Continuous-time Duality for Super-replication with Transient Price Impact

TL;DR

This paper develops a duality framework for super-replication in continuous-time markets with transient price impact, incorporating liquidity considerations and proving optimal strategies for options and utility maximization.

Contribution

It introduces a novel duality involving a liquidity-weighted norm and extends super-replication theory to models with transient price impact and market resilience.

Findings

Optimal buy-and-hold strategies for call options are established.

A verification theorem for utility maximization is proved.

A new duality involving a liquidity-weighted norm is developed.

Abstract

We establish a super-replication duality in a continuous-time financial model where an investor's trades adversely affect bid- and ask-prices for a risky asset and where market resilience drives the resulting spread back towards zero at an exponential rate. Similar to the literature on models with a constant spread, our dual description of super-replication prices involves the construction of suitable absolutely continuous measures with martingales close to the unaffected reference price. A novel feature in our duality is a liquidity weighted $L^2$-norm that enters as a measurement of this closeness and that accounts for strategy dependent spreads. As applications, we establish optimality of buy-and-hold strategies for the super-replication of call options and we prove a verification theorem for utility maximizing investment strategies.