Far Field Boundary Conditions for Black--Scholes Equations

Kangro, Raul; Nicolaides, R. A.

doi:10.1137/s0036142999355921

Cited by 206 publications

(126 citation statements)

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“…For put options, the contract becomes worthless as the price of any of the underlying assets tends to infinity. Therefore, we employ the following far-field boundary conditions [21]:…”

Section: Multi-asset American Option Pricingmentioning

confidence: 99%

A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

2014

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A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications. Computing, http://dx.doi.org/10.1007/s10915-014-9935-9 Journal of ScientificAccess to the published version may require subscription. Abstract Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF-PUM) proposed here. The objective of this paper is to establish that RBF-PUM is viable for parabolic PDEs of convection-diffusion type. The stability and accuracy of RBF-PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection-diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF-PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF-PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.

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“…For put options, the contract becomes worthless as the price of any of the underlying assets tends to infinity. Therefore, we employ the following far-field boundary conditions [21]:…”

Section: Multi-asset American Option Pricingmentioning

confidence: 99%

A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

2014

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“…then we can be sure that w(x, τ ), the solution of the truncated domain problem, gives us a call option value that is within K/A from the correct value [11].…”

Section: This Estimation Tells Us That Ifmentioning

confidence: 99%

“…Please refer to [11] for more details about the far-field boundary conditions for the Black-Scholes equations.…”

Section: Appendix a Far Field Boundary Conditionmentioning

confidence: 99%

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An Adaptive Finite Difference Method Using Far-Field Boundary Conditions for the Black-Scholes Equation

Jeong¹,

Kim

et al. 2014

Bulletin of the Korean Mathematical Society

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Abstract. We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid generation: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demonstrate the accuracy and efficiency of the proposed method. The results show that the computational time is reduced substantially with the accuracy being maintained.

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“…The nonlinear correction of the volatility in (1) is a function of the second derivative, so we assume that the influence of the nonlinearity at the boundary can be neglected for large R. The error caused by boundary conditions imposed on an artificial boundary for a class of Black-Scholes equations has been studied rigorously in [21]. We use a Dormand-Prince-4-5 Runge-Kutta scheme to solve the ordinary differential equation (3) and a cubic spline interpolation to obtain the values of Ψ for arbitrary arguments.…”

Section: Definition Of the Schemementioning

confidence: 99%

Convergence of a high-order compact finite difference scheme for a nonlinear Black–Scholes equation

Düring¹,

Fournié

Jüngel³

2004

ESAIM: M2AN

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Abstract.A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.Mathematics Subject Classification. 49L25, 65M06, 65M12.

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Far Field Boundary Conditions for Black--Scholes Equations

Cited by 206 publications

References 2 publications

A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

An Adaptive Finite Difference Method Using Far-Field Boundary Conditions for the Black-Scholes Equation

Convergence of a high-order compact finite difference scheme for a nonlinear Black–Scholes equation

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