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American options are priced numerically using a space- and time-adaptive finite difference method. The generalized Black-Scholes operator is discretized on a Cartesian structured but non-equidistant grid in space. The space- and time-discretizations are adjusted such that a predefined tolerance level on the local discretization error is met. An operator splitting technique is used to separately handle the early exercise constraint and the solution of linear systems of equations from the finite difference discretization of the linear complementarity problem. In numerical experiments three variants of the adaptive time-stepping algorithm with and without local time-stepping are compared.

Stock options are priced numerically using space- and time-adaptive finite difference methods. European options on one and several underlying assets are considered. These are priced with adaptive numerical algorithms including a second order method and a more accurate method. For American options we use the adaptive technique to price options on one stock with and without stochastic volatility. In all these methods emphasis is put on the control of errors to fulfill predefined tolerance levels. The adaptive second order method is compared to an alternative discretization technique using radial basis functions. This method is not adaptive but shows potential in option pricing for one and several underlying assets. A finite difference method and a Monte Carlo method are applied to a new financial contract called Turbo warrant. A comparison of these two methods shows that for the case considered the finite difference method is superior.

In this paper we develop a high-order adaptive finite difference space-discretization for the Black-Scholes (B-S) equation. The final condition is discontinuous in the first derivative yielding that the effective rate of convergence is two, both for low-order and high-order standard finite difference (FD) schemes. To obtain a sixth-order scheme we use an extra grid in a limited space- and time-domain. The new sixth-order method is called FD6G2. The FD6G2-method is combined with space- and time-adaptivity to further enhance the method. To obtain solutions of high accuracy in several dimensions the adaptive FD6G2-method is superior to both standard and adaptive second-order FD-methods.

In this paper, we have derived a radial basis function (RBF) based method for the pricing of financial contracts by solving the Black-Scholes partial differential equation. As an example of a financial contract that can be priced with this method we have chosen the multi-dimensional European basket call option. We have shown numerically that our scheme is second order accurate in time and spectrally accurate in space for constant shape parameter. For other, non-optimal choices of shape parameter values, the resulting convergence rate is algebraic. We propose an adaptive node point placement that improves the accuracy compared with a uniform distribution. Compared with an adaptive finite difference method, the RBF method is 20-40 times faster in one and two space dimensions and has approximately the same memory requirements.

We numerically price the financial contracts named turbo warrant that were released early in 2005. They have been studied mathematically in [Eriksson05] where explicit pricing formulas for the Geometric Brownian motion were derived. For more general underlying stochastic processes we have no analytical formulas and numerical methods are necessary. In this work two different methods are compared, stochastic pricing using a Monte Carlo method and a deterministic PDE approach using finite differences. The methods are evaluated in terms of numerical efficiency, computation time and accuracy. In the numerical experiments the geometric Brownian motion has been used as underlying stochastic process. Our results show that for low accuracy the methods are almost equal in efficiency but for higher accuracy the finite difference method is much more efficient.