Pricing Contingent Claims from Arrow-Debreu State Prices
Summary
The document considers two equivalent ways to represent a complete market: state-contingent Arrow-Debreu securities and a set of complex securities whose payoff vectors span all possible states. It asks how to express the equilibrium price of each complex security using state prices and its payoffs across states.
The proposed method is to replicate each Arrow-Debreu payoff using a portfolio of the complex securities. In a complete market, the full-rank payoff matrix makes such replication possible, and the price of a security follows from the prices assigned to its state-contingent payoffs. The exercise’s displayed formula also includes a risk-free discount factor, though the question’s notation and timing convention are not fully clarified. No worked solution or numerical evidence is included, so readers should check whether the state prices are already discounted before applying another discount factor. The core lesson is the link between replication, completeness, and no-arbitrage pricing.
Key ideas
- A complete market allows state-contingent payoffs to be replicated using the available securities.
- A full-rank payoff matrix supports replication across all states.
- A security’s value can be derived from state prices weighted by its payoff in each state.
- Check the timing and discounting convention to avoid applying a discount factor twice.
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Full text
# Arrow-Debreu Equilibrium Pricing
# Arrow-Debreu Equilibrium Pricing
I have this problem in asset pricing that I don't know how to solve. Here it is: Consider an economy with a complete set of Securities and $N$ states of the world Tomorrow. Assume that there are two alternative set of assets: either a) there is a complete set of Arrow-Debreu Securities with price $q_{\theta}$ for the one with nonzero payoff in state $\theta$ or b) there is a complete set of complex Securities (and no A-D security) denoted by $j=1,...,N$ with price $q_j$.
Consider the equilibrium of the economy with the two different asset structures (and assume it is unique). $(...,\hat{q}_{\theta},...)$ is the equilibrium of the economy with a complete set of A-D Securities; $(...,\bar{q}_{j},...)$ is the equilibrium in the economy with a set of complete Securities, $j=1,...,N$.
Show that, for each asset $j$, $$\bar{q}_{j}=\frac{1}{1+r_f}\sum_{\theta}\hat{q}_{\theta} R_{j \theta}$$
My approach:
Since markets are complete, $R$ denotes the $N\times N$ full-ranked matrix of payoffs for the securities, with typical column $$ R_j = [R_{j1},..., R_{jN}] $$
Let $\bar{q_j} = [q_{j1},...,q_{jN}]$ be the vector of asset prices. Then, it is possible to determine a vector $\psi$ such that: $$ \bar{q_j} = \psi_{\theta} R_{\theta} $$ Where $\psi_{\theta}$ is the price of the portfolio of complex securities delivering a unit of A-D security. Therefore, $\bar{q}_j$ can be explicited as a summation, and discounting at period one with the risk free rate, yields $$ \bar{q}_{j}=\frac{1}{1+r_f}\sum_{\theta}\hat{q}_{\theta} R_{j \theta} $$
Now, I don't know if my solution is correct, because to me it seems too much naive... I hope that someone can help!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.