Operator splitting methods for pricing American options under stochastic volatility

Abstract

We consider the numerical pricing of American options under Heston's stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank-Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations. Mathematics Subject Classification (2000)35K85 · 65M06 · 65M55 · 65Y20 · 91B28

Radial basis function partition of unity operator splitting method for pricing multi-asset American options

“…The main idea of this method is to split the time integration step into two fractional steps, where the partial differential equation is integrated together with an auxiliary variable λ in the first step, and then the solution and λ are updated to fulfill the American option constraint in the second step. Such a splitting scheme introduces an error, which turns out to be sufficiently small to not influence the order of the time integration scheme [10]. It is also possible in each time step to ignore the free boundary and then apply the American constraint explicitly [7], but this approach reduces the order of the time integration, while the splitting method uses the auxiliary variable λ to improve the accuracy.…”

“…It is also possible in each time step to ignore the free boundary and then apply the American constraint explicitly [7], but this approach reduces the order of the time integration, while the splitting method uses the auxiliary variable λ to improve the accuracy. More details about the operator splitting method for pricing American options can be found in [10] and references therein.…”

“…At the first fractional step the linear system is solved and then, at the second fractional step, the solution is corrected to satisfy the American option constraint V ≥ Φ and the auxiliary variable λ n is updated. As it was shown in [10] the order of the splitting is the same as the order of the time integration scheme, i.e., second order. The splitting does not reduce the order of accuracy since the American option constraint is treated in a separate fractional time step.…”

Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model

“…The boundary conditions attached to the stock are the following; At S = 0 Dirichlet boundary condition [6]. ; 0, T, U = VW 01 At S = S XYZ [12], lim Z→_ ; S, T, U = 0 Note that ∞ in our case is S XYZ .…”

Adaptive finite differences and IMEX time-stepping to price options under Bates model

“…For American options an LCP with the same operator needs to be solved at each time step. The LCP can be approximated and solved using various methods including an operator splitting method [12], [15], [30], penalty methods [39], [9], and multigrid methods [7], [22], [26], [14], [35]. Here we use the operator splitting method to approximate the solutions of LCPs as it is accurate and easy to implement.…”