Why are the state prices $\psi_j$ not probabilities, but the transformed vector $q_j$ is?

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.