A Parallel Implementation on GPUs of ADI Finite Difference Methods for Parabolic PDEs with Applications in Finance

2010 Workshop on High Performance Computational Finance at SC10 (WHPCF)

2010

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Abstract-We develop highly efficient parallel pricing methods on Graphics Processing Units (GPUs) for multi-asset American options via a Partial Differential Equation (PDE) approach. The linear complementarity problem arising due to the free boundary is handled by a penalty method. Finite difference methods on uniform grids are considered for the space discretization of the PDE, while classical finite differences, such as Crank-Nicolson, are used for the time discretization. The discrete nonlinear penalized equations at each timestep are solved using a penalty iteration. A GPU-based parallel Alternating Direction Implicit Approximate Factorization technique is employed for the solution of the linear algebraic system arising from each penalty iteration. We demonstrate the efficiency and accuracy of the parallel numerical methods by pricing American options written on three assets.

Section: B Gpu Implementationmentioning

confidence: 99%

Section: B Gpu Implementationmentioning

confidence: 99%

“…For brevity, we only present the main steps of the parallel algorithm for the ADI-AF scheme (13). A more detailed discussion of the similar ADI algorithm can be found in [5]. 1)…”

Section: B Gpu Implementationmentioning

confidence: 99%

“…(Hence, the loading of components of the vectors u m,(κ) and u * used for the checking of the stopping criterion in Step a.5 is fully coalesced.) See [5] for a more detailed discussion.…”

Section: B Gpu Implementationmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Pricing multi-asset American options on Graphics Processing Units using a PDE approach

2010 Workshop on High Performance Computational Finance at SC10 (WHPCF)

2010

Self Cite

An efficient graphics processing unit‐based parallel algorithm for pricing multi‐asset American options

Concurrency and Computation

2011

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SUMMARY We develop highly efficient parallel PDE‐based pricing methods on graphics processing units (GPUs) for multi‐asset American options. Our pricing approach is built upon a combination of a discrete penalty approach for the linear complementarity problem arising because of the free boundary and a GPU‐based parallel alternating direction implicit approximate factorization technique with finite differences on uniform grids for the solution of the linear algebraic system arising from each penalty iteration. A timestep size selector implemented efficiently on GPUs is used to further increase the efficiency of the methods. We demonstrate the efficiency and accuracy of the parallel numerical methods by pricing American options written on three assets. Copyright © 2011 John Wiley & Sons, Ltd.

Graphics processing unit pricing of exotic cross‐currency interest rate derivatives with a foreign exchange volatility skew model

Concurrency and Computation

2012

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We present a graphics processing unit (GPU) parallelization of the computation of the price of exotic crosscurrency interest rate derivatives via a partial differential equation (PDE) approach. In particular, we focus on the GPU-based parallel pricing of long-dated foreign exchange (FX) interest rate hybrids, namely power reverse dual currency (PRDC) swaps with Bermudan cancelable features. We consider a three-factor pricing model with FX volatility skew, which results in a time-dependent parabolic PDE in three spatial dimensions. Finite difference methods on uniform grids are used for the spatial discretization of the PDE, and the alternating direction implicit (ADI) technique is employed for the time discretization. We then exploit the parallel architectural features of GPUs together with the Compute Unified Device Architecture framework to design and implement an efficient parallel algorithm for pricing PRDC swaps. Over each period of the tenor structure, we divide the pricing of a Bermudan cancelable PRDC swap into two independent pricing subproblems, each of which can efficiently be solved on a GPU via a parallelization of the ADI timestepping technique. Numerical results indicate that GPUs can provide significant increase in performance over CPUs when pricing PRDC swaps. An analysis of the impact of the FX skew on such derivatives is provided. PRDC swaps. The use of a local volatility model provides better modeling for the skewness of the FX rate and at the same time avoids introducing more stochastic factors into the model.The ever growing interest in cross-currency interest rate derivatives in general, and PRDC swaps in particular, has created a need for efficient pricing and hedging strategies for them. The popular choice for pricing PRDC swaps is Monte-Carlo (MC) simulation, but this approach has several major disadvantages, including slow convergence, and the limitation that the price is obtained at a single point only in the domain as opposed to the global character of the partial differential equation (PDE) approach. In addition, accurate hedging parameters, such as delta and gamma, are generally harder to compute via an MC approach than via a PDE approach. On the other hand, when pricing PRDC swaps by the PDE approach, each stochastic factor in the pricing model gives rise to a spatial variable in the PDE. Because of the 'curse of dimensionality' associated with highdimensional PDEs-not to mention additional complexity due to multiple cash flows and exotic features, such as Bermudan cancelability-the pricing of such derivatives via the PDE approach is not only mathematically challenging but also very computationally intensive.Over the last few years, the rapid evolution of graphics processing units (GPUs) into powerful, cost efficient, programmable computing architectures for general purpose computations has provided application potential beyond the primary purpose of graphics processing. In the area of computational finance, although there has been great interest in utilizing GPUs in developing eff...