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## Monday, October 13, 2014

### another (2nd) numeraire paradox

(This scenario is actually a 2-period world, well-covered in [[math of financial modeling and inv mgmt]]. However, this is NOT the simplest problem using a bank account or bond as numeraire. )

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
G_T(w2) = \$12.

Under G-measure, we are given RN PrG (w1) = PrG (w2) = 50%. It seems at time-0 (right now) G_0 must be 9, but it turns out to be 7!

Key - this RNPG is inferred from (and must be consistent with) the current market price of another asset [1]. In fact I believe any asset's current price must be consistent with this G-measure RNPG. I guess the discounted expected payout equals the time-0 price.

Now can there be a 0% interest bank account B? In other words, is it possible to have B_T = B_0 = \$1? Well, this actually implies a PrG (w1) = 5/7 (Verified!), not 50%. So this bank account's current price is inconsistent with whatever asset used in [1] above. Arbitrage? I guess so.

I think it's useful to work out (from the [1] asset's current price) the bond current price  Z_0 = \$0.875. This implies a predicable drift rate. I would say all assets (G, X, Z etc) have the same drift rate as the bond numeraire.

Next, it's useful to work out that under Z-measure the RN Prz (w1) = 66.66% and Prz (w2) = 33.33%, very different RNPG values.

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

1) 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 and more tricky. Given such a numeraire with multiple outcomes, it's useful to infer the bond numeraire.

2) When I must work with such a numeraire, I usually have

G_T(w1),
G_T(w2),
G_0,
X_T(w1),
X_T(w2)

* If I also have X_0 then I can back out Risk Neutral PrG(w1) and PrG(w2)
* alternatively, I can use X as numeraire and back out PrX(w1) and PrX(w1)
* If on the other hand we are given some of the PG numbers, then we can compute X_0 i.e. price the asset X.

[1] Here's one such asset X_0 = 70 and X(w1) = 60 and X(w2) = 120.