Higher-Order Finite Element Solutions of Option Prices

Kinks and jumps in the payoff function of option contracts prevent an effective implementation of higher-order numerical approximation methods. Moreover, the derivatives (the greeks) are not easily determined around such singularities, even with standard lower-order methods. This paper suggests a transformation to turn the original ill-conditioned pricing problem into a well-behaved numerical problem. For a standard 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 to approximately 10 are obtained for prices as well as for first and second order derivatives(delta and gamma). Unlike similar studies, numerical approximation errors are measured both as weighted averages and in the supnorm over a state space including time-to-maturities down to a split second.