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I am trying to find out how to solve state prices, but I do not know what I am supposed to do, my professor has given a solution to this problem as being (0.060 0.417 0.476), but I can't figure out how he gets there and he says the deadline for asking questions are over.

Problem information

Any help is much appreciated

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1 Answer 1

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You seek the price of a hedging portfolio which replicates the payoff of an Arrow-Debreu asset.

State 1

We seek $w_1,w_2,w_3$ such that

\begin{align} \begin{pmatrix} 1.05 & 1.8 & 1\\ 1.05 & 1 & 1\\ 1.05 &1 &1.1 \end{pmatrix}\begin{pmatrix} w_1\\ w_2\\ w_3 \end{pmatrix}=\begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}. \end{align}

With a little help, the solution is $w_1=-\frac{25}{21}$, $w_2=\frac{5}{4}$ and $w_3=0$.

Thus, because the price of each asset is one ($p_i=1$), the state price for state 1 is \begin{align} q_1 = p_1w_1 + p_2w_2+p_3w_3 = \frac{5}{84}\approx0.060. \end{align}

The other states

To find the price of state 2, you have to solve \begin{align} \begin{pmatrix} 1.05 & 1.8 & 1\\ 1.05 & 1 & 1\\ 1.05 &1 &1.1 \end{pmatrix}\begin{pmatrix} w_1\\ w_2\\ w_3 \end{pmatrix}=\begin{pmatrix} 0 \\ 1 \\ 0\end{pmatrix} \end{align} and set $q_2=w_1+w_2+w_3$.

For state price 3, you have to look at \begin{align} \begin{pmatrix} 1.05 & 1.8 & 1\\ 1.05 & 1 & 1\\ 1.05 &1 &1.1 \end{pmatrix}\begin{pmatrix} w_1\\ w_2\\ w_3 \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix} \end{align}
and again set $q_3=w_1+w_2+w_3$.

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  • $\begingroup$ Thanks a lot for the super solution @Kevin, I finally get it now. $\endgroup$
    – Emil
    Jan 27, 2022 at 9:54

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