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A note on the exact discretization for a Cauchy-Euler equation: application to the Black-Scholes equation. (English) Zbl 1326.65095

Summary: We construct the exact finite difference representation for a second-order, linear, Cauchy-Euler ordinary differential equation. This result is then used to construct new non-standard finite difference schemes for the Black-Scholes partial differential equation.

MSC:

65L12 Finite difference and finite volume methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems
65L05 Numerical methods for initial value problems involving ordinary differential equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
91G60 Numerical methods (including Monte Carlo methods)
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[1] DOI: 10.1086/260062 · Zbl 1092.91524 · doi:10.1086/260062
[2] DOI: 10.2307/3003143 · doi:10.2307/3003143
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[4] DOI: 10.1080/1023619021000000807 · doi:10.1080/1023619021000000807
[5] DOI: 10.2307/2321656 · Zbl 0498.34049 · doi:10.2307/2321656
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[7] DOI: 10.1080/10236198.2013.771635 · Zbl 1300.65055 · doi:10.1080/10236198.2013.771635
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