What is an Arrow-Debreu security and how does it relate to the state price vector?

The solution for the state price vector is unique because the coefficient matrix of the resulting linear pricing system has a nonzero determinant. A nonzero determinant guarantees that the system of linear equations has exactly one unique solution for the unknown state prices.

Conditions: The market model has three states and three traded assets.; The pricing system is a 3x3 linear system.; The coefficient matrix of the system is nonsingular (determinant is nonzero).

To derive the linear pricing equations, you impose the no-arbitrage condition that the current price of each traded asset must equal the sum of its state-contingent payoffs weighted by the corresponding state prices. This produces one linear equation per traded asset, with the state prices acting as the unknowns.

Conditions: One-period model with a finite set of states.; Prices of all traded assets are known today.; State-contingent payoffs for each asset are known.; The unknowns are the state prices (Arrow-Debreu security prices).

You convert a state price vector into risk-neutral probabilities by multiplying each state price by the gross risk-free factor $(1+i)$, where $i$ is the per-period risk-free interest rate. This scaling transforms the monetary prices of Arrow-Debreu securities into a valid probability distribution that sums to 1.

Conditions: The state prices $\psi_j$ are already determined.; $i$ is the known per-period risk-free interest rate.; The market is a one-period finite-state model.

The arbitrage-free price of a European put option is calculated by first determining its state-contingent payoffs, decomposing this payoff vector into the Arrow-Debreu basis, and then taking the dot product of the payoff vector with the state price vector. This relies on the linearity of pricing in a finite-state market.

Conditions: The market is a one-period finite-state model.; The state price vector $[\psi_1, \psi_2, \dots]$ is known.; The option is European and expires at the end of the period.; The underlying asset's state-contingent payoffs and the strike price are known.

The state price vector for this specific three-state market model is $[24/55, 4/55, 2/5]$. This vector is found by solving the linear system of pricing equations derived from the given asset payoffs and their common current price of 100.

Conditions: One-period securities market model.; Three possible scenarios (states).; Risk-free bank account with 10% interest (payoff 110 in all states).; Current price of each security is 100.; Security X payoffs: 220, 55, 0.; Security Y payoffs: 0, 0, 250.