Partial differential equation (PDE) methods for pricing financial derivatives are quite popular due to their global character, ability to approximate hedging parameters, such as delta and gamma, and efficiency, at least for reasonably low dimension problems. However, various techniques that are highly associated with the numerical solution of traditional PDEs, such as adaptive mesh techniques and high-order methods, have not been widely used yet for financial PDE problems.

We consider certain financial derivative pricing problems such as valuing American options and options with jump-diffusion. We develop adaptive and high-order methods for such problems using a PDE approach. The fundamental underlying PDE for all these problems is the well-known Black-Scholes PDE, with each problem also having additional special properties. The American option pricing problem is associated with a partial differential inequality and a free-boundary problem, which is handled by a penalty iterative method. Pricing options with jump-diffusion in the asset price model involves a partial integro-differential equation (PIDE), which is handled by a fixed-point iteration.

Both finite differences and finite elements are considered for the space discretization, while classical finite differences, such as Crank-Nicolson, are used for the time discretization. The high-order discretization in space is based on an optimal finite element collocation method, the main computational requirements of which are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth-order. To control the space error, we use adaptive gridpoint distribution based on an error equidistribution principle. A time stepsize selector is used to further increase the efficiency of the methods.

Numerical examples show that our methods converge fast and provide highly accurate options prices, Greeks, and early exercise boundaries.