In this paper, we present two methods in order to calibrate the local volatility with American put options. Both calibration methods use a least-square formulation and a descent algorithm. Pricing is done by solving parabolic variational inequalities, for which solution procedures by active set methods are discussed.
The first strategy consists in computing the optimality conditions and the descent direction needed by the optimization loop. This approach has been implemented both at the continuous and discrete levels. It requires a careful analysis of the underlying variational inequalities and of their discrete counterparts. In the numerical example presented here (American options on the FTSE index), the squared volatility is parameterized by a bicubic spline.
In the second approach, which works in low dimension, the descent directions are computed with Automatic Differentiation of computer programs implemented in C++.
"Volatility calibration with American options." Methods Appl. Anal. 11 (4) 533 - 556, December 2004.