High order ADI schemes for option pricing problems
In financial engineering the pricing of options and derivatives is a major field of interest. The behavior of assets is in general described via stochastic differential equations (SDEs). With the help of the Ito formula options prices can be shown to be solutions of parabolic partial differential equations (PDEs). Especially if option prices are driven by a basket of assets or other variables, e.g. volatility, interest rates, etc., the governing PDE exhibits several spatial dimensions and it becomes rather difficult to solve these problems. The curse of dimensionality shows its effects very quickly and leads to excessively long run-times or the problems even become unsolvable due to memory constraints.
In our research we employ advanced numerical techniques to solve high dimensional PDEs arising in computational finance. In order to lower the computational workload we use sparse grids to reduce the number of involved grid points while maintaining a high accuracy. Figure 1 shows the solution of an option pricing problem on a full and a sparse grid, where the spatial domain has been transformed logarithmical. One directly observes that the sparse grid consists of significantly less grid nodes. Furthermore we use a high order compact discretization in the spatial domain to achieve fourth order accuracy on a compact finite difference stencil. In the time domain efficient Alternating-Direction-Implicit (ADI) schemes are used to decompose the discretization matrix into tridiagonal ones, which can be solved in linear run-time.
Beside designing efficient schemes, we focus in our work on a thorough analysis of the numerical properties to provide accurate, stable and efficient solvers for practitioners. In the case of high-order ADI schemes [1, 2] we were able to prove that the stability regions of the derived schemes coincide with their second order counterpart, which are widely used in financial engineering departments. Thus, the accuracy can be increased from order two to four without any restrictions on the stability.
 C. Hendricks, M. Ehrhardt, M. Günther, High-order ADI schemes for diffusion equations with mixed derivatives in the combination technique, Appl. Numer. Math. 101 (2016) 36-52.
 C. Hendricks, C. Heuer, M. Ehrhardt, M. Günther, High-order ADI schemes for parabolic equations in the combination technique with application in finance, in preparation (2015).