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Space-time adaptive finite difference method for European multi-asset options. (English) Zbl 1154.91462

Summary: The multi-dimensional Black-Scholes equation is solved numerically for a European call basket option using a priori-a posteriori error estimates. The equation is discretized by a finite difference method on a Cartesian grid. The grid is adjusted dynamically in space and time to satisfy a bound on the global error. The discretization errors in each time step are estimated and weighted by the solution of the adjoint problem. Bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations. Comparisons are made with Monte Carlo and quasi-Monte Carlo methods in one dimension, and the performance of the method is illustrated by examples in one, two, and three dimensions.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
91G20 Derivative securities (option pricing, hedging, etc.)

Software:

TOMS659
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References:

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