= Solution
European <put-call parity> gives
$$
\boxed{C_E(0)=P_E(0)+15-15/R^2=\tfrac13+15-\tfrac{48}5=\tfrac{86}{15}.}
$$
For the American call, there are no dividends and the riskless rate is positive, so early exercise sacrifices the benefit of deferring the strike. Directly, at stock 30 its continuation value is $(4/5)[(1/6)30+(5/6)21]=18$, exceeding its exercise value 15; at stock 12 it is $(4/5)(5/6)=2/3$, exceeding exercise value zero. At time zero continuation is $86/15>0$. Thus \b[$C_A(0)=C_E(0)=86/15$], with no optimal early exercise. European parity must not be applied using the American put price $3/2$.
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