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findSPV

# findSPV - Stat/Actsc 446/846 Mathematical Models in Finance...

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Stat/Actsc 446/846: Mathematical Models in Finance – Fall 2011 Finding state-price vectors when M > N There are two main approaches for finding state-price vectors when M > N . Approach 1: If we’re given a specific matrix S (1 , Ω) and vector of initial prices ~ S (0), we can write out the system of N equations, parameterize the solution and see if we can find solutions that are positive. Approach 2: This is the approach used to prove that an arbitrage-free market admits at least one state-price vector, when M > N . The main idea is to add Arrow-Debreu (AD) securities in a way that preserves the no-arbitrage property. We illustrate the two approaches with an example. Suppose ~ S (0) = (1 , 1) and S (1 , Ω) = 1 0 1 1 1 0 1 3 . Approach 1: Let ~ ψ = ( ψ 1 , . . . , ψ 4 ). Then we need to see if we can solve ψ 1 + ψ 2 + ψ 3 + ψ 4 = 1 ψ 2 + 3 ψ 4 = 1 using a positive vector ~ ψ . Let s = ψ 3 , then ψ 4 = (1 - s ) / 3 and if t = ψ 1 , then ψ 2 = 1 - s - t - (1 - s ) / 3 = 2 / 3 - (2 / 3) s - t . So as long as s < 1 and t < (2 / 3)(1 - s ), then ψ j > 0 for all j . So for instance if s = t = 1 /

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findSPV - Stat/Actsc 446/846 Mathematical Models in Finance...

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