implies a predicable drift rate.

I would say under bond numeraire, all assets (G, X, Z etc) have the same drift rate as the bond numeraire. For example, under the Z-numeraire, G has the same drift as Z.

Q: under Z-measure, what’s G’s drift?
A: $7 -> $8

It’s also useful to work out under Z-measure the Pr(w1) = 66.66% and Pr(w2) = 33.33%. This is using the G_0, G_T numbers.

Now can there be a 0-interest bank account B? In other words, could B_T = B_0 = $1? No, since such prices imply a G-measure Pr(w1) like 5/7 (Verified!) So this bank account’s current price is inconsistent with whatever asset used in [1] above.

The most common numeraires (bank accounts and discount bonds) have just one “outcome”. (In a more advanced context, bank account outcome is uncertain, due to stoch interest rates.) This stylized example is different. Given a numeraire with multiple outcomes, it’s useful to infer the bond numeraire. It’s generally easier to work with one-outcome numeraires. I feel it’s even better if we know the exact terimnal price and the current price of this numeraire — I guess only the discount bond meet this requirement.

I like this stylized 1-period, 2-outcome world.
Q1: Given Z_T, Z_0, G_0, G_T [2], can i work out the G-Pr (i.e. distro under G-numeraire)? can i swap the roles and work out the Z-Pr ?
A: I think we can work out both distros and they aren’t identical !

Q2: Given G_0 and the G_T possible values[2] without Z prices, can we work out the G-Pr (i.e. distro under G-numeraire)?
A: no we don’t have a numeraire. In a high vs a low interest-rate world, the Pr implied by G_T would be different

[2] these are like pre-set enum values. We only know these values in this unrealistic world.

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