The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied. Sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. These conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem. The efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the American constraint explicitly. Convergence rates are studied as the timestep and mesh size tend to zero. It is observed that an implicit treatment of the American constraint does not converge quadratically (as the timestep is reduced) if constant timesteps are used. A timestep selector is suggested which restores quadratic convergence.

No abstract

No abstract

The pricing equations derived from uncertain volatility models in finance are often cast in the form of nonlinear partial differential equations. Implicit timestepping leads to a set of nonlinear algebraic equations which must be solved at each timestep. To solve these equations, an iterative approach is employed. In this paper, we prove the convergence of a particular iterative scheme for one factor uncertain volatility models. We also demonstrate how non-monotone discretization schemes (such as standard Crank-Nicolson timestepping) can converge to incorrect solutions, or lead to instability. Numerical examples are provided.

We explore the pricing of Asian options by n umerically solving the the associated partial di erential equations. We demonstrate that numerical PDE techniques commonly used in nance for standard options are inaccurate in the case of Asian options and illustrate modi cations which alleviate this problem. In particular, the usual methods generally produce solutions containing spurious oscillations. We adapt ux limiting techniques originally developed in the eld of computational uid dynamics in order to rapidly obtain accurate solutions. We s h o w that ux limiting methods are total variation diminishing (and hence free of spurious oscillations) for non-conservative P D E s s u c h as those typically encountered in nance, for fully explicit, and fully and partially implicit schemes. We also modify the van Leer ux limiter so that the second-order total variation diminishing property is preserved for non-uniform grid spacing.