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Consistent price systems and face-lifting pricing under transaction costs. (English) Zbl 1133.91422

Summary: In markets with transaction costs, consistent price systems play the same role as martingale measures in frictionless markets. We prove that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs. This result applies to a large class of Markovian and non-Markovian models, including geometric fractional Brownian motion.
Using the constructed price systems, we show, under very general assumptions, the following “face-lifting” result: the asymptotic superreplication price of a European contingent claim \(g(S_T)\) equals \(\hat g(S_{0})\), where \(\hat g\) is the concave envelope of \(g\) and \(S_t\) is the price of the asset at time \(t\). This theorem generalizes similar results obtained for diffusion processes to processes with conditional full support.

MSC:

91G80 Financial applications of other theories
60G99 Stochastic processes
91B24 Microeconomic theory (price theory and economic markets)
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