Limits of Semistatic Trading Strategies

TL;DR

This paper proves that pointwise limits of semistatic trading strategies remain semistatic in discrete time, clarifying the conditions under which this stability holds and contrasting with previous counterexamples.

Contribution

It establishes the stability of semistatic strategies under pointwise limits in general models, addressing prior counterexamples and linking to function decomposability in Schrödinger bridges.

Findings

Pointwise limits of semistatic strategies are semistatic.

Stability holds in two-period models and under certain conditions in multi-period models.

Contrasts with previous counterexamples due to integrability issues.

Abstract

We show that pointwise limits of semistatic trading strategies in discrete time are again semistatic strategies. The analysis is carried out in full generality for a two-period model, and under a probabilistic condition for multi-period, multi-stock models. Our result contrasts with a counterexample of Acciaio, Larsson and Schachermayer, and shows that their observation is due to a failure of integrability rather than instability of the semistatic form. Mathematically, our results relate to the decomposability of functions as studied in the context of Schr\"odinger bridges.