Stochastic Processes and Financial Mathematics
(part two)
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17.5 Incomplete markets \(\offsyl \)
In Theorem 15.2.5 we proved that we could replicate any contingent claim \(X\) (providing that \(\E ^\Q [X]\) exists, which is not a major restriction).
Essentially, replication relies on having enough independently tradeable commodities that we can create a portfolio that fully replicates the randomness built into the contingent claim \(X\).
In some markets, the number of (independent) sources of random information is much greater than then number of (independent) commodities that are traded. They are known as incomplete markets. In
practice, it is not particularly easy to judge if a given market fits into this class, but very many do.
In an incomplete market we can’t hedge every contingent claim. Consequently, we also can’t deduce arbitrage free prices for every contingent claim. Instead, it is possible to construct arbitrage based arguments to
say that particular relationships exist between contingent claims must exist; statements of the form ‘if the price of \(X\) is this then the price of \(Y\) must be that’. Then, by observing some carefully chosen
prices from the market, we have enough information (when combined with the usual arbitrage free pricing methods) to uniquely determine the price of any contingent claim. The ‘extra’ information, that is observed
via prices, essentially involves quantifying how risk averse the market is towards particular asset classes.
We won’t go into any details on this procedure in these notes. A good source of further information is Chapter 15 of the book ‘Arbitrage Theory in Continuous Time’ by Bjork.