Lecture 3

that is it is impossible to nd a portfolio that pays

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Unformatted text preview: cient, we ¯ ¯ have V ∩ D = ∅. Therefore by the separating hyperplane theorem, there exists 0 ̸= λ ∈ RI such that ⟨λ, v ⟩ ≤ ⟨λ, u + d⟩ for any v ∈ V and d ∈ RI . Letting ¯ ++ di → ∞, it must be λi ≥ 0, so λ ∈ RI . Letting d → 0, we get ⟨λ, v ⟩ ≤ ⟨λ, u⟩, ¯ + which implies I I λi u(xi ) ≤ i=1 λi u(¯i ) x i=1 for any feasible {xi }. Hence {xi } maximizes W over all feasible allocations. ¯ 4.2 Asset pricing Consider an economy with two periods, denoted by t = 0, 1. Suppose that at t = 1 the state of the economy can be one of s = 1, . . . , S . There are J assets in the economy, indexed by j = 1, . . . , J . One share of asset j trades for price qj at time 0 and pays Vsj in state s. (It can be Vsj < 0, in which case the holder of one share of asset j must deliver −Vsj > 0 in state s.) Let q = (q1 , . . . , qJ ) the vector of asset prices and V = (Vsj ) be the matrix of asset payoffs. Define W = W (q, V ) = −q ′ V 2014W Econ 172B Operations Research (B) Alexis Akira Toda be the (1 + S ) × J matrix of net payments of one share of each asset in each state. Here, state 0 is defined by time 0 and the presence of −q = (−q1 , . . . , −qJ ) means that in order to receive Vsj in state s one must purchase one share of asset j at time 0, thus paying qj (receiving −qj ). Let θ ∈ RJ be a portfolio. (θj is the number of shares of asset j an investor buys. θj < 0 corresponds to shortselling.) The net payments of the portfolio θ is the vector −q ′ θ Wθ = ∈ R1+S . Vθ Here the investor pays q ′ θ at t = 0 for buying the portfolio θ, and receives (V θ)s in state s at t = 1. Let ⟨W ⟩ = W θ θ ∈ RJ be the set of payoffs generated by all portfolios, called the asset span. We say that the asset span ⟨W ⟩ exhibits no-arbitrage if ⟨W ⟩ ∩ R1+S = {0} . + That is, it is impossible to find a portfolio that pays a non-negative amount in every state and a positive amount in at least one state. Theorem 10 (Fundamental Theorem of Asset Pricing). The asset span ⟨W ⟩ exhibits no-arbitrage if and only if there exists π ∈ RS such that [1, π ′ ]W = 0. ++ In this case, the asset prices age given by S qj = πs Vsj . s=1 πs > 0 is called the state price in state s. Proof. Suppose that such a π exists. If 0 ̸= w = (w0 , . . . , wS ) ∈ R1+S , then + S [1, π ′ ]w = w0 + πs ws > 0, s=1 so w ∈ ⟨W...
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This document was uploaded on 02/18/2014 for the course ECON 172b at UCSD.

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