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Republic of South Africa


Spectral methods have been actively developed in the last decades. The main advantage of these methods is that they yield exponential order accuracy if the function is smooth enough. However, for discontinuous functions, their accuracy deteriorates to low accuracy due to the Gibbs phenomenon. The main purpose of this paper is to show that high order accuracy can be recovered from spectral approximation contaminated with the Gibbs phenomenon if proper workarounds are applied. In this paper, we review some spectral method convergence remedies including spectral collocation grid stretching method (SCGSM), spectral collocation discontinuity inclusion method (SCDIM), and spectral collocation domain decomposition method (SCDDM) in pricing options. We first perform barycentric interpolations on European vanilla, bull spread, and butterfly option payoffs, solve numerically the Black Scholes partial differential equation (PDE) with the proposed workarounds of barycentric spectral methods and then perform numerical comparisons. In this paper, the SDDM appears to be the most accurate workaround when solving a Black Scholes PDE with different payoffs

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