Limit Theorems for Partial Hedging Under Transaction Costs

TL;DR

This paper establishes limit theorems for shortfall risk minimization of American options with path-dependent payoffs under proportional transaction costs in the Black--Scholes model, extending previous results to incomplete markets.

Contribution

It introduces the first limit theorems for shortfall risk minimization of American options with transaction costs in continuous models.

Findings

Shortfall risk is a limit of binomial model risks.

Existence of an optimal portfolio strategy in continuous time.

Extension of limit theorems to markets with transaction costs.

Abstract

We study shortfall risk minimization for American options with path dependent payoffs under proportional transaction costs in the Black--Scholes (BS) model. We show that for this case the shortfall risk is a limit of similar terms in an appropriate sequence of binomial models. We also prove that in the continuous time BS model for a given initial capital there exists a portfolio strategy which minimizes the shortfall risk. In the absence of transactions costs (complete markets) similar limit theorems were obtained in Dolinsky and Kifer (2008, 2010) for game options. In the presence of transaction costs the markets are no longer complete and additional machinery required. Shortfall risk minimization for American options under transaction costs was not studied before.