= Risk-neutral measure
{title2=$Q\sim P,\quad S/B\text{ is a }Q\text{-martingale}$}
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= Equivalent martingale measure
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A <risk-neutral measure> $Q$ is a <probability measure> equivalent to the physical <probability measure> $P$, meaning they agree on which events have zero <probability>. For assets without intermediate payments, prices expressed in the chosen <numéraire> must be <martingales> under $Q$. For a positive <bank account> $B$ and a <stock> without dividends, this requires $S/B$ to be a <martingale>; with dividends, use <discounted dividend gains>. In an unrestricted finite-state one-period <financial market>, existence of such a measure is equivalent to absence of <arbitrage>, by the <fundamental theorem of asset pricing>.
Physical <probability> $p=P(\text{up})$ describes the modeled frequency or belief about an up move. The <risk-neutral probability> $q=Q(\text{up})$ is constrained by traded prices. In the worked <binomial market>, $q=5/8$ even if $p=1/2$ or $p=4/5$. Calling $Q$ risk-neutral does not assume that investors have no <risk aversion>; it changes the weights used for <risk-neutral pricing>.
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