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Answers for “为什么状态价格 $\psi_j$ 不是概率,而转换后的向量 $q_j$ 是?”

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Understand why

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The state prices ψj\psi_j represent the prices of Arrow-Debreu securities, which are monetary values and do not necessarily sum to 1. The transformed vector qjq_j represents risk-neutral probabilities, obtained by multiplying each state price by the gross risk-free factor (1+i)(1+i).

Conditions: The market is a one-period finite-state model.; ψj\psi_j are the solved state prices (Arrow-Debreu security prices).; ii is the per-period risk-free interest rate.; qjq_j is defined as qj=(1+i)ψjq_j = (1+i)\psi_j.

Find a method

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You convert a state price vector into risk-neutral probabilities by multiplying each state price by the gross risk-free factor (1+i)(1+i), where ii 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 ψj\psi_j are already determined.; ii is the known per-period risk-free interest rate.; The market is a one-period finite-state model.

Meet the concept

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An Arrow-Debreu security is a theoretical financial instrument that pays exactly 1 unit of currency in one specific state of the world and 0 in all other states. The state price vector is the collection of the current prices of all such Arrow-Debreu securities for the finite set of states in the market model.

Conditions: The market model has a finite set of states.; Payoffs are normalized to 1 in one state and 0 elsewhere.; The model is a one-period setting.