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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 However, the price $R_u = R_d = e^{rT}$ includes the interest, so no interest has been withdrawn—the interest has been reinvested—prior to the maturity T of the option.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...)

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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.

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  • $\begingroup$ Thanks for this counterexample. Is there a deeper explanation (economic or mathematical) on this restriction? $\endgroup$ – Vim Feb 24 at 13:13
  • $\begingroup$ 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. $\endgroup$ – dm63 Feb 24 at 19:20
  • $\begingroup$ Anything can be a numeraire. This is really a tautological statement about the implied risk-neutral measure. $\endgroup$ – mathtick Apr 7 at 11:35

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