GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
The numerical solution of second-order boundary value problems on nonuniform meshes
Mathematics of Computation
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Algorithm 659: Implementing Sobol's quasirandom sequence generator
ACM Transactions on Mathematical Software (TOMS)
Quasi-Monte Carlo methods in numerical finance
Management Science
Iterative methods for solving linear systems
Iterative methods for solving linear systems
Accurate and efficient pricing of vanilla stock options via the Crandall--Douglas scheme
Applied Mathematics and Computation
Pricing European multi-asset options using a space-time adaptive FD-method
Computing and Visualization in Science
Improved radial basis function methods for multi-dimensional option pricing
Journal of Computational and Applied Mathematics
Pricing American options using a space-time adaptive finite difference method
Mathematics and Computers in Simulation
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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.