American option pricing and penalty methods

Abstract

The main purpose of this thesis is to study penalty approaches to American option pricing problems. We consider penalty approaches to pricing plain American options, American options with jump diffusion processes and two-asset American options. Convergence properties of these methods are investigated. Also, the numerical schemes - finite element method and fitted finite volume method - for solving the penalized PDE are developed. Finally, an augmented Lagrangian method is applied to solving the plain American option pricing. Empirical tests are carried out to illustrate the effectiveness and usefulness of our methods. For plain American option pricing, based on the theory of variational inequalities, a monotonic penalty approach is developed and its convergence properties are established in some appropriate infinite dimensional spaces. We derive the convergence rate of the combination of two power penalty functions. This convergence rate gives a unified result on that of higher and lower order penalty functions. After that, a fitted finite volume method is applied to finding the numerical solution of the penalized nonlinear PDE. We then test this method empirically, and compare it with projected successive over relaxation method (PSOR for short). We conclude that the monotonic penalty method is roughly comparable with the PSOR method, but is more desirable for its robustness under changes in market parameters, and furthermore the effect of the time reserving of the monotonic penalty method becomes significantly enhanced as the number of space steps increases. Pricing American options with jump diffusion processes can be formulated as a partial integro-differential complementarity problem. We propose a power penalty approach for solving this complementarity problem. The convergence analysis of this method is established in some appropriate infinite dimensional spaces. Then, using the finite element method, we propose a numerical scheme to solve the penalized problem and carry out the numerical tests to illustrate the efficiency of our method. The two-asset American option pricing problem is formulated as a continuous complementarity problem involving a two dimensional Black-Scholes operator. By using a power penalty method, the two-asset American option model is reformulated as a two dimensional nonlinear parabolic PDE. By introducing a weighted Sobolev space and the corresponding norm, the coerciveness and continuity of the bilinear operator in the variational problem are derived. Hence, the unique solvability of the original and penalized problems is established. The convergence rate of the power penalty method is obtained in some appropriate infinite dimensional spaces. Moreover, to overcome the computational difficulty of the convection-dominated Black-Scholes operator, a novel fitted finite volume method is proposed to solve the penalized nonlinear two dimensional PDE. We perform numerical tests empirically to illustrate the efficiency of our new method. Finally, based on the fitted finite volume discretization, an algorithm is developed by applying an augmented Lagrangian method (ALM for short) to pricing the plain American option. Convergence properties of ALM are considered. By empirical numerical experiments, we conclude that ALM is more effective than penalty method and Lagrangian method, and comparable with the PSOR method. Furthermore, numerical results show that ALM is more robust in terms of computation time under the changes in market parameters: interest rate and volatility.