Efficient numerical methods for pricing American options

Abstract

In this thesis we study efficient numerical methods for pricing American options. We apply option pricing models which are based on the Black and Scholes theory and Heston’s stochastic volatility model. Prices for American options are modelled by linear complementarity problems with one-dimensional and two-dimensional parabolic partial differential operators. The use of numerical methods is unavoidable because of the complexity of these option pricing problems. Large scale option trading gives a motivation to develop efficient numerical procedures for solving American option pricing problems. In this work we apply a finite difference method to the discretization. After discretization, a sequence of discrete linear complementarity problems should be solved in order to obtain prices for American options. This thesis is built around two types of splitting methods. In the articles of this thesis one is referred as the operator splitting method and the other one as the componentwise splitting method. Operator splitting methods are first applied for solving basic American option pricing models and then they are applied to a solution of a model with a stochastic volatility assumption. The idea in these operator splitting methods is that at each time step a treatment of an obstacle constraint and a solution of a system of linear equations are made in separate fractional steps. Particularly, the advantage of these methods is shown when a stochastic volatility model is used. Componentwise splitting methods are applied for a solution of the American option pricing problem with a stochastic volatility setting and shown to be highly efficient. In a basic form of this splitting a discrete linear complementarity problem is divided in such a way that three linear complementarity problems with tridiagonal matrices need to be solved. The efficiency of this splitting method is based on the use of a direct solver at each fractional step. Strang symmetrization is used to increase the accuracy of this splitting method. The efficiency of the proposed numerical techniques is demonstrated with several numerical experiments. This thesis ends with an article considering a numerical solution of the American option pricing problem with the stochastic volatility assumption where an extensive comparison of efficiency of numerical methods are presented.