Hedging with physical or cash settlement under transient multiplicative price impact

TL;DR

This paper develops a model for superhedging European options in an illiquid market with transient, multiplicative price impact, considering physical or cash settlement, extending classical no-arbitrage pricing to illiquid settings.

Contribution

It introduces a framework for superhedging under transient multiplicative price impact with settlement options, generalizing existing models to include non-linear impact and resilience functions.

Findings

Superhedging prices governed by transient impact and settlement type.

Model extends no-arbitrage pricing to illiquid markets with impact.

Framework accommodates non-linear impact and resilience functions.

Abstract

We solve the superhedging problem for European options in an illiquid extension of the Black-Scholes model, in which transactions have transient price impact and the costs and the strategies for hedging are affected by physical or cash settlement requirements at maturity. Our analysis is based on a convenient choice of reduced effective coordinates of magnitudes at liquidation for geometric dynamic programming. The price impact is transient over time and multiplicative, ensuring non-negativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black-Scholes model with relative price impact being proportional to the volume of shares traded, where the transience for impact on log-prices is being modelled like in Obizhaeva-Wang \cite{ObizhaevaWang13} for nominal prices. More generally, we allow for non-linear price impact and resilience functions. The viscosity solutions describing the minimal superhedging price are governed by the transient character of the price impact and by the physical or cash settlement specifications. Pricing equations under illiquidity extend no-arbitrage pricing a la Black-Scholes for complete markets in a non-paradoxical way (cf.\ {\c{C}}etin, Soner and Touzi \cite{CetinSonerTouzi10}) even without additional frictions, and can recover it in base cases.