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If I understand well, you have a market with 3 states: up, flat or down. You have 3 instruments: The stock The risk-free rate (50%) The option If you can create a portfolio today with these 3 instruments that can replicate de payoff of the option you have to price, then the law of one price tells you that the price of the option should be the price of ...


The state price vector are the prices of securities which pay \$1 if and only if that state of the world occurs. This is just a question of being able to replicate the payoffs $$ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} $$ with payoff vectors $\vec{b} = [1,1,1]^T$ and ...

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