= Solution
The <European call option> with strike $5$ pays $2,0,0$ at terminal stock prices $7,5,3$. Its lower-node value is zero, while its upper-node value is
$$
V_u=R^{-1}\left(\frac78\cdot2\right)=\frac{14}{9}.
$$
A second step of <backward option pricing> gives
$$
V_0=R^{-1}\frac{13}{16}\frac{14}{9}
=\boxed{\frac{91}{81}}.
$$
By the <fundamental theorem of asset pricing>, this discounted expectation under the unique risk-neutral probabilities is the unique no-arbitrage price.
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