Operator splitting methods for American option pricing

Ikonen, Samuli; Toivanen, Jari

doi:10.1016/j.aml.2004.06.010

Cited by 170 publications

(110 citation statements)

References 5 publications

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“…The projected SOR (PSOR) method is the most well known while the penalty method and projected multigrid methods have been applied more recently for option pricing. The operator splitting method and the componentwise splitting method have been proposed by the authors in [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Efficient numerical methods for pricing American options under stochastic volatility

Ikonen

Toivanen

2007

Numerical Methods Partial

141

131

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Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two-dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M-matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one-dimensional differential operators that are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered methods. The accuracies of all methods appear to be similar. Thus, the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid-based methods have similar rapid grid independent convergence rates. They are about two or three times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on finer grids it is several times slower.

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Section: Introductionmentioning

confidence: 99%

Efficient numerical methods for pricing American options under stochastic volatility

Ikonen

Toivanen

2007

Numerical Methods Partial

141

131

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“…In contrast to the penalty approach, the operator splitting method [9] deals with the BlackScholes equation in the linear complementarity form (2.8)-(2.10). The main idea of this method is to split the time integration step into two fractional steps, where the partial differential equation is integrated together with an auxiliary variable λ in the first step, and then the solution and λ are updated to fulfill the American option constraint in the second step.…”

Section: The Operator Splitting Methodsmentioning

confidence: 99%

“…For many of the benchmark problems RBF-PUM was competitive with finite difference (FD) methods, but for the American option problems it performed significantly worse. In the project RBF-PUM employed the penalty method [13] for handling the free exercise boundary, while the FD methods used the operator splitting (OS) formulation [9].…”

Section: Introductionmentioning

confidence: 99%

Radial basis function partition of unity operator splitting method for pricing multi-asset American options

Shcherbakov

2016

Bit Numer Math

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PostprintThis is the accepted version of a paper published in BIT Numerical Mathematics. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination. AbstractThe operator splitting method in combination with finite differences has been shown to be an efficient approach for pricing American options numerically. Here, the operator splitting formulation is extended to the radial basis function partition of unity method. An approach that has previously often been used together with radial basis function methods to deal with the free boundary arising in American option pricing is to solve a penalised version of the Black-Scholes equation. It is shown that the operator splitting technique outperforms the penalty approach when used with the radial basis function partition of unity method. Numerical experiments are performed for one, two and three underlying assets. The advantage of the operator splitting technique grows with the number of dimensions.

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“…4 Ikonen and Toivanen (2004b) provide an additional example of a splitting method, in which the di¤usion operator and early exercise constraint are decoupled into separate time steps. Ikonen and Toivanen (2004a) describe the method in a simpler Black-Scholes setting. 5 Technically, Chiarella et al (2008) consider the problem of option valuation under stochastic volatility with jumps, in which case the equation to discretize is of an integro-partial di¤erential form.…”

Section: Underlying Stock Modelmentioning

confidence: 99%

A fast Fourier transform technique for pricing American options under stochastic volatility

Zhylyevskyy

2009

Rev Deriv Res

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This paper develops a non-finite-difference-based method of American option pricing under stochastic volatility by extending the Geske-Johnson compound option scheme. The characteristic function of the underlying state vector is inverted to obtain the vector's density using a kernel-smoothed fast Fourier transform technique. The method produces option values that are closely in line with the values obtained by finite-difference schemes. It also performs well in an empirical application with traded S&P 100 index options. The method is especially well suited to price a set of options with different strikes on the same underlying asset, which is a task often encountered by practitioners. Abstract This paper develops a non-…nite-di¤erence-based method of American option pricing under stochastic volatility by extending the Geske-Johnson compound option scheme. The characteristic function of the underlying state vector is inverted to obtain the vector's density using a kernel-smoothed fast Fourier transform technique. The method produces option values that are closely in line with the values obtained by …nite-di¤erence schemes. It also performs well in an empirical application with traded S&P 100 index options. The method is especially well suited to price a set of options with di¤erent strikes on the same underlying asset, which is a task often encountered by practitioners.

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Operator splitting methods for American option pricing

Cited by 170 publications

References 5 publications

Efficient numerical methods for pricing American options under stochastic volatility

Efficient numerical methods for pricing American options under stochastic volatility

Radial basis function partition of unity operator splitting method for pricing multi-asset American options

A fast Fourier transform technique for pricing American options under stochastic volatility

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