Ware, AntonyLi, Yu2024-11-082024-11-082024-11-07Li, Y. (2024). A Weighted Multilevel Monte Carlo method (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.https://hdl.handle.net/1880/120053This thesis begins by introducing the Weighted Multilevel Monte Carlo (WMLMC) method in a one-dimensional context, expanding upon the established Multilevel Monte Carlo (MLMC) method. The focus is on demonstrating that the WMLMC method provides even greater computational savings compared to the traditional MLMC method, which has already shown superior efficiency over the standard Monte Carlo (MC) approach under similar conditions. In the second part, we apply a mixed Partial Differential Equation (PDE)/Weighted Multilevel Monte Carlo (WMLMC) method to the pricing of Double-No-Touch options. We compare these results with those obtained using a mixed PDE/Multilevel Monte Carlo (MLMC) method. The analysis reveals a significant reduction in total computational costs when substituting the MLMC method with the more efficient WMLMC method. This finding prompts us to explore the broader applicability of the WMLMC method as a superior alternative to the MLMC method across various domains, anticipating substantial cost savings in computational tasks traditionally dominated by the MLMC approach. The adoption of the WMLMC method is expected to optimize resource utilization and enhance computational performance, making it a promising avenue for future research and application in diverse fields beyond options pricing. The third part integrates the WMLMC method with one-step Richardson Extrapolation (RE), resulting in a considerable boost in efficiency from a convergence standpoint. A comparative analysis highlights this computational advantage, demonstrating that the WMLMC method with one-step RE achieves the greatest reduction in computational cost. This approach could potentially be extended to combine WMLMC with multi-step RE for even greater efficiency. In the final section, we present the Weighted Multi-Index Multilevel Monte Carlo (WMIMLMC) method designed for a multi-dimensional setting, building on the Multi-Index Monte Carlo (MIMC) method. By utilizing high-order mixed differences instead of first-order differences, the WMIMLMC method significantly reduces the variance of hierarchical differences while adhering to similar assumptions as its predecessors. We provide a 2-dimensional example to showcase the computational advantages of the WMIMLMC method.enUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.Monte Carlo MethodEducation--Mathematicsdoctoral thesis