Convergence of Numerical Schemes for Parabolic Equations Arising in Finance Theory

Barles, Guy; Daher, Ch.; Romano, Marc

doi:10.1142/s0218202595000085

Cited by 114 publications

(88 citation statements)

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“…It is easy to carry over the existence and uniqueness proof with only small changes, so we omit the proof. The convergence of the solution on a bounded domain to the solution on the half-space has been studied in [5,22] for the linear case, i.e. a = 0.…”

Section: Analytical Convergence Resultsmentioning

confidence: 99%

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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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Section: Analytical Convergence Resultsmentioning

confidence: 99%

“…Our main goal in this article is to prove the convergence of the numerical solution obtained by the compact scheme R3C to the unique viscosity solution of (1)- (5). In the literature, unlike for many standard finite difference schemes, there are very few results concerning the convergence of high-order compact finite difference schemes.…”

Section: Introductionmentioning

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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“…As pointed out in [7], we can expect any errors incurred by imposing approximate boundary conditions at S = S max to be small in areas of interest if S max is sufficiently large.…”

Section: Boundary Conditionsmentioning

confidence: 94%

Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance

Forsyth¹,

Labahn²

2007

JCF

150

187

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Many nonlinear option pricing problems can be formulated as optimal control problems, leading to Hamilton-Jacobi-Bellman (HJB) or Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods which ensure convergence to the financially relevant solution, which in this case is the viscosity solution. In addition, for the HJB type equations, we can guarantee convergence of a Newton-type (Policy) iteration scheme for the nonlinear discretized algebraic equations. However, in some cases, the Newton-type iteration cannot be guaranteed to converge (for example, the HJBI case), or can be very costly (for example for jump processes). In this case, we can use a piecewise constant control approximation. While we use a very general approach, we also include numerical examples for the specific interesting case of option pricing with unequal borrowing/lending costs and stock borrowing fees.

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“…(1) the explicit DPEXP scheme [3]; (2) the DPADI scheme [25] using the same number of steps for the space and time discretizations; (3) the KR scheme [20]; (4) the binomial BEG scheme [8]; (5) the centered explicit finite volume scheme FVCEEXP (defined in this paper by (21), (22), (26)), using a time step equal to 0.31 times the stability condition and rectangular control volumes; (6) the centered implicit finite volume scheme FVCEIMP (defined in this paper by (21), (22), (25)), using a time step equal to the stability condition of the explicit scheme, a preconditioning ILU(0) GMRES method to solve linear systems (see [23]) and rectangular control volume; (7) the upwind implicit finite volume scheme FVUPIMP (defined in this paper by (21), (22), (23)), using a time step equal to twice the stability condition of the explicit scheme, a preconditioning ILU(0) GMRES method to solve linear systems and rectangular control volumes.…”

Section: Applications and Open Problemsmentioning

confidence: 99%

Finite volume methods for the valuation of American options

Berton¹,

Eymard²

2006

ESAIM: M2AN

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Abstract.We consider the use of finite volume methods for the approximation of a parabolic variational inequality arising in financial mathematics. We show, under some regularity conditions, the convergence of the upwind implicit finite volume scheme to a weak solution of the variational inequality in a bounded domain. Some results, obtained in comparison with other methods on two dimensional cases, show that finite volume schemes can be accurate and efficient.Mathematics Subject Classification. 65M12.

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Convergence of Numerical Schemes for Parabolic Equations Arising in Finance Theory

Abstract: We present several results and methods concerning the convergence of numerical schemes for problems arising in Finance theory. This paper is essentially expository but we present however new results concerning some problems of rate of convergence.

Cited by 114 publications

References 0 publications

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

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

Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance

Finite volume methods for the valuation of American options

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