Crash hedging strategies are derived as solutions of non–linear differential equations which itself are consequences of an equilibrium strategy which make the investor indifferent to uncertain (down) jumps. This is done in the situation where the investor has a logarithmic utility and where the market coefficients after a possible crash may change. It is scrutinized when and in which sense the crash hedging strategy is optimal. The situation of an investor with incomplete information is considered as well. Finally, introducing the crash horizon, an implied volatility is derived.