Scaling Limits for Super--replication with Transient Price Impact

TL;DR

This paper establishes a probabilistic scaling limit for super-replication costs in a binomial model with transient price impact, extending previous PDE-based results to path-dependent options.

Contribution

It introduces a new probabilistic framework for the scaling limit of super-replication costs with transient price impact, broadening applicability to path-dependent options.

Findings

Scaling limit keeps market depth constant while resilience increases inversely with trading interval.

Limit coincides with PDE-based results for models with temporary impact.

Framework extends to path-dependent options.

Abstract

We prove a scaling limit theorem for the super-replication cost of options in a Cox--Ross--Rubinstein binomial model with transient price impact. The correct scaling turns out to keep the market depth parameter constant while resilience over fixed periods of time grows in inverse proportion with the duration between trading times. For vanilla options, the scaling limit is found to coincide with the one obtained by PDE methods in [12] for models with purely temporary price impact. These models are a special case of our framework and so our probabilistic scaling limit argument allows one to expand the scope of the scaling limit result to path-dependent options.