| For the pricing of options on equity shares, the Black-Scholes equation has become an indispensable tool for agents on the financial market. Under the assumption that the value of the underlying share evolves in time according to a stochastic differential equation and some further assumptions on the financial market, the equation can be derived by an application of Ito's calculus. It represents a deterministic second-order parabolic differential equation backward in time with the price of the option as the unknown and the interest rate and the volatility entering the equation as coefficient functions. Since analytical solutions in explicit form are only available in special cases, in general the equation must be solved by numerical methods based on appropriate discretizations in time and in space where the spatial variable is the value of the share. This can be done by finite difference techniques or finite element methods with respect to suitable partitions of the time interval and the spatial domain. If the volatility depends on the independent variables, sudden changes of the volatility may imply rapid local changes of the solution as well so that a solution-dependent time-stepping and space-meshing is appropriate in order to keep the computational work at a moderate level while maintaining the accuracy of the computed approximate solution. During the past thirty years, such an adaptive choice of the discretizations in time and in space based on reliable a posteriori estimators of the global discretization error has been developed for finite element methods and achieved some state of maturity for standard partial differential equations. This dissertation is devoted to an application of adaptive finite element methods to the numerical solution of the Black-Scholes equation. |
