A high-order front-tracking finite difference method for pricing American options under jump-diffusion models

Toivanen, Jari

doi:10.21314/jcf.2010.222

Cited by 21 publications

(7 citation statements)

References 21 publications

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“…Several numerical approaches exist such as operator splitting [26], front tracking [39], employing a penalty method [7], and solving a linear complementary problem via projected successive over-relaxation [23] or by the use of Lagrange multipliers [10]. Unfortunately, when pricing higher dimensional problems, these methods are typically accurate but very time-consuming.…”

Section: Including Constraints In Case Of American-style Optionsmentioning

confidence: 99%

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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Section: Including Constraints In Case Of American-style Optionsmentioning

confidence: 99%

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

Heinecke

Schraufstetter

Bungartz

2012

International Journal of Computer Mathematics

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“…Furthermore, the COS method can be combined with an extrapolation strategy on the Bermudan options for American option pricing. On the other hand, PIDE-based methods are also used for pricing American options recently [25,26]. Therefore, the current work is a stepping-stone to American option pricing and more advanced models like the stochastic volatility with correlated and contemporaneous jumps in return and variance (SVCJ) model [6].…”

Section: Discussionmentioning

confidence: 99%

Fast exponential time integration scheme for option pricing with jumps

Lee

Liu

Sun

2010

Numerical Linear Algebra App

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SUMMARYA fast exponential time integration scheme is considered for pricing European and double barrier options in jump-diffusion models. After spatial discretization, the option pricing problem is transformed into the product of a matrix exponential and a vector, while the matrix bears a Toeplitz structure. The shift-andinvert Arnoldi method is then employed for fast approximation to such operation. Owing to the Toeplitz structure, the computational cost can be reduced by the fast Fourier transform. Furthermore, the discretized form of option pricing problem satisfies a given condition such that the error bound of the shift-and-invert Arnoldi approximation is unrelated to the norm of the matrix. Numerical tests are carried out to compare the proposed method with other option pricing methods.

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“…Let denote the value of an American put option with strike price on the underlying asset and time . It is known that the price under a jump-diffusion model satis�es the following partial integrodifferential complementarity problem [14][15][16][17]:…”

Section: The Continuous Problemmentioning

confidence: 99%

“…d'Halluin et al [12,13] developed a second-order accurate numerical method with a �xed-point iteration method and an implicit �nite difference scheme along with a penalty method for pricing American options under jump diffusion processes. Toivanen et al [14][15][16] introduced a high-order front-�xing �nite difference method and an arti�cial volatility scheme along with an iterative method for pricing American options under jumpdiffusion models. Zhang and Wang [17,18] proposed �tted �nite volume schemes coupled with the Crank-�icolson time stepping method for pricing options under jump diffusion processes.…”

Section: Introductionmentioning

confidence: 99%

A Finite Difference Scheme for Pricing American Put Options under Kou's Jump-Diffusion Model

Huang

Cen

2013

Journal of Function Spaces and Applications

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We present a stable finite difference scheme on a piecewise uniform mesh along with a penalty method for pricing American put options under Kou's jump-diffusion model. By adding a penalty term, the partial integrodifferential complementarity problem arising from pricing American put options under Kou's jump-diffusion model is transformed into a nonlinear parabolic integro-differential equation. Then a finite difference scheme is proposed to solve the penalized integrodifferential equation, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit-explicit time stepping technique. This leads to the solution of problems with a tridiagonal M-matrix. It is proved that the difference scheme satisfies the early exercise constraint. Furthermore, it is proved that the scheme is oscillation-free and is second-order convergent with respect to the spatial variable. The numerical results support the theoretical results.

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A high-order front-tracking finite difference method for pricing American options under jump-diffusion models

Cited by 21 publications

References 21 publications

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

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

Fast exponential time integration scheme for option pricing with jumps

A Finite Difference Scheme for Pricing American Put Options under Kou's Jump-Diffusion Model

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