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Higher-Order Finite Element Solutions of Option Prices

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Kinks and jumps in the payoff function of option contracts prevent an effectiveimplementation of higher-order numerical approximation methods. Moreover, thederivatives (the greeks) are not easily determined around such singularities, even withstandard lower-order methods. This paper suggests a transformation to turn the originalill-conditioned pricing problem into a well-behaved numerical problem. For astandard test case, both vanilla- and binary call price functions are approximated with(tensor) B-splines of up to 10’th order. Polynomial convergence rates of orders up toapproximately 10 are obtained for prices as well as for first and second order derivatives(delta and gamma). Unlike similar studies, numerical approximation errors aremeasured both as weighted averages and in the supnorm over a state space includingtime-to-maturities down to a split second.KEYWORDS: Numerical option pricing, Transformed state spaces, Higher-orderB-splines.

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Peter Raahauge

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