Geodesic
Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing
Serdica Mathematical Journal, Tome 35 (2010) no. 1, pp. 75-88.

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The paper is devoted to pricing options characterized by discontinuities in the initial conditions of the respective Black-Scholes partial differential equation. Finite difference schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We analyze the drawbacks of the Crank-Nicolson scheme that is most frequently used numerical method in Finance because of its second order accuracy. We propose an alternative scheme that is free of spurious oscillations and satisfy the positivity requirement, as it is demanded for the financial solution of the Black-Scholes equation.
Keywords: Black-Scholes Equation, Finite Difference Schemes, Jacobi Matrix, M-Matrix, Nonsmooth Initial Conditions, Positivity-Preserving 
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Milev, Mariyan; Tagliani, Aldo. Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing. Serdica Mathematical Journal, Tome 35 (2010) no. 1, pp. 75-88. http://geodesic.mathdoc.fr/item/SMJ2_2010_35_1_a4/