The Sustainable Black-Scholes Equations.

Summary In incomplete markets, a basic Black-Scholes perspective has to be complemented by the valuation of market imperfections. Otherwise this results in Black-Scholes Ponzi schemes, such as the ones at the core of the last global financial crisis, where always more derivatives need to be issued for remunerating the capital attracted by the already opened positions. In this paper we consider the sustainable Black-Scholes equations that arise for a portfolio of options if one adds to their trade additive Black-Scholes price, on top of a nonlinear funding cost, the cost of remunerating at a hurdle rate the residual risk left by imperfect hedging. We assess the impact of model uncertainty in this setup.