Numerical techniques for the valuation of basket options and their Greeks

Hager, Corinna; Hüeber, S.; Wohlmuth, Barbara

doi:10.21314/jcf.2010.217

Cited by 16 publications

(18 citation statements)

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“…Next, the implicit second-order Crank-Nicolson scheme is used for time discretization, and the weak form of (11) is constructed via multiplication with a test function and integration over Ω. Appropriate Dirichlet boundary conditions P i D at time t i , possibly depending on the choice of ψ, are enforced on the boundary of Ω (see [42] for details). Finally, we introduce a simplicial shaperegular triangulation T h of Ω and approximate P i on the space V h ⊂ H 1 (Ω) spanned by the lowest order conforming finite element basis functions φ j , j ∈ N h = J h ∪ D h , where J h and D h are the sets of all inner or boundary nodes of T h , respectively.…”

Section: Mathematical Finance: Basket American Optionsmentioning

confidence: 99%

“…with M h and A h being the mass and the stiffness matrix associated with V h , respectively, the latter steming from the weak form of the differential operator L (see [42] for details). Due to (12), the matrix D h is diagonal and can thus easily be inverted.…”

Section: Mathematical Finance: Basket American Optionsmentioning

confidence: 99%

“…There are several possibilities to define such nonlinear complementarity (NCP) functions (see, e.g., [19] and the references therein). Here, we use the functions employed in [34,42,44] which can be written in terms of the nodal trial values…”

Section: Reformulation As Equality Constraintsmentioning

confidence: 99%

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Semismooth Newton methods for variational problems with inequality constraints

Hager

Wohlmuth

2010

GAMM-Mitteilungen

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Inequality constraints occur in many different fields of application, e.g., in structural mechanics, flow processes in porous media or mathematical finance. In this paper, we make use of the mathematical structure of these conditions in order to obtain an abstract computational framework for problems with inequality conditions. The constraints are enforced locally by means of Lagrange multipliers which are defined with respect to dual basis functions. The reformulation of the inequality conditions in terms of nonlinear complementarity functions leads to a system of semismooth nonlinear equations that is solved by a generalized version of Newton's method for semismooth problems. By this, both nonlinearities in the pde model and inequality constraints are treated within a single Newton iteration which converges locally superlinear. The scheme can efficiently be implemented in terms of an active set strategy with local static condensation of the non-essential variables. Numerical examples from different fields of application illustrate the generality and the robustness of the method.

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Section: Mathematical Finance: Basket American Optionsmentioning

confidence: 99%

Section: Mathematical Finance: Basket American Optionsmentioning

confidence: 99%

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Semismooth Newton methods for variational problems with inequality constraints

Hager

Wohlmuth

2010

GAMM-Mitteilungen

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“…Sparse grids have been applied to a whole range of different research fields such as physics, visualization, finance and econometrics (see, e.g. [10,5,14,26,38]). Using the original formulation of Smolyak [35], Krüger and Kübler [22] were the first to solve dynamic economic models using sparse grids.…”

Section: Introductionmentioning

confidence: 99%

Using Adaptive Sparse Grids to Solve High-Dimensional Dynamic Models

Brumm

Scheidegger

2013

SSRN Journal

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We present a flexible and scalable method to compute global solutions of high-dimensional stochastic dynamic models. Within a time-iteration setup, we interpolate policy functions using an adaptive sparse grid algorithm with piecewise multi-linear (hierarchical) basis functions. As the dimensionality increases, sparse grids grow considerably slower than standard tensor product grids. In addition, the grid scheme we use is automatically refined locally and can thus capture steep gradients or even non-differentiabilities. To further increase the maximum problem size we can handle, our implementation is fully hybrid parallel, i.e. using a combination of distributed and shared memory parallelization schemes. This parallelization enables us to efficiently use high-performance computing architectures. Our algorithm scales up nicely to more than one thousand parallel processes. To demonstrate the performance of our method, we apply it to high-dimensional international real business cycle models with capital adjustment costs and irreversible investment.Keywords: Adaptive Sparse Grids, High-Performance Computing, International Real Business Cycles, Occasionally Binding Constraints JEL Classification: C63, C68, F41 * We are very grateful to Felix Kübler for helpful discussions and support. We thank Ken Judd, Karl Schmedders and seminar participants at University of Zürich, Stanford University, University of Chicago, Argonne National Laboratory, and CEF 2013 in Vancouver for valuable comments. Moreover, we thank Xiang Ma for very instructive email discussions regarding the workings of adaptive sparse grids. We are grateful for the support of Olaf Schenk, Antonio Messina and Riccardo Murri concerning HPC related issues. We acknowledge CPU time granted on the University of Zürich's 'Schrödinger' HPC cluster. Johannes Brumm gratefully acknowledges financial support from the ERC.

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“…Sparse grids reduce the number of grid points needed from the order O N d to O N · (log N) d−1 , while the accuracy of the interpolation only slightly deteriorates in the case of sufficiently 60 smooth functions [8]. Sparse grids go back to Smolyak [14] and have been applied to a whole range of different research fields such as physics, visualization, data mining, Hamilton-Jacobi Bellman (HJB) equations, mathematical finance, insurance, and econometrics [15,16,8,17,18,19,20,21,22]. Krueger and Kubler [3] and Judd et al [23] solve dynamic economic models using sparse grids with global polynomials 65 as basis functions.…”

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confidence: 99%

Scalable high-dimensional dynamic stochastic economic modeling

Brumm

Mikushin

Scheidegger

et al. 2015

Journal of Computational Science

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We present a highly parallelizable and flexible computational method to solve high-dimensional stochastic dynamic economic models. Solving such models often requires the use of iterative methods, like time iteration or dynamic programming. By exploiting the generic iterative structure of this broad class of economic problems, we propose a parallelization scheme that favors hybrid massively parallel computer architectures. Within a parallel nonlinear time iteration framework, we interpolate policy functions partially on GPUs using an adaptive sparse grid algorithm with piecewise linear hierarchical basis functions. GPUs accelerate this part of the computation one order of magnitude thus reducing overall computation time by 50%. The developments in this paper include the use of a fully adaptive sparse grid algorithm and the use of a mixed MPI-Intel TBB-CUDA/Thrust implementation to improve the interprocess communication strategy on massively parallel architectures. Numerical experiments on "Piz Daint" (Cray XC30) at the Swiss National Supercomputing Centre show that high-dimensional international real business cycle models can be efficiently solved in parallel. To our knowledge, this performance on a massively parallel petascale architecture for such nonlinear high-dimensional economic models has not been possible prior to present work. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe present a highly parallelizable and flexible computational method to solve highdimensional stochastic dynamic economic models. Solving such models often requires the use of iterative methods, like time iteration or dynamic programming. By exploiting the generic iterative structure of this broad class of economic problems, we propose a parallelization scheme that favors hybrid massively parallel computer architectures.Within a parallel nonlinear time iteration framework, we interpolate policy functions partially on GPUs using an adaptive sparse grid algorithm with piecewise linear hierarchical basis functions. GPUs accelerate this part of the computation one order of magnitude thus reducing overall computation time by 50%. The developments in this paper include the use of a fully adaptive sparse grid algorithm and the use of a mixed MPI-Intel TBB-CUDA/Thrust implementation to improve the interprocess communication strategy on massively parallel architectures. Numerical experiments on "Piz Daint" (Cray XC30) at the Swiss National Supercomputing Centre show that highdimensional international real business cycle models can be efficiently solved in parallel. To our knowledge, this performance on a massively ...

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Numerical techniques for the valuation of basket options and their Greeks

Cited by 16 publications

References 0 publications

Semismooth Newton methods for variational problems with inequality constraints

Semismooth Newton methods for variational problems with inequality constraints

Using Adaptive Sparse Grids to Solve High-Dimensional Dynamic Models

Scalable high-dimensional dynamic stochastic economic modeling

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