Standard Methods for Standard Options

The Chapter 4 elaborates the numerical solution of the Black-Scholes equation for European plain-vanilla options, and of the corresponding inequalities for the American-style case. Following the Black-Scholes model, this chapter is confined to constant coefficients. This allows to solve an equivalent partial-differential equation of the simplest parabolic type. Several finite-difference schemes are explained, as well as numerical stability. Boundary conditions are introduced, which lead to obstacle problems in the American-style case and to a formulation as linear complementarity problem. The solution of the free boundary problem is tackled by the Brennan-Schwartz approach and by a projected SOR-method. Error control and accuracy are discussed. The final part of Chapter 4 is devoted to analytic methods. This includes the interpolation method, the quadratic approximation, the analytic method of lines, and quadrature methods for an integral representation. Finally we discuss criteria for the comparison of different methods and for judging their efficiency.