Option Pricing with Market Impact and Non-Linear Black and Scholes PDEs

We propose a few variations around a simple model in order to take into account the market impact of the option seller when hedging an option. This "retro-action" mechanism turns the linear Black and Scholes PDE into a non-linear one. This model allows also to retrieve some earlier results of \cite{CheriSonTouz}. Numerical simulations are then performed.