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Unformatted text preview: 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 were given a specific matrix S (1 , ) 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 , ) = 1 0 1 1 1 0 1 3 . Approach 1: Let ~ = ( 1 ,..., 4 ). Then we need to see if we can solve 1 + 2 + 3 + 4 = 1 2 + 3 4 = 1 using a positive vector ~ . Let s = 3 , then 4 = (1- s ) / 3 and if t = 1 , then 2 = 1...
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This note was uploaded on 03/17/2012 for the course ACTSC 446 taught by Professor Adam during the Spring '09 term at Waterloo.
- Spring '09