Grossinho and Morais Boundary Value Problems 2013, 2013:146 http://www.boundaryvalueproblems.com/content/2013/1/146 RESEARCH Open Access A fully nonlinear problem arising in financial modelling Maria do Rosário Grossinho 1* and Eva Morais 1,2 * Correspondence: mrg@iseg.utl.pt 1 ISEG, CEMAPRE - Technical University of Lisbon, Rua do Quelhas 6, Lisboa, 1200-781, Portugal Full list of author information is available at the end of the article Abstract We state existence and localisation results for a fully nonlinear boundary value problem using the upper and lower solutions method. With this study we aim to contribute to a better understanding of some analytical features of a problem arising in financial modelling related to the introduction of transaction costs in the classical Black-Scholes model. Our result concerns stationary solutions which become interesting in finance when the time does not play a relevant role such as, for instance, in perpetual options. 1 Introduction In , Fisher Black and Myron Scholes suggested a model that became fundamental for the valuation of financial derivatives in a complete frictionless market. Along with the no- arbitrage possibilities, the classical Black-Scholes model assumes that in order to replicate exactly the returns of a certain derivative, the hedging portfolio is continuously adjusted by transactioning the underlying asset of the derivative. This fact can only happen if no transaction costs exist when buying or selling financial assets. Otherwise, a continuous adjustment would imply that those costs, such as taxes or fees, would become infinitely large. Hence the introduction of transaction costs in the model is a problem that has been mo- tivating the work of several authors and has led to the study of new models that generalise the classical Black-Scholes model. In this paper, we aim to give a contribution to better understanding of some analytical features of the problem. We are concerned with the existence and localisation results for the nonlinear second-order Dirichlet boundary problem ⎧ ⎨ ⎩ x  (V ′′ )  + px  V ′′ + qxV ′ = qV in ]c, d[, V (c)= V c , V (d)= V d , () where  < c < d and p, q are positive constants (p and q can be assumed nonnegative, but the assumption is neither interesting from the mathematical viewpoint nor reasonable for applications to finance). We also consider that V c ≤ V d . This assumption turns out to be quite natural in some financial settings, for instance, if we are dealing with call options. This problem is related to the study of stationary solutions of a nonlinear parabolic equa- tion that models the valuation of a call option in presence of transaction costs. These sta- tionary solutions give the option value V as a function of the stock price, which can be © 2013 Grossinho and Morais; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.