Hedging of covered options with linear market impact and gamma constraint

TL;DR

This paper investigates the super-replication of covered European options in a linear impact market with gamma constraints, providing theoretical insights, construction methods, and numerical schemes.

Contribution

It introduces a novel approach using stochastic target and PDE smoothing techniques to characterize super-replication prices under market impact and gamma constraints.

Findings

Super-replication price characterized as viscosity solution of a non-linear PDE

Construction of epsilon-optimal hedging strategies

Proposed numerical resolution scheme for practical implementation

Abstract

Within a financial model with linear price impact, we study the problem of hedging a covered European option under gamma constraint. Using stochastic target and partial differential equation smoothing techniques, we prove that the super-replication price is the viscosity solution of a fully non-linear parabolic equation. As a by-product, we show how $\epsilon$-optimal strategies can be constructed. Finally, a numerical resolution scheme is proposed.