= Risk-neutral pricing
{title2=$V_t=B_t\mathbb E_Q[X/B_T\mid\mathcal F_t]$}
= Risk-neutral valuation
{synonym}
In a finite-state discrete-time <financial market>, a replicable terminal <financial payoff> $X$ has the displayed value under any <equivalent martingale measure> $Q$ for the <bank account> <numéraire>. Here $\mathcal F_t$ represents current information and <conditional expectation> averages over future states. The <self-financing portfolio>'s discounted <portfolio wealth> is a <martingale>, so its current value equals the discounted <expectation> of its terminal value. In an <incomplete market>, nonreplicable claims need not have a unique price; continuous-time extensions require further conditions on <admissible trading strategies> and <integrable> discounted wealth.
For the <European call option> in the example,
$$
C_0=\frac{(5/8)20+(3/8)0}{21/20}=\frac{250}{21}\approx11.90476.
$$
Discounting the physical <expected value> instead gives $200/21\approx9.52381$ when $p=1/2$, or $320/21\approx15.23810$ when $p=4/5$. Those amounts describe discounted physical <expectations>, not the <replicating portfolio>'s cost. Holding $S_0,u,d,R$ fixed is crucial: changing beliefs can influence traded prices in a richer <financial market> model.
\Image[/pricing-probability-vs-belief.png]
{title=Physical expectations vary with beliefs while the one-step replication price stays fixed}
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