Stable up-downwind finite difference methods for solving Heston stochastic volatility equations.
This dissertation explores effective and efficient computational methodologies for solving two-dimensional Heston stochastic volatility option pricing models with multiple financial engineering applications. Dynamically balanced up-downwind finite difference methods taking care of cross financial derivative terms in the partial differential equations involved are implemented and rigorously analyzed. Semidiscretization strategies are utilized over variable data grids for highly vibrant financial market simulations. Moving mesh adaptations are incorporated with experimental validations. The up-downwind finite difference schemes derived are proven to be numerically stable with first order accuracy in approximations. Discussions on concepts of the stability and convergence are fulfilled. Simulation experiments are carefully designed and carried out to illustrate and validate our conclusions. Multiple convergence and relative error estimates are obtained via computations with reality data. The novel new methods developed are highly satisfactory with distinguished simplicity and straightforwardness in programming realizations for option markets, especially when unsteady stocks’ markets are major concerns. The research also reveals promising directions for continuing accomplishments in financial mathematics and computations.