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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.