# Perfect hedging under endogenous permanent market impacts

## TL;DR

This paper develops a model for perfect hedging in markets with endogenous permanent impacts, where trades influence prices through a utility-based market maker, enabling exact replication of derivatives.

## Contribution

It introduces a nonlinear market impact model based on utility indifference, establishing conditions for perfect hedging and deriving PDE-based pricing in Markovian cases.

## Key findings

- Derivation of a nonlinear stochastic integral for P&L
- Establishment of a completeness condition for perfect hedging
- Solution of pricing and hedging via semi-linear PDEs in Markovian settings

## Abstract

We model a nonlinear price curve quoted in a market as the utility indifference curve of a representative liquidity supplier. As the utility function we adopt a g-expectation. In contrast to the standard framework of financial engineering, a trader is no more price taker as any trade has a permanent market impact via an effect to the supplier's inventory. The P&L of a trading strategy is written as a nonlinear stochastic integral. Under this market impact model, we introduce a completeness condition under which any derivative can be perfectly replicated by a dynamic trading strategy. In the special case of a Markovian setting the corresponding pricing and hedging can be done by solving a semi-linear PDE.