| Abstract [eng] |
Today the Heston model is one of the most popular models in Mathematical Finance. However, the distribution of the Heston process is explicitly known only in the form of the characteristic function, therefore, as in the case of many other stochastic models, numerical approximation methods play a very important role. Unfortunately classical approximation methods do not provide any usefull solution at all. For example, the Euler approximation takes negative values with positive probability, and various attempts to modify the approximation to make it possitive seem to produce approximations converging rather slowly. In the thesis we suggest to construct discretization schemes for the Heston model by discrete random variables using „split-step“ and moment matching techniques. „Split-step“ technique lets to divide the model into deterministic and stochastic parts, so that we need to construct a discretization scheme for the stochastic part only, as the deterministic part is easily solvable. Moment matching helps to construct discrete random variables so that their moments match the moments of weak approximation of the order we aim at. In the thesis we present two new discretization schemes that use, at each step, only generation of discrete random variables and prove that they are strongly potential first- and second-order weak approximations for the solution of the log-Heston model. |
