The pricing model for American lookback options can be characterised as a
two-dimensional free boundary problem. The main challenge in this problem is the free
boundary, which is also the main concern for financial investors. We use a standard technique
to reduce the pricing model to a one-dimensional linear complementarity problem
on a bounded domain and obtain a corresponding variational inequality. The inequality
is discretised by finite differences and finite elements in the temporal and spatial directions,
respectively. By enforcing inequality constraints related to the options using
Lagrange multipliers, the discretised variational inequality is reformulated as a set of
semi-smooth equations, which are solved by a primal-dual active set method. One of
the major advantages of our algorithm is that we can obtain the option values and the
free boundary simultaneously, and numerical simulations show that our approach is as
efficient as some other methods.
American lookback option, linear complementarity problem, variational inequality, finite element method, primal-dual active set method.