A novel fitted finite volume method for the Black-Scholes equation governing option pricing

Wang, Song

doi:10.1093/imanum/24.4.699

Cited by 132 publications

(103 citation statements)

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“…In the next section, we will discuss the discretization of (5) and (8) based on the fitted finite volume discretization ( [11,9]) in space and implicit scheme in time.…”

Section: The Model and The Penalty Approachmentioning

confidence: 99%

“…For brevity, we omit the derivation of the fitted finite volume method which can be found in ( [11,9]) and only list the final discretized forms.…”

Section: Discretizationmentioning

confidence: 99%

“…(9) (respectively (13)) at i = 1 and N + 1 respectively with the the nodal values of the corresponding Dirichlet boundary conditions and moving the relevant terms to the right-hand side of (9) (respectively (13)), M can be made an (N − 1) × (N − 1) matrix and (9) (respectively (13)) is a system for…”

Section: Discretizationmentioning

confidence: 99%

“…Recently, the power penalty method has shown its great potential for valuating American options accurately and efficiently ( [14,15,11,9,3]). In this method a linear complementarity problem is approximated by a nonlinear equation (or equation system) by adding a power penalty term, called an l k (k > 0) penalty.…”

Section: Introductionmentioning

confidence: 99%

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Numerical performance of penalty method for American option pricing

Zhang

Yang

Wang

et al. 2010

Optimization Methods and Software

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This paper is devoted to studying the numerical performance of a power penalty method for a linear parabolic complementarity problem arising from American option valuation. The penalized problem is a nonlinear parabolic partial differential equation (PDE). A fitted finite volume method and an implicit time-stepping scheme are used for respectively the spatial and time discretizations of the PDE. The rate of convergence of the penalty methods with respect to the penalty parameters is investigated both theoretically and numerically. The numerical robustness and computational effectiveness of the penalty method with respect to the market parameters are also studied and compared with those from an existing popular method, PSOR.

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“…In the next section, we will discuss the discretization of (5) and (8) based on the fitted finite volume discretization ( [11,9]) in space and implicit scheme in time.…”

Section: The Model and The Penalty Approachmentioning

confidence: 99%

“…For brevity, we omit the derivation of the fitted finite volume method which can be found in ( [11,9]) and only list the final discretized forms.…”

Section: Discretizationmentioning

confidence: 99%

Section: Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical performance of penalty method for American option pricing

Zhang

Yang

Wang

et al. 2010

Optimization Methods and Software

Self Cite

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“…Beginning in 1973, it was described that a mathematical framework for finding the fair price of a European option by Black and Scholes [1,2], several numerical methods have been presented for the cases where analytic solutions are neither available nor easily computable. See more details about numerical methods such as finite difference method (FDM) [3,4,5,6,7,8,9,10,11,12,13], finite element method [14,15,16], finite volume method [17,18,19], and a fast Fourier transform [20,21,22,23,24]. For convenience, we use the closed-form of the Black-Scholes equation in this work.…”

Section: Introductionmentioning

confidence: 99%

Path Averaged Option Value Criteria for Selecting Better Options

Kim¹,

Yoo²,

Son³

et al. 2016

Journal of the Korea Society for Industrial and Applied Mathema

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ABSTRACT. In this paper, we propose an optimal choice scheme to determine the best option among comparable options whose current expectations are all the same under the condition that an investor has a confidence in the future value realization of underlying assets. For this purpose, we use a path-averaged option as our base instrument in which we calculate the time discounted value along the path and divide it by the number of time steps for a given expected path. First, we consider three European call options such as vanilla, cash-or-nothing, and assetor-nothing as our comparable set of choice schemes. Next, we perform the experiments using historical data to prove the usefulness of our proposed scheme. The test suggests that the pathaveraged option value is a good guideline to choose an optimal option.

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An efficient numerical approach for solving three‐dimensional Black‐Scholes equation with stochastic volatility

Ngondiep

2024

Math Methods in App Sciences

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This paper develops an efficient combined interpolation/finite element approach for solving a three‐dimensional Black‐Scholes problem with stochastic volatility. The technique consists to approximate the time derivative by interpolation whereas the space derivatives are approximated using the finite element method. Both stability and error estimates of the new algorithm are deeply analyzed in the ‐norm. The proposed method is explicit, unconditionally stable, temporal second‐order accurate and fourth‐order convergence in space. This result suggests that the constructed scheme is faster and more efficient than a broad range of numerical methods widely studied in the literature for the Black‐Scholes models. Some numerical experiments are carried out to confirm the theoretical analysis.

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A novel fitted finite volume method for the Black-Scholes equation governing option pricing

Cited by 132 publications

References 0 publications

Numerical performance of penalty method for American option pricing

Numerical performance of penalty method for American option pricing

Path Averaged Option Value Criteria for Selecting Better Options

An efficient numerical approach for solving three‐dimensional Black‐Scholes equation with stochastic volatility

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