# How to find state prices?

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.

Any help is much appreciated

## 1 Answer

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 $$q_3=w_1+w_2+w_3$$.

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