Option pricing with a direct adaptive sparse grid approach

Bungartz, Hans‐Joachim; Heinecke, Alexander; Pflüger, Dirk; Schraufstetter, Stefanie

doi:10.1016/j.cam.2011.09.024

Cited by 28 publications

(32 citation statements)

References 11 publications

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“…In [4], we were able to show that this approach lets us reduce the number of degrees of freedom notably while achieving comparable accuracies.…”

Section: Weighted Refinementmentioning

confidence: 92%

“…A simple, well-known, and frequently used technique is the transformation to logarithmic coordinates. We applied this transformation already in [4]. However, this eliminates only the variable coefficients in Equation (1), but the computational effort is still high in case of multiple correlated underlyings due to the many different terms the Black-Scholes PDE (1) consists of.…”

Section: Principal Axis Transformationmentioning

confidence: 97%

“…(1) G := itinitial, regular sparse grid (2) p( x) := itfunction for initial condition: payoff (3) itrefineGrid(G, u, p( x)) (4) for τ = 0 to T do (5) itsolveTimeStep(G, u, τ ) (6) itcalculateHedges(G, u) (if required) (7) itrestructureGrid(G, u) (if required) (8) τ ← τ + τ (9) end for (10) bfreturn u, G (see Section 5.2 for details). Therefore, we are not able to calculate an ILU decomposition as the matrix is only given implicitly.…”

Section: Solver Structurementioning

confidence: 99%

“…In [4], we presented a different approach. Here, the Black-Scholes PDE is discretized by finite elements on spatially adaptive sparse grids in order to optimally adopt to a given option that should be priced.…”

Section: Introduction and Related Workmentioning

confidence: 98%

See 3 more Smart Citations

A highly parallel Black–Scholes solver based on adaptive sparse grids

Heinecke

Schraufstetter

Bungartz

2012

International Journal of Computer Mathematics

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In this paper, we present a highly efficient approach for numerically solving the Black-Scholes equation in order to price European and American basket options. Therefore, hardware features of contemporary high performance computer architectures such as non-uniform memory access and hardware-threading are exploited by a hybrid parallelization using MPI and OpenMP which is able to drastically reduce the computing time. In this way, we achieve very good speed-ups and are able to price baskets with up to six underlyings. Our approach is based on a sparse grid discretization with finite elements and makes use of a sophisticated adaption. The resulting linear system is solved by a conjugate gradient method that uses a parallel operator for applying the system matrix implicitly. Since we exploit all levels of the operator's parallelism, we are able to benefit from the compute power of more than 100 cores. Several numerical examples as well as an analysis of the performance for different computer architectures are provided.

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“…In [4], we were able to show that this approach lets us reduce the number of degrees of freedom notably while achieving comparable accuracies.…”

Section: Weighted Refinementmentioning

confidence: 92%

Section: Principal Axis Transformationmentioning

confidence: 97%

Section: Solver Structurementioning

confidence: 99%

Section: Introduction and Related Workmentioning

confidence: 98%

See 2 more Smart Citations

A highly parallel Black–Scholes solver based on adaptive sparse grids

Heinecke

Schraufstetter

Bungartz

2012

International Journal of Computer Mathematics

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“…An other important application is the solution of high-dimension partial differential equations (PDEs) [6]. We re-measured a highly-parallel solver of the Black-Scholes equation [4] based on spatially adaptive sparse grids using a finite element discretization [7]. The BlackScholes equation is used to price (European) basket options on d underlyings S = (S 1 , .…”

Section: Sparse Grids -Solving Pdesmentioning

confidence: 99%

Exploiting State-of-the-Art x86 Architectures in Scientific Computing

Heinecke

Auckenthaler

Trinitis

2012

2012 11th International Symposium on Parallel and Distributed Computing

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In recent years, general purpose x86 architectures have undergone significant modifications towards high performance computing capabilities. Lately, technologies like wider vector units or Fused Multiply-Add (FMA) instruction, which were mainly known from GPU arcitectures, have been introduced. In this paper, we examine the performance of current x86 architectures, namely Intel Sandy Bridge and AMD Bulldozer, for four different parallel workloads with different properties. These properties comprise optimally cache-blocked algorithms as well as adaptive grid structures resulting in memory latency and bandwidth bound executions. The achieved performance on both architectures is very promising, and, if extrapolated towards upcoming server silicon, can be regarded as on par with current high-end GPU based accelerators.

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Fast Sparse Grid Operations Using the Unidirectional Principle: A Generalized and Unified Framework

Holzmüller

Pflüger

2021

Lecture Notes in Computational Science and Engineering

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Option pricing with a direct adaptive sparse grid approach

Cited by 28 publications

References 11 publications

A highly parallel Black–Scholes solver based on adaptive sparse grids

A highly parallel Black–Scholes solver based on adaptive sparse grids

Exploiting State-of-the-Art x86 Architectures in Scientific Computing

Fast Sparse Grid Operations Using the Unidirectional Principle: A Generalized and Unified Framework

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