Why must we only use a non-dividend paying asset as a numeraire? I read in Shreve's book (number 2):"Associated with each numeraire, we shall have a risk-neutral measure. When making this association, we shall take only non-dividend-paying assets as numeraires."I'm not sure I understand why this is the case and why it must also be therefore non-dividend paying?!If we have a dividend paying stock then in the real world measure:[$]\mathrm{d}S = S(\mu - q) \mathrm{d}t + S \sigma \mathrm{d}W[$]where [$] q [$] is the continuously compounded dividend rate (to avoid confusion with the symbol [$]d[$]). Why can I now not use [$]S[$] this as numeraire?Any help would be much appreciated.

When you change numeraire you are basically switching the probability measure in which you take expectations. Say you price some contract [$]C[$] with payoff [$]f(S_T)[$] in the risk neautral measure [$]\mathbb{Q}[$] where [$]B_t=e^{-rt}[$] is your numeraire:[$]C(t)/B(t)=\mathbb{E}_t[\frac{f(S_T)}{B_T}][$]Then look to re-write the expectation in a different, but equivalent, measure (denoted by a tilde). We can use any strictly positive, normalised [$]\mathbb{Q}[$] martingale [$]X_t[$], ie a Radon-Nikodym derivative, to define this new measure:[$]\tilde{\mathbb{E}}_t[Y_s]=\mathbb{E}_t[Y_s X_s]\Leftrightarrow \tilde{\mathbb{E}}_t[Y_s(X_s)^{-1}]=\mathbb{E}_t[Y_s][$]for any random variable [$]Y[$].Assuming [$]S_t/B_t[$] is a [$]\mathbb{Q}[$] martingale, we can then use [$]X_t = S_t/B_t \cdot B_0/S_0[$] and write[$]C(0)/B(0)=\tilde{\mathbb{E}}_t[\frac{f(S_T)}{B_T}\frac{B_T}{S_T}\frac{S_0}{B_0}]\Rightarrow [$][$]C(0)/S(0)=\tilde{\mathbb{E}}_t[\frac{f(S_T)}{S_T}[$]].The point of all this rambling is that your dividend paying stock over the bank account is not a [$]\mathbb{Q}[$] martingale (it drifts downwards at rate [$]d[$]), so not a valid Radon-Nikodym derivative, and cannot be used to define an equivalent measure to [$]\mathbb{Q}[$]. All we know is that all pure assets or self-funding trading strategies are [$]\mathbb{Q}[$] martingales. The dividend paying stock is not self funding as it generates dividend cashflows. Instead you would have to define a strategy, for example "hold the stock and re-invest all dividends in the stock". That vaue of that strategy is a [$]\mathbb{Q}[$] martingale and could be used as a numeraire.

That's useful, thanks. So if we defined the portfolio of the stock paying dividend with dividends reinvested in the stock, then we could reasonably use that as a good choice of numeraire?

In fact, double thanks. Looking back at Shreve 2 Pg 213-215 it all makes great sense. I obviously forgot that a valid Radon-Nikodym derivative process should be a martingale in the [$]\mathbb{Q}[$] measure and clearly as you showed above, this isn't satisfied.