Adaptive and high-order methods for valuing American options

Abstract

We develop space-time adaptive and high-order methods for valuing American options using a partial differential equation (PDE) approach. The linear complementarity problem arising due to the free boundary is handled by a penalty method. Both finite difference and finite element methods are considered for the space discretization of the PDE, 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.

A decomposition approach via Fourier sine transform for valuing American knock-out options with rebates

“…Essentially, more points are automatically distributed to regions that the option price lacks regularity, such as those around the optimal exercise prices and the barrier, to minimize the error. As shown in Christara and Dang (2011), the adaptive FD technique is significantly more efficient than both the uniform and pre-determined nonuniform FD methods.…”

“…This adaptiveFD method is built upon the highly efficient adaptive techniques developed in Christara and Dang (2011) for American vanilla options. In the adaptiveFD, the penalty method of Forsyth and Vetzal (2002) is employed to handle the non-linear PDE that arises.…”

“…Like the adaptive methods developed in Christara and Dang (2011), the complexity of adaptiveFD, at each penalty iteration, consists of solving a tridiagonal linear system of size "# S points" × "# S points", where "# S points" denotes the number of grid points in the S-direction. Hence, the total computation cost of adaptiveFD can be modeled by the formula total cost = # penalty iter.…”

Pricing multi-asset American options on Graphics Processing Units using a PDE approach

“…subject to the initial and boundary conditions (2) and (3), respectively. The penalty parameter ζ effectively ensures that the solution satisfies u − u * ≥ −ǫ for 0 < ǫ ≪ 1.…”

An Adaptive Finite Difference Method Using Far-Field Boundary Conditions for the Black-Scholes Equation

“…Moreover, the computational grid is refined in blocks and the grid and time step change at every discrete time point in [14]. The authors in [6] developed space-time adaptive and high-order methods for valuing American options using PDE approach. In [15], the grid and time step sizes were chosen dynamically to satisfy a bound on the global error at the expiry date.…”