Remarks on the Nonlinear Black-Scholes Equations with the Effect of Transaction Costs

We deal with the solvability and a weak formulation of nonlinear partial differential equations of Black-Scholes type for the pricing of options in the presence of transaction costs. Examples include the Hoggard–Whalley–Wilmott equation, which is introduced to model portfolios of European type options with transaction costs based on the idea of Leland. The cost structure gives rise to nonlinear terms. We show the existence and the uniqueness of solutions both in the classical and the weak sense, where the notion of weak solutions is introduced.