Penalty methods for American options with stochastic volatility

The American early exercise constraint can be viewed as transforming the original linear two dimensional stochastic volatility option pricing PDE into a PDE with a nonlinear source term. Several methods are described for enforcing the early exercise constraint by using a penalty source term in the discrete equations. The resulting nonlinear algebraic equations are solved using an approximate Newton iteration. The solution of the Jacobian is obtained using an incomplete LU (ILU) preconditioned conjugate gradient-like (PCG) method. Some example computations are presented for option pricing problems based on a stochastic volatility model, including an exotic American chooser option written on a put and call with discrete double knockout barriers and discrete dividends.