Abstract

The Monte Carlo forward Euler method with uniform time stepping is the standard technique to compute an approximation of the expected payoff of a solution of an Itoˆ Stochastic Differential Equation (SDE). For a given accuracy requirement TOL, the complexity of this technique for well behaved problems, that is the amount of computational work to solve the problem, is O(TOL^{-3}).

A new hybrid adaptive Monte Carlo forward Euler algorithm for SDEs with non- smooth coefficients and low regular observables is developed in this thesis. This adaptive method is based on the derivation of a new error expansion with computable leading-order terms. The basic idea of the new expansion is the use of a mixture of prior information to determine the weight functions and posterior information to compute the local error. In a number of numerical examples the superior efficiency of the hybrid adaptive algorithm over the standard uniform time stepping technique is verified. When a non-smooth binary payoff with either Geometric Brownian Motion (GBM) or drift singularity type of SDEs is considered, the new adaptive method achieves the same complexity as the uniform discretization with smooth problems. Moreover, the new developed algorithm is extended to the Multilevel Monte Carlo (MLMC) forward Euler setting which reduces the complexity from O(TOL^{-3}) to O(TOL^{-2}(log(TOL))^{2}). For the binary option case with the same type of Itoˆ SDEs, the hybrid adaptive MLMC forward Euler recovers the standard multilevel computational cost O(TOL−2(log(TOL))2). When considering a higher order Mil- stein scheme, a similar complexity result was obtained by Giles using the uniform time stepping for one dimensional SDEs. The difficulty to extend Giles’ Milstein MLMC method to the multidimensional case is an argument for the flexibility of our new constructed adaptive MLMC forward Euler method which can be easily adapted to this setting. Similarly, the expected complexity O(TOL^{−2}(log(TOL))^{2}) is reached for the multidimensional case and verified numerically.