No-Arbitrage and Hedging with Liquid American Options

Since most of the traded options on individual stocks are of American type, it is of interest to generalize the results obtained in semistatic 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 [Bayraktar E, Zhou Z (2016) Arbitrage, hedging and utility maximization using semi-static trading strategies with American options. Ann. Appl. Probab. 26(6):3531-3558.] to the case where the investor can also short American options. Following Bayraktar and Zhou [Bayraktar E, Zhou Z (2016) Arbitrage, hedging, and utility maximization using semistatic trading strategies with American options. Ann. Appl. Probab. 26(6):3531-3558.], we assume that the longed American options are divisible. As for the shorted American options, we show that divisibility plays no role regarding arbitrage property and hedging prices. Then, using the method of enlarging probability spaces proposed in Deng and Tan [Deng S, Tan X (2016) Duality in nondominated discrete-time models for Americain options. ArXiv e-prints.], we convert the shorted American options to European options and establish the FTAP and subhedging and superhedging dualities in the enlarged space both with and without model uncertainty.