A Fourth Order L-stable Method for the Black-Scholes Model with Barrier Options

Voss, D.A.; Kazmi, Saira; He, Haibo

doi:10.1007/3-540-44842-x_21

Cited by 6 publications

(8 citation statements)

References 14 publications

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“…However, for such high-dimensional Black-Scholes PDEs our algorithms are CPU demanding and require the use of sophisticated numerical tools. Further information about this issue can be found in [23,15,10,9].…”

Section: Resultsmentioning

confidence: 97%

“…This raises the question whether suitable higher order schemes can be designed, see Voss et al [23] and Khaliq et al [15].…”

Section: Discretizationmentioning

confidence: 98%

“…Consequently, it is necessary to design efficient numerical schemes for such problems. Stable higher order methods for the Black and Scholes equation have been introduced by Voss et al [23] and Khaliq et al [15]. Mesh-free methods based on radial basis functions may also reduce the computational efforts significantly, see Fasshauer et al [9].…”

Section: Introductionmentioning

confidence: 98%

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Penalty methods for the numerical solution of American multi-asset option problems

Nielsen

Skavhaug

Tveito

2008

Journal of Computational and Applied Mathematics

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We derive and analyze a penalty method for solving American multi-asset option problems. A small, non-linear penalty term is added to the Black-Scholes equation. This approach gives a fixed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Explicit, implicit and semi-implicit finite difference schemes are derived, and in the case of independent assets, we prove that the approximate option prices satisfy some basic properties of the American option problem. Several numerical experiments are carried out in order to investigate the performance of the schemes. We give examples indicating that our results are sharp. Finally, the experiments indicate that in the case of correlated underlying assets, the same properties are valid as in the independent case.

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Section: Resultsmentioning

confidence: 97%

“…This raises the question whether suitable higher order schemes can be designed, see Voss et al [23] and Khaliq et al [15].…”

Section: Discretizationmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Penalty methods for the numerical solution of American multi-asset option problems

Nielsen

Skavhaug

Tveito

2008

Journal of Computational and Applied Mathematics

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“…Well-known θ-methods, whose members include the Backward Euler and Crank-Nicolson methods, have also been implemented as time-stepping procedures for pricing financial derivatives. The literature contains many documented instances which show that the Crank-Nicolson method is prone to producing spurious oscillations while the Backward Euler method maintains strong stability properties (see, for example, [20,25,26]). …”

Section: Introductionmentioning

confidence: 98%

Adaptive θ-methods for pricing American options

Khaliq

Voss

Kazmi

2008

Journal of Computational and Applied Mathematics

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We develop adaptive θ -methods for solving the Black-Scholes PDE for American options. By adding a small, continuous term, the Black-Scholes PDE becomes an advection-diffusion-reaction equation on a fixed spatial domain. Standard implementation of θ-methods would require a Newton-type iterative procedure at each time step thereby increasing the computational complexity of the methods. Our linearly implicit approach avoids such complications. We establish a general framework under which θ-methods satisfy a discrete version of the positivity constraint characteristic of American options, and numerically demonstrate the sensitivity of the constraint. The positivity results are established for the single-asset and independent two-asset models. In addition, we have incorporated and analyzed an adaptive time-step control strategy to increase the computational efficiency. Numerical experiments are presented for one-and two-asset American options, using adaptive exponential splitting for two-asset problems. The approach is compared with an iterative solution of the two-asset problem in terms of computational efficiency.

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“…[14], where European options based on three underlying assets are solved numerically. Stable higher order methods for the Black and Scholes equation have been introduced by [15] and [16], mesh-free methods based on RBFs may also reduce the computational efforts significantly, see [17]. In [4] Jo and Kim combined the operator splitting method with parallel computation technique to solve the multi-dimensional Black-Scholes equations.…”

Section: Introductionmentioning

confidence: 99%

A numerical study of RBFs-DQ method for multi-asset option pricing problems

Ranjbar

Khodayari

2018

B Soc Paran Mat

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In this paper, we propose a numerical scheme to solve multi-dimensional Black-Scholes equation using the global radial basis functions-based differential quadrature (RBFs-DQ) method. Before applying the method, it is needed to remove mixed derivatives from the Black-Scholes equation by making an appropriate change of variables. Then, any spatial derivativeis are approximated by a linear weighted sum of all the function values in the whole physical domain. In the RBFs-DQ method the weighting coefficients are computed by RBFs. The method is very easy to implement and the non-singularity is ensured. The proposed method combines the advantages of the conventional DQ method and the RBFs. It also preserves mesh-free feature of RBFs.

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A Fourth Order L-stable Method for the Black-Scholes Model with Barrier Options

Cited by 6 publications

References 14 publications

Penalty methods for the numerical solution of American multi-asset option problems

Penalty methods for the numerical solution of American multi-asset option problems

Adaptive θ-methods for pricing American options

A numerical study of RBFs-DQ method for multi-asset option pricing problems

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