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Nonstandard finite difference schemes with application to finance: option pricing. (English) Zbl 1224.65205

Summary: 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 the 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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
91G60 Numerical methods (including Monte Carlo methods)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
91B24 Microeconomic theory (price theory and economic markets)
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
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