A penalty-based method from reconstructing smooth local volatility surface from American options

This paper is devoted to develop a robust penalty-based method of reconstructing smooth local volatility surface from the observed American option prices. This reconstruction problem is posed as an inverse problem: given a finite set of observed American option prices, find a local volatility function such that the theoretical option prices matches the observed ones optimally with respect to a prescribed performance criterion. The theoretical American option prices are governed by a set of partial differential complementarity problems (PDCP). We propose a penalty-based numerical method for the solution of the PDCP. Typically, the reconstruction problem is ill-posed and a bicubic spline regularization technique is thus proposed to overcome this difficulty. We apply a gradient-based optimization algorithm to solve this nonlinear optimization problem, where the Jacobian of the cost function is computed via finite difference approximation. Two numerical experiments: a synthetic American put option example and a real market American put option example, are performed to show the robustness and effectiveness of the proposed method to reconstructing the unknown volatility surface.