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Low volatility options and numerical diffusion of finite difference schemes. (English) Zbl 1224.65206

Summary: We explore the numerical diffusion introduced by two nonstandard finite difference schemes applied to the Black-Scholes partial differential equation for pricing discontinuous payoff and low volatility options. Discontinuities in the initial conditions require applying nonstandard non-oscillating finite difference schemes such as the exponentially fitted finite difference schemes suggested by D. Duffy and the Crank-Nicolson variant scheme of Milev-Tagliani. We present a short survey of these two schemes, investigate the origin of the respective artificial numerical diffusion and demonstrate how it could be diminished.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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)
91G60 Numerical methods (including Monte Carlo methods)
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
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