 Solution

Let $c$ be the initial call price. Write the risk-neutral probabilities of stock prices $5,3,2$ as $q_5,q_3,q_2$. The stock's martingale equation and the normalization give $q_2=2q_5$ and $q_3=1-3q_5$. The call pays only in the first state, so its martingale equation is $q_5=c$. ThusAll three objective probabilities are positive. This defines an equivalent martingale measure precisely when $0<c<1/3$, and part (iii) proves that every such price excludes arbitrage.

To prove necessity without appealing to an existence theorem, consider the other prices explicitly. If $c\le 0$, buy one call and hold $-c$ units of cash. Its initial value is zero and terminal value is $(S_1-4{)}^{+}-c$, nonnegative and positive with positive probability. If $c\ge 1/3$, take holdings $(h_B,h_S,h_C)=(c-1,1/3,-1)$. The initial value is $c-1+1-c=0$ and terminal values at $5,3,2$ are $(c-1/3,c,c-1/3)$. They are nonnegative and the middle-state value is strictly positive. This includes the endpoint $c=1/3$. Consequently