# A mathematical programming with equilibrium constraints approach to the implied volatility surface of American options

## Jacqueline Huang and Jong-Shi Pang

#### Abstract

ABSTRACT

This paper presents a novel approach to the computation of an implied volatility surface of American options written on risky assets. The approach is based on the simple observation that this computational problem is the inverse of the forward-pricing problem of American options. As detailed by Huang and Pang (1998), the latter forward problem can be modeled by a discretized partial differential linear complementarity system. As such, the inverse problem, i.e., the implied volatility problem, becomes an instance of a mathematical problem with equilibrium constraints (MPEC), which is a class of constrained optimization problem with a finite-dimensional parametric linear complementarity system as part of its constraints. Two methods for solving an MPEC are described and applied to the problem of computing an implied volatility surface of American options. Some computational results on experimental data are reported.