We propose a few variations around a simple model in order to take into account the market impactof 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 \mbox{%DIFAUXCMD CheriSonTouz .
We consider a model of linear market impact, and address the problem of replicating a contingent claim in this framework. We derive a non-linear Black-Scholes Equation that provides an exact replication strategy. This equation is fully non-linear and singular, but we show that it is well posed, and we prove existence of smooth solutions for a large class of final payoffs, both for constant and local volatility. To obtain regularity of the solutions, we develop an original method based on Legendre transforms. The close connections with the problem of hedging with it gamma constraints studied by Cheridito, Soner and Touzi and with the problem of hedging under it liquidity costs are discussed. We also derive a modified Black-Scholes formula valid for asymptotically small impact parameter, and finally provide numerical simulations as an illustration .
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propose a few variations around a simple model in order to take into account the market impactof the option seller when hedging an option. This "retro-action" mechanism turns the linear Black and Scholes PDE into a
consider a model of linear market impact, and address the problem of replicating a contingent claim in this framework. We derive a
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one. This model allows also to retrieve some earlier results of \mbox{%DIFAUXCMD CheriSonTouz
Black-Scholes Equation that provides an exact replication strategy. This equation is fully non-linear and singular, but we show that it is well posed, and we prove existence of smooth solutions for a large class of final payoffs, both for constant and local volatility. To obtain regularity of the solutions, we develop an original method based on Legendre transforms. The close connections with the problem of hedging with it gamma constraints studied by Cheridito, Soner and Touzi and with the problem of hedging under it liquidity costs are discussed. We also derive a modified Black-Scholes formula valid for asymptotically small impact parameter, and finally provide numerical simulations as an illustration