In the first part of this thesis, we consider B-spline based methods for pricing American options in the Black-Scholes and Heston model. The difference between these two models is the assumption on the volatility of the underlying asset. While in the Black-Scholes model the volatility is assumed to be constant, the Heston model includes a stochastic volatility variable. The underlying problems are formulated as parabolic variational inequalities. Recall that, in finance, to determine optimal risk strategies, one is not only interested in the solution of the variational inequality, i.e., the option price, but also in its partial derivatives up to order two, the so-called Greeks. A special feature for these option price problems is that initial conditions are typically given as piecewise linear continuous functions. Consequently, we have derived a spatial discretization based on cubic B-splines with coinciding knots at the points where the initial condition is not differentiable. Together with an implicit time stepping scheme, this enables us to
achieve an accurate pointwise approximation of the partial derivatives up to order two. For the efficient numerical solution of the discrete variational inequality, we propose a monotone multigrid method for (tensor product) B-splines with possible internal coinciding knots. Corresponding numerical results show that the monotone multigrid method is robust with respect to the refinement level and mesh size.
In the second part of this thesis, we consider the pricing of a European option in the uncertain volatility model. In this model the volatility of the underlying asset is a priori unknown and is assumed to lie within a range of extreme values. Mathematically, this problem can be formulated as a one dimensional parabolic Hamilton-Jacobi-Bellman equation and is also called Black-Scholes-Barenblatt equation. In the resulting non-linear equation, the diffusion coefficient is given by a volatility function which depends pointwise on the second derivative. This kind of non-linear partial differential equation does not admit a weak H^1-formulation. This is due to the fact that the non-linearity depends pointwise on the second derivative of the solution and, thus, no integration by parts is possible to pass the partial derivative onto a test function. But in the discrete setting this pointwise second derivative can be approximated in H^1 by L^1-normalized B-splines. It turns out that the approximation of the volatility function leads to discontinuities in the partial derivatives. In order to improve the approximation of the solution and its partial derivatives for cubic B-splines, we develop a Newton like algorithm within a knot insertion step. Corresponding numerical results show that the convergence of the solution and its partial derivatives are nearly optimal in the L^2-norm, when the location of volatility change is approximated with desired accuracy
