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

Abstract

This paper is devoted to develop a robust penalty-based method of reconstructingsmooth local volatility surface from the observed American optionprices. This reconstruction problem is posed as an inverse problem:given a finite set of observed American option prices, find a localvolatility function such that the theoretical option prices matchesthe observed ones optimally with respect to a prescribed performancecriterion. The theoretical American option prices are governed bya set of partial differential complementarity problems (PDCP). Wepropose a penalty-based numerical method for the solution of the PDCP.Typically, the reconstruction problem is ill-posed and a bicubic splineregularization technique is thus proposed to overcome this difficulty.We apply a gradient-based optimization algorithm to solve this nonlinearoptimization problem, where the Jacobian of the cost function is computedvia finite difference approximation. Two numerical experiments: asynthetic American put option example and a real market American putoption example, are performed to show the robustness and effectivenessof the proposed method to reconstructing the unknown volatility surface.