Finite Difference Method for the Multi-Asset Black–Scholes Equations

In this paper, we briefly review the finite difference method (FDM) for the Black–Scholes (BS) equations for pricing derivative securities and provide the MATLAB codes in the Appendix for the one-, two-, and three-dimensional numerical implementation. The BS equation is discretized non-uniformly in space and implicitly in time. The two- and three-dimensional equations are solved using the operator splitting method. In the numerical tests, we show characteristic examples for option pricing. The computational results are in good agreement with the closed-form solutions to the BS equations.

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A practical finite difference method for the three-dimensional Black–Scholes equation

European Journal of Operational Research ◽

10.1016/j.ejor.2015.12.012 ◽

2016 ◽

Vol 252 (1) ◽

pp. 183-190 ◽

Cited By ~ 6

Author(s):

Junseok Kim ◽

Taekkeun Kim ◽

Jaehyun Jo ◽

Yongho Choi ◽

Seunggyu Lee ◽

...

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Difference Method ◽

Three Dimensional ◽

Black Scholes

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Finite Difference Method for the Hull–White Partial Differential Equations

Mathematics ◽

10.3390/math8101719 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1719

Author(s):

Yongwoong Lee ◽

Kisung Yang

Keyword(s):

Partial Differential Equations ◽

Finite Difference ◽

Finite Difference Method ◽

Differential Equations ◽

Difference Method ◽

Trading Volume ◽

Operator Splitting ◽

Splitting Method ◽

Analytic Solutions ◽

Partial Differential

This paper reviews the finite difference method (FDM) for pricing interest rate derivatives (IRDs) under the Hull–White Extended Vasicek model (HW model) and provides the MATLAB codes for it. Among the financial derivatives on various underlying assets, IRDs have the largest trading volume and the HW model is widely used for pricing them. We introduce general backgrounds of the HW model, its associated partial differential equations (PDEs), and FDM formulation for one- and two-asset problems. The two-asset problem is solved by the basic operator splitting method. For numerical tests, one- and two-asset bond options are considered. The computational results show close values to analytic solutions. We conclude with a brief comment on the research topics for the PDE approach to IRD pricing.

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Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

Journal of Function Spaces ◽

10.1155/2021/9984473 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Sangkwon Kim ◽

Chaeyoung Lee ◽

Wonjin Lee ◽

Soobin Kwak ◽

Darae Jeong ◽

...

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Numerical Experiments ◽

Operator Splitting ◽

Three Dimensional ◽

Nonuniform Grid ◽

Call Option ◽

Splitting Scheme ◽

European Call Option ◽

Black Scholes

In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-order α in the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

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