Finite volume methods for the valuation of American options

Berton, Julien; Eymard, Robert

doi:10.1051/m2an:2006011

Cited by 9 publications

(5 citation statements)

References 21 publications

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“…For a comprehensive study of finite difference schemes as well as finite element schemes in this context we refer to Achdou and Pironneau [1]. In relation with the obstacle problem, a finite volume method is also studied in Berton and Eymard [3]. In connection with the present work, Windcliff, Forsyth, and Vetzal applied in [33] a second order backward differentiation formula (BDF2) scheme to shout options, which can be understood as a sequence of interdependent American option type problems.…”

Section: Introductionmentioning

confidence: 99%

Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term

Bokanowski

Debrabant

2020

IMA Journal of Numerical Analysis

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Abstract Finite difference schemes, using backward differentiation formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term of the form $$\begin{equation*}\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v- \varphi(t,x))= f(t,x).\end{equation*}$$For the scheme building on the second-order BDF formula, we discuss unconditional stability, prove an $L^2$-error estimate and show numerically second-order convergence, in both space and time, unconditionally on the ratio of the mesh steps. In the analysis an equivalence of the obstacle equation with a Hamilton–Jacobi–Bellman equation is mentioned, and a Crank–Nicolson scheme is tested in this context. Two academic problems for parabolic equations with an obstacle term with explicit solutions and the American option problem in mathematical finance are used for numerical tests.

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Section: Introductionmentioning

confidence: 99%

Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term

Bokanowski

Debrabant

2020

IMA Journal of Numerical Analysis

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“…Then the system (10) has a unique solution for any Q ⊂ M T and for any g = g(x) which is bounded and Lipschitz continuous in x. Furthermore, let w be given by (5). Let w R,R1 τ,h denote the solution to (11).…”

Section: Resultsmentioning

confidence: 99%

“…Lemma 4.3. Let w be given by (5), let g be a bounded Lipschitz continuous function of x and let Assumption 2.1 be satisfied. Furthermore, let R 1 > 0 be given.…”

Section: Approximations In Cylindrical Domainsmentioning

confidence: 99%

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Error estimates for finite difference approximations of American put option price

Šiška¹

2011

Preprint

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Finite difference approximations to multi-asset American put option price are considered. The assets are modelled as a multi-dimensional diffusion process with variable drift and volatility. Approximation error of order one quarter with respect to the time discretisation parameter and one half with respect to the space discretisation parameter is proved by reformulating the corresponding optimal stopping problem as a solution of a degenerate Hamilton-Jacobi-Bellman equation. Furthermore, the error arising from restricting the discrete problem to a finite grid by reducing the original problem to a bounded domain is estimated.

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“…For a comprehensive study of finite difference schemes as well as finite element schemes in this context we refer to Achdou and Pironneau [1]. In relation with the obstacle problem, a finite volume method is also studied in Berton and Eymard [3]. In connection with the present work, Windcliff, Forsyth, and Vetzal applied in [32] a second order backward differentiation formula (BDF2) scheme to shout options, which can be understood as a sequence of interdependent American option type problems.…”

Section: Introductionmentioning

confidence: 99%

Backward Differentiation Formula finite difference schemes for diffusion equations with an obstacle term

Bokanowski,

Debrabant

2018

Preprint

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Finite difference schemes, using Backward Differentiation Formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term, of the formFor the scheme building on BDF2, we discuss unconditional stability, prove an L 2 -error estimate and show numerically second order convergence, in both space and time, unconditionally on the ratio of the mesh steps. In the analysis, an equivalence of the obstacle equation with a Hamilton-Jacobi-Bellman equation is mentioned, and a Crank-Nicolson scheme is tested in this context. Two academic problems for parabolic equations with an obstacle term with explicit solutions and the American option problem in mathematical finance are used for numerical tests.

show abstract

Finite volume methods for the valuation of American options

Cited by 9 publications

References 21 publications

Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term

Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term

Error estimates for finite difference approximations of American put option price

Backward Differentiation Formula finite difference schemes for diffusion equations with an obstacle term

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