Adaptive θθ-methods for pricing American options

We develop adaptive θθ-methods for solving the Black–Scholes PDE for American options. By adding a small, continuous term, the Black–Scholes PDE becomes an advection–diffusion-reaction equation on a fixed spatial domain. Standard implementation of θθ-methods would require a Newton-type iterative procedure at each time step thereby increasing the computational complexity of the methods. Our linearly implicit approach avoids such complications. We establish a general framework under which θθ-methods satisfy a discrete version of the positivity constraint characteristic of American options, and numerically demonstrate the sensitivity of the constraint. The positivity results are established for the single-asset and independent two-asset models. In addition, we have incorporated and analyzed an adaptive time-step control strategy to increase the computational efficiency. Numerical experiments are presented for one- and two-asset American options, using adaptive exponential splitting for two-asset problems. The approach is compared with an iterative solution of the two-asset problem in terms of computational efficiency.