Why can only non-dividend paying assets serve as numeraire?

In Kerry Back, A Course in Derivative Securities, Sect. 1.4 (page 29), the author stated the FTAP in the following form (in boldface):

If there are no arbitrage opportunities, then for each (non-dividend-paying) asset, there exists a probability measure such that the ratio of any other (non-dividend-paying) asset price to the first (numeraire) asset price is a martingale.

And he commented that

We have applied this statement to the risk-free asset, which pays dividends (interest). However, the price $$R_u = R_d = e^{rT}$$ includes the interest, so no interest has been withdrawn—the interest has been reinvested—prior to the maturity T of the option. This is what we mean by a “non-dividend-paying” asset. In general, we will apply the formulas developed in this and the following section to dividend-paying assets by considering the portfolios in which dividends are reinvested.

In such spirits, when we want to use a dividend paying stock $$S_t$$ as numeraire we should actually use the "reinvested" asset $$e^{qt}S_t$$ as the numeraire instead.

My question is, what would be the consequences of really using a div-paying asset, such as $$S_t$$ (when $$q>0$$) as numeraire? Is it possible to make the corresponding martingale measure not exist, for example? (But in my opinion, the existence of a martingale measure seems only to rely on positivity...)

1 Answer

Well, consider using $$S_t$$ as the numeraire and let the asset be the reinvested stock $$S_te^{qt}$$. Then this ratio equals $$e^{qt}$$ so can never be a martingale.

• Thanks for this counterexample. Is there a deeper explanation (economic or mathematical) on this restriction? – Vim Feb 24 '19 at 13:13
• The example of a discrete dividend might help : the ratio of some asset to the numeraire will systematically jump upwards on the date that the numeraire pays a dividend. That’s not a martingale. Basically a dividend paying asset is a different class of asset than a reinvested one. The residual price of the dividend paying asset can’t perform as well as the non paying one, on average. – dm63 Feb 24 '19 at 19:20
• Anything can be a numeraire. This is really a tautological statement about the implied risk-neutral measure. – mathtick Apr 7 '19 at 11:35