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 However, the price $$R_u = R_d = e^{rT}$$ includes the interest, so no interest has been withdrawn—the interest has been reinvested—prior to the maturity T of the option.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

## 2 Answers

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

I am not familiar with the deep mathematical intricacies of advanced no-arbitrage theory, an extremely technical subject. However, from reading literature reviews, I suspect this is an historical legacy of the research path that led to the most general versions of no-arbitrage theory.

If you consider dividend-paying assets whose dividends are not continuously reinvested, then you actually need to model the dynamics of your asset using a jump process to represent the out-coming (dividend, coupon) cash flow: indeed, if $$S_t$$ is your stock price process, then when a dividend of amount $$D$$ is paid at $$t^-$$, then the price of the stock should immediately jump downards: $$S_{t^+}=S_{t^-}-D$$

It turns out introducing jump processes does seem to compound considerably the mathematical difficulty of formalising no-arbitrage theory. The seminal paper proving the most general version of the no-arbitrage theorem is "A general version of the fundamental theorem of asset pricing" (Delbaen and Schachermayer, 1993). In page 2, they write (my emphasis):

We believe that the main theorem (Theorem 1.1 below) of this paper contributes to both theories mathematics as well as economics. In economic terms the theorem contains essentially two messages. First that it is possible to characterise the existence of an equivalent martingale measure for a general class of processes in terms of the concept of no free lunch with vanishing risk, a concept to be defined below. In this notion the aspect of vanishing risk bears economic relevance. The second message is that - in a general setting - there is no way to avoid general stochastic integration theory. If the model builder accepts the possibility that the price process has jumps at all possible times, he needs a sophisticated integration theory, going beyond the theory for "simple integrands". In particular the integral of unbounded predictable processes of general nature has to be used. [...]

Further, in pages 4 and 5 they write (my emphasis):

To relate our work to earlier results, let us summarise the present state of the art. The case when the time set is finite is completely settled in Dalang et al. (1989) and the use of simple or even elementary integrands is no restriction at all (see Schachermayer (1992), Kabanov and Kramkov (1993) and Rogers (1993) for elementary proofs). For the case of discrete but infinite time sets, the problem is solved in Schachermayer (1993). The case of continuous and bounded processes in continuous time, is solved in Delbaen (1992). In these two cases the theorems are stated in terms of simple integrands and limits of sequences and by using the concept of no free lunch with bounded risk. We shall review these issues in Sect. 6.

In the general case, i.e. a time set of the form $$[0,\infty[$$ or $$[0, 1]$$ and with a possibility of random jumps, the situation is much more delicate. [...]