21 Pages Posted: 28 May 2016
Date Written: May 27, 2016
Since most of the traded options on individual stocks is of American type it is of interest to generalize the results obtained in semi-static trading to the case when one is allowed to statically trade American options. However, this problem has proved to be elusive so far because of the asymmetric nature of the positions of holding versus shorting such options.
Here we provide a unified framework and generalize the fundamental theorem of asset pricing (FTAP) and hedging dualities in Bayraktar and Zhou (2015, to appear in Annals of Applied Probability, also available at http://ssrn.com/abstract=2569256) to the case where the investor can also short American options. Following this earlier work, we assume that the longed American options are divisible. As for the shorted American options, we show that the divisibility plays no role regarding arbitrage property and hedging prices. Then using the method of enlarging probability spaces proposed in Deng and Tan (2016, ArXiv e-prints), we convert the shorted American options to European options, and establish the FTAP and sub- and super-hedging dualities in the enlarged space both with and without model uncertainty.
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