A penalty method for a fractional order parabolic variational inequality governing American put option valuation

A power penalty method is proposed for a parabolic variational inequality or linear complementarity problem (LCP) involving a fractional order partial derivative arising in the valuation of American options whose underlying stock prices follow a geometric Lévy process. We first approximate the LCP with a nonlinear fractional partial differential equation (fPDE) with a penalty term. We then prove that the solution to the nonlinear fPDE converges to that of the LCP in a Sobolev norm at an exponential rate depending on the parameters used in the penalty term. Numerical results are presented to demonstrate the convergence rates and usefulness of the penalty method for pricing American put options of this type.