Skip to content
All library documents

Calculating State Prices by Replicating Arrow–Debreu Payoffs

Article Quant Q&A · Author: Emil
Read the original

Summary

The document explains how to derive state prices from asset payoffs by constructing portfolios that replicate Arrow–Debreu securities. For each state, solve a linear system in which the payoff matrix multiplied by portfolio weights equals a payoff of one in that state and zero in the others. The state price is then the cost of the replicating portfolio, calculated from its weights and the assets’ current prices.

A three-asset example demonstrates the procedure for the first state and gives its resulting price; the remaining two states are specified as analogous systems. The answer does not show the arithmetic for those latter solutions, nor does it discuss broader assumptions such as market completeness or arbitrage-free pricing. The method therefore illustrates the linear-algebra calculation in a finite-state setting, while leaving those conditions implicit.

Key ideas

  • A state price is the current cost of a portfolio that replicates a unit payoff in one state and zero in others.
  • Find the replicating portfolio by solving a linear system based on asset payoffs.
  • Compute the state price as the weighted sum of asset prices using the portfolio weights.
  • The example gives the procedure for one state and sets up equivalent systems for the others.

Tags

Derivatives pricingStatisticsPortfolio construction
Full text
# How to find state prices?


# 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

## Answer by Kevin (score 2, accepted)

https://quant.stackexchange.com/a/69656

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 $q_3=w_1+w_2+w_3$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.