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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 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.