A fast numerical method for the valuation of American lookback put options

• We convert the American lookback option pricing problem into a LCP on a bounded domain by the numeraire transformation and domain truncation technique.
• The variational inequality (VI) form corresponding to the bounded LCP is obtained skillfully by some discussions.
• We discretize the resulting VI by a finite element method, and use a projection and contraction method to solve the discretized VI related to options.

A fast and efficient numerical method is proposed and analyzed for the valuation of American lookback options. American lookback option pricing problem is essentially a two-dimensional unbounded nonlinear parabolic problem. We reformulate it into a two-dimensional parabolic linear complementary problem (LCP) on an unbounded domain. The numeraire transformation and domain truncation technique are employed to convert the two-dimensional unbounded LCP into a one-dimensional bounded one. Furthermore, the variational inequality (VI) form corresponding to the one-dimensional bounded LCP is obtained skillfully by some discussions. The resulting bounded VI is discretized by a finite element method. Meanwhile, the stability of the semi-discrete solution and the symmetric positive definiteness of the full-discrete matrix are established for the bounded VI. The discretized VI related to options is solved by a projection and contraction method. Numerical experiments are conducted to test the performance of the proposed method.