The Scaling Limit of Superreplication Prices with Small Transaction Costs in the Multivariate Case

TL;DR

This paper extends the understanding of superreplication prices under small transaction costs from single-asset models to multi-asset models, revealing how the limiting prices depend on transaction costs and the chosen reference model.

Contribution

It generalizes Kusuoka's single-asset scaling limit results to a multivariate setting, incorporating multiple assets and their interaction with transaction costs.

Findings

Derived the multivariate scaling limit of superreplication prices.

Showed the limiting price involves a G-expectation with a volatility range.

Demonstrated dependence of the limit on the reference model and transaction costs.

Abstract

Kusuoka [ Limit Theorem on Option Replication Cost with Transaction Costs, Ann. Appl. Probab. 5, 198--221, (1995).] showed how to obtain non-trivial scaling limits of superreplication prices in discrete-time models of a single risky asset which is traded at properly scaled proportional transaction costs. This article extends the result to a multi-variate setup where the investor can trade in several risky assets. The $G$-expectation describing the limiting price involves models with a volatility range around the frictionless scaling limit that depends not only on the transaction costs coefficients but also on the chosen complete discrete-time reference model.