Abstracts

We consider the numerical pricing of European multi-asset options. The standard way to price such options is to use a Monte-Carlo method to solve the stochastic differential equations for the underlying assets. Due to the slow convergence of these methods we solve the high-dimensional Black-Scholes equation. A straight-forward discretization of this equation leads to the so-called curse of dimensionality. One way to circumvent this curse is to use a dimension reduction technique based on a principal component analysis and asymptotic expansions, [1,2,3]. This way, we only have to solve 1 one-dimensional problem and d-1 two-dimensional problems where d is the number of dimensions (= number of underlying assets) to the original problem. This is in general a large reduction in the complexity of the problem.
The remaining problems are discretized in space using second-order finite differences. To further reduce the arithmetic complexity of the algorithm we introduce adaptivity in space by estimating the discretization error using Richardson-extrapolation, [4]. Thereby, we can place gridpoints where they are most needed for accuracy reasons.
In time we use a discontinuous Galerkin discretization. This type of discretizations have shown to be highly efficient for this type of problems [5,6].
We will present numerical results demonstrating the efficiency of the suggested method for options issued on correlated assets.

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[2] E. Ekedahl, E. Hansander and E. Lehto, Dimension Reduction for the Black-Scholes Equation - Alleviating the Curse of Dimensionality, Dept. of Information Technology, Uppsala University, 2007.
[3] P. Ghafari, Dimension Reduction and Adaptivity to Price Basket Options, U.U.D.M. project report; 2013:3, Dept. of Mathematics, Uppsala University, 2013.
[4] J. Persson and L. von Sydow, Pricing European multi-asset options using a space-time adaptive FD-method, Comput. Vis. Sci., 10(2007), pp. 173-183.
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