No-arbitrage and hedging with liquid American options

TL;DR

This paper extends the fundamental theorem of asset pricing and hedging dualities to include the trading of American options, both long and short, providing a unified framework for arbitrage and hedging in markets with liquid American options.

Contribution

It generalizes FTAP and hedging dualities to cases involving short American options, using space enlargement techniques to handle the asymmetry of positions.

Findings

Unified framework for American options trading

FTAP and hedging dualities extended to short American options

Model uncertainty incorporated in the enlarged space

Abstract

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 arXiv:1502.06681 (to appear in Annals of Applied Probability) to the case where the investor can also short American options. Following arXiv:1502.06681, 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 arXiv:1604.05517, 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.