The Black-Scholes Equation in Presence of Arbitrage

The celebrated Black-Scholes PDE, allowing to price a derivative in term of the underlying, relies on the no arbitrage assumption. In this work, we utilize a market model, where portfolio rebalancing and discounting are seen... [ view full abstract ]

The celebrated Black-Scholes PDE, allowing to price a derivative in term of the underlying, relies on the no arbitrage assumption. In this work, we utilize a market model, where portfolio rebalancing and discounting are seen as a parallel transport in some geometric space, whose curvature quantifies the arbitrage possibilities. On this basis we derived an extension of Black-Scholes' PDE, where a non linear term depending explicitly on the arbitrage measure appears. We provide an approximated solution for the price of a European call option by means of perturbation theory with respect to this arbitrage measure.