Why when considering numéraires, one cannot use a dividend paying asset to define a risk neutral measure?

Here's where I got my question : (Shreve - Stochastic Calculus For Finance II)

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    $\begingroup$ I am not entirely sure what you mean. A dividend paying asset $S_t$ (such as a stock) can be used to define an equivalent pricing measure $\Bbb{Q}^S$ under which prices of self-financing strategies - when expressed into the stock numéraire - are martingales. It's actually one way of obtaining the $N(d_1)$ term in the BS formula and interpreting it as $\Bbb{Q}^S(S_T > K)$ (while $N(d_2)$ is $\Bbb{Q}^B(S_T > K)$ where $B_t$ is the risk-free money market account hence $\Bbb{Q}^B$ the risk-neutral measure). $\endgroup$ – Quantuple Sep 13 '18 at 14:02
  • $\begingroup$ The only caveat IMO is that a numéraire should be a traded asset with an always positive price (which is OK under the dividend yield extension of BS, but needs greater care when dividends are discrete and fixed rather than proportional). $\endgroup$ – Quantuple Sep 13 '18 at 14:05
  • $\begingroup$ Thank you. I edited my post with what's in Shreve's book. So if you say that you can define a measure with respect to $S$ paying a dividend, you mean that you can find a that measure but it's not a risk neutral one? $\endgroup$ – user30614 Sep 13 '18 at 15:54

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