  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.
