Arbitrage, hedging and utility maximization using semi-static trading strategies with American options

TL;DR

This paper develops a comprehensive framework for arbitrage, hedging, and utility maximization in markets with semi-static trading strategies involving American options, addressing both model certainty and uncertainty.

Contribution

It introduces duality results and a fundamental theorem of asset pricing for markets with American options under both model certainty and non-dominated model uncertainty.

Findings

Established FTAP for markets with American options.

Derived dualities for hedging prices of European and American options.

Extended results to markets with model uncertainty using discretization and minimax techniques.

Abstract

We consider a financial market where stocks are available for dynamic trading, and European and American options are available for static trading (semi-static trading strategies). We assume that the American options are infinitely divisible, and can only be bought but not sold. In the first part of the paper, we work within the framework without model ambiguity. We first get the fundamental theorem of asset pricing (FTAP). Using the FTAP, we get the dualities for the hedging prices of European and American options. Based on the hedging dualities, we also get the duality for the utility maximization. In the second part of the paper, we consider the market which admits non-dominated model uncertainty. We first establish the hedging result, and then using the hedging duality we further get the FTAP. Due to the technical difficulty stemming from the non-dominancy of the probability measure set, we use a discretization technique and apply the minimax theorem.