Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations

Jeong, Darae; Seo, Seungsuk; Hwang, Hyeongseok; Lee, Dongsun; Choi, Yongho; Kim, Junseok

doi:10.1155/2015/359028

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…We focus on the well-known model in financial mathematics, the so-called Black-Scholes partial differential equation, which is a very particular and important case of the diffusion model to describe the price of the options [11,12]. The option prices derived from several underlying assets satisfy the multidimensional Black-Scholes equations [1,3,5].…”

Section: Multidimensional Black-scholes Equation and Its Reductionmentioning

confidence: 99%

Projection of the two-dimensional Black-Scholes equation for options with underlying stock and strike prices in two different currencies

Chacón‐Acosta¹,

Salas

2021

Rev. Mex. Fís.

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The two-variable Black-Scholes equation is used to study the option exercise price of two different currencies. Due to the complexity of dealing with several variables, reduction methods have been implemented to deal with these problems. This paper proposes an alternative reduction by using the so-called Zwanzig projection method to one-dimension, successfully developed to study the diffusion in confined systems. In this case, the option price depends on the stock price and the exchange rate between currencies. We assume that the exchange rate between currencies will depend on the stock price through some model that bounds such dependence, which somehow influences the final option price.As a result, we find a projected one-dimensional Black-Scholes equation similar to the so-called Fick-Jacobs equation for diffusion on channels. This equation is an effective Black-Scholes equation with two different interest rates, whose solution gives rise to a modified Black-Scholes formula. The properties of this solution are shown and were graphically compared with previously found solutions, showing that the corresponding difference is bounded.

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Section: Multidimensional Black-scholes Equation and Its Reductionmentioning

confidence: 99%

Projection of the two-dimensional Black-Scholes equation for options with underlying stock and strike prices in two different currencies

Chacón‐Acosta¹,

Salas

2021

Rev. Mex. Fís.

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“…We note that there are many boundary conditions for the BS equation such as Dirichlet, Neumann, linear, PDE, and payoff-consistent boundary conditions. See [27,28] for the detailed numerical treatment of the above-mentioned boundary conditions. Furthermore, there are no-boundary condition [6] and hybrid boundary condition [22].…”

Section: Gridmentioning

confidence: 99%

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

et al. 2020

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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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“…The boundary condition at = is set to be V( , ) = 0. A detailed discussion of the choice of the linear boundary conditions can be found in [11]. Normally, the error in the computed option price due to the domain truncation is negligible [12].…”

Section: The Continuous Problemmentioning

confidence: 99%

A Robust Spline Collocation Method for Pricing American Put Options

Cen

2019

Discrete Dynamics in Nature and Society

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In this paper a robust numerical method is proposed for pricing American put options. The Black-Scholes differential operator in the original form is discretized by using a quadratic spline collocation method on a piecewise uniform mesh for the spatial discretization and the implicit Euler scheme for the time discretization. The position of collocation points is chosen so that the spline difference operator satisfies the discrete maximum principle, which guarantees that the scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the scheme is second-order convergent with respect to the spatial variable and first-order convergent with respect to the time variable. Numerical results demonstrate that the scheme is stable and accurate.

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Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations

Cited by 7 publications

References 22 publications

Projection of the two-dimensional Black-Scholes equation for options with underlying stock and strike prices in two different currencies

Projection of the two-dimensional Black-Scholes equation for options with underlying stock and strike prices in two different currencies

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

A Robust Spline Collocation Method for Pricing American Put Options

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