In more complex situations linear algebra offers a

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analysis. In more complex situations, linear algebra offers a useful frameworkfor identifying replicating portfolios.Consider a general context where there areMpossible outcomes and wewish to replicate a payoff vectorb∈ <MusingNavailable contingent claimswith payoffs encoded in a matrixP∈ <M×N. A portfoliox∈ <Nreplicatesthe payoffsbif and only ifPx=b.To obtain a replicating portfolio, onecan use software tools to solve this equation. If there are no solutions to thisequation, there is no portfolio that replicates
b.Can every possible payoff vector be replicated by contingent claims avail-able in a market?
2.4.3Pricing and ArbitrageWe now turn our focus from payoff to prices.Consider a market withNcontingent claims, and letρ∈ <Nbe the row vector of prices. Suppose thata new contingent claim is introduced, but since it has not yet been traded,there is no market price. How should we price this new contract?

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