Monday, December 19, 2016

pricing - FTAP a-la Harrison, Kreps and Pliska


I was reading the papers co-authored by Harrison, Kreps and Pliska, that initiated the formal research on the connection between pricing, martingale measures, arbitrage and completeness. I have some issues relating some of the ideas, though, and would be happy if somebody could help me.



In a more or less abstract setting, one considers an ordered topological space X of cash flows, a subspace MX of traded instruments and a positive linear functional π:MR. The 1st FTAP that relates absence of arbitrage to the existence of EMM concerns continuous extension of π to X which is still positive. As an example, one can consider M={m=m0+T0HtdSt} where H is some "good" strategy and S is a vector of traded assets. In such case, one defines π(m)=m0 as an initial cash endowment required to get m at the maturity time T.


I guess, the existence of a unique positive extension of π to the whole X implies the completeness of the market: that is, we can price everything in a consistent way and there is a unique way to do this. Apparently, this automatically leads to the representability of any xX as x0+T0HtdSt.


At the same time, I do not understand, how the existence of at least one continuous positive extension may be related to the absence of arbitrage. That is, I would imagine an arbitrage being rather existence of two strategies H and H such that m'_0 + \int_0^T H'_t\;\mathrm dS_t = m''_0 + \int_0^T H''_t\;\mathrm dS_t with m'_0 \neq m''_0. But this rather says \pi is not uniquely defined even on M instead of it is not possible to extend \pi to X. Does that mean that the latter two statements are equivalent?




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