We consider as given a discrete time financial market with a risky asset and options written on that asset, and we determine both the sub- and superhedging prices of an American option in the model independent framework of [B. Bouchard and M. Nutz, Ann. Appl. Probab., 25 (2015), pp. 823-859]. We obtain the duality of results for the sub- and superhedging prices, as well as the existence of optimal hedging strategies. For the subhedging prices we discuss whether the sup and inf in the dual representation can be exchanged (a counterexample shows that this is not true in general). For the superhedging prices we discuss several alternative definitions and argue that our choice is more reasonable. Then, assuming that the path space is compact, we construct a discretization of the path space and demonstrate the convergence of the hedging prices at the optimal rate. The latter result would be useful for numerical computation of the hedging prices. Our results generalize those of [Y. Dolinsky, Electron. Commun. Probab., 19 (2014), 19] to the case when static positions in (finitely many) European options can be used in the hedging portfolio.
- American options
- Model independent pricing
- Semistatic hedging strategies