The space of outcomes of semi-static trading strategies need not be closed

TL;DR

This paper demonstrates that the set of possible outcomes from semi-static trading strategies may lack closure properties, complicating optimal investment analysis in mathematical finance.

Contribution

It reveals that the space of outcomes for semi-static strategies is not necessarily closed when all European options are available, contrasting with the classical dynamic trading case.

Findings

The outcome space can be non-closed under certain conditions.

This non-closure impacts the existence of optimal strategies.

Contrast with classical dynamic trading results.

Abstract

Semi-static trading strategies make frequent appearances in mathematical finance, where dynamic trading in a liquid asset is combined with static buy-and-hold positions in options on that asset. We show that the space of outcomes of such strategies can have very poor closure properties when all European options for a fixed date $T$ are available for static trading. This causes problems for optimal investment, and stands in sharp contrast to the purely dynamic case classically considered in mathematical finance.